Content-Type: multipart/mixed; boundary="-------------0609291329753" This is a multi-part message in MIME format. ---------------0609291329753 Content-Type: text/plain; name="06-272.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-272.keywords" renormalization, Riemannian manifolds, flow equations, heat kernel ---------------0609291329753 Content-Type: application/x-tex; name="riemann.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="riemann.tex" \documentclass[12pt]{article} \pagestyle{plain} \hoffset=-1cm \voffset=-1cm \usepackage{amsmath,epsf} \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc} \usepackage{epsf} \usepackage{amsfonts,amssymb} \usepackage[usenames,dvipsnames]{color} \pagenumbering{arabic} \renewcommand{\textwidth} {16cm} \renewcommand{\textheight} {22cm} \renewcommand{\oddsidemargin} {1.5cm} \renewcommand{\baselinestretch} {1.2} \renewcommand{\baselinestretch} {1.2} \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\vep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\ka}{\kappa} \newcommand{\la}{\lambda} \newcommand{\si}{\sigma} \newcommand{\Si}{\Sigma} \newcommand{\om}{\omega} \newcommand{\Om}{\Omega} \newcommand{\vp}{\varphi} \newcommand{\La}{\Lambda} \newcommand{\ze}{\zeta} \newcommand{\Lao}{\Lambda_0} \newcommand{\Lam}{\Lambda_m} \newcommand{\lam}{\lambda_m} \newcommand{\uvp}{\underline{\varphi}} \newcommand{\uhvp}{\underline{\hat{\varphi}}} \newcommand{\ua}{\underline{a}} \newcommand{\ub}{\underline{b}} \newcommand{\uk}{\underline{k}} \newcommand{\ul}{\underline{l}} \newcommand{\un}{\underline{n}} \newcommand{\up}{\underline{p}} \newcommand{\uq}{\underline{q}} \newcommand{\ur}{\underline{r}} \newcommand{\hp}{\hat{p}} \newcommand{\hq}{\hat{q}} \newcommand{\pv}{\vec p} \newcommand{\qv}{\vec q} \newcommand{\xv}{\vec x} \newcommand{\yv}{\vec y} \newcommand{\ox}{\overline{x}} \newcommand{\oy}{\overline{y}} \newcommand{\oz}{\overline{z}} \newcommand{\cA}{{\cal A}} \newcommand{\cB}{{\cal B}} \newcommand{\cC}{{\cal C}} \newcommand{\cD}{{\cal D}} \newcommand{\cH}{{\cal H}} \newcommand{\ocD}{\overline{\cD}} \newcommand{\cL}{{\cal L}} \newcommand{\cP}{{\cal P}} \newcommand{\cS}{{\cal S}} \newcommand{\cT}{{\cal T}} \newcommand{\cZ}{{\cal Z}} \newcommand{\hPh}{{\hat \Phi}} \newcommand{\pa}{\partial} \newcommand{\ve}[1]{\vec{#1}} \newcommand{\ti}[1]{\tilde{#1}} \newcommand{\h}[1]{\hat{#1}} \newcommand{\qed}{\hfill \rule {1ex}{1ex}\\ } \newcommand{\sq}{\fbox{} } \newcommand{\eq}{\begin{equation}} \newcommand{\eqe}{\end{equation}} \newcommand{\nom}{|\,\omega(x_1)|} \newcounter{saveeqn} \newcommand{\alpheqn}{\setcounter{saveeqn}{\value{equation}} \setcounter{equation}{0} \addtocounter{saveeqn}{1} \renewcommand{\theequation}{\mbox{\arabic{saveeqn}\alph{equation}}}} \newcommand{\reseteqn}{\setcounter{equation}{\value{saveeqn}}% \renewcommand{\theequation}{\arabic{equation}}} \begin{document} \message{reelletc.tex (Version 1.0): Befehle zur Darstellung |R |N, Aufruf= % z.B. \string\bbbr} % % % Sonderzeichen \message{reelletc.tex (Version 1.0): Befehle zur Darstellung |R |N, Aufruf= % z.B. \string\bbbr} \font \smallescriptscriptfont = cmr5 \font \smallescriptfont = cmr5 at 7pt \font \smalletextfont = cmr5 at 10pt \font \tensans = cmss10 \font \fivesans = cmss10 at 5pt \font \sixsans = cmss10 at 6pt \font \sevensans = cmss10 at 7pt \font \ninesans = cmss10 at 9pt \newfam\sansfam \textfont\sansfam=\tensans\scriptfont\sansfam=\sevensans \scriptscriptfont\sansfam=\fivesans \def\sans{\fam\sansfam\tensans} %---------------------------------------------------------- \def\bbbr{{\rm I\!R}} %reelle Zahlen \def\bbbn{{\rm I\!N}} %natuerliche Zahlen \def\bbbE{{\rm I\!E}} %Einheitsmatrix by I. Zoller \def\bbbm{{\rm I\!M}} \def\bbbh{{\rm I\!H}} \def\bbbk{{\rm I\!K}} \def\bbbd{{\rm I\!D}} \def\bbbp{{\rm I\!P}} \def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}} \def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}} \def\bbbe{{\mathchoice {\setbox0=\hbox{\smalletextfont e}\hbox{\raise 0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.3pt height0.7\ht0\hss}\box0}} {\setbox0=\hbox{\smalletextfont e}\hbox{\raise 0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.3pt height0.7\ht0\hss}\box0}} {\setbox0=\hbox{\smallescriptfont e}\hbox{\raise 0.1\ht0\hbox to0pt{\kern0.5\wd0\vrule width0.2pt height0.7\ht0\hss}\box0}} {\setbox0=\hbox{\smallescriptscriptfont e}\hbox{\raise 0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.2pt height0.7\ht0\hss}\box0}}}} \def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}} \def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}} \def\bbbs{{\mathchoice {\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}} \def\bbbz{{\mathchoice {\hbox{$\sans\textstyle Z\kern-0.4em Z$}} {\hbox{$\sans\textstyle Z\kern-0.4em Z$}} {\hbox{$\sans\scriptstyle Z\kern-0.3em Z$}} {\hbox{$\sans\scriptscriptstyle Z\kern-0.2em Z$}}}} \noindent \title{ Renormalization Proof for Massive $\vp_4^4$ Theory on Riemannian Manifolds} \author{Christoph Kopper\footnote{\ kopper@cpht.polytechnique.fr} \\ Centre de Physique Th{\'e}orique, CNRS, UMR 7644\\ Ecole Polytechnique\\ F-91128 Palaiseau, France \and Volkhard F. M{\"u}ller\footnote{\ vfm@physik.uni-kl.de}\\ Fachbereich Physik, Technische Universit{\"a}t Kaiserslautern\\ D-67653 Kaiserslautern, Germany } \date{} \maketitle \begin{abstract} In this paper we present an inductive renormalizability proof for massive $\vp_4^4$ theory on Riemannian manifolds, based on the Wegner-Wilson flow equations of the Wilson renormalization group, adapted to perturbation theory. The proof goes in hand with bounds on the perturbative Schwinger functions which imply tree decay between their position arguments. An essential prerequisite are precise bounds on the short and long distance behaviour of the heat kernel on the manifold. With the aid of a regularity assumption (often taken for granted) we also show, that for suitable renormalization conditions the bare action takes the minimal form, that is to say, there appear the same counter terms as in flat space, apart from a logarithmically divergent one which is proportional to the scalar curvature. \end{abstract} \newpage \noindent \section{Introduction } Among the different schemes deviced to prove the perturbative renormalizability of a local quantum field theory, the one based on the Wegner-Wilson differential flow equations of the Wilson renormalization group shows a distinctive characteristic: it circumvents completely the combinatoric complexity of generating Feynman diagrams and the subsequent cumbersome analysis of Feynman integrals with in general overlapping divergences. Initiated by Polchinski [Pol], this approach to renormalization has now been adapted to a wide variety of physically interesting instances. Partial reviews of the rigorous work which started from [KKS] may be found in [Kop1], [Sal], [Kop2], [M\"u]. It is tempting to extend the approach via Wilson's flow equation further to prove the perturbative renormalizability of a quantum field theory defined on curved spacetime. There is a caveat, however. Using functional integration, one actually deals with a quantum field theory defined on a "Euclidean section" of curved spacetime, i.e. on a Riemannian manifold. In contrast to flat space there is no Wick rotation of Lorentzian curved spacetime, in general. Nevertheless, beyond static spacetimes, on particular nonstatic ones the analytic continuation of a quantum field theory to a corresponding Euclidean formulation has been rigorously shown recently: Bros, Epstein and Moschella [BEM] considered a quantum field theory on the anti-de Sitter (AdS) spacetime within a Wightman-type approach. As a consequence of certain spectral assumptions they show that the $n$-point correlation functions admit an analytic continuation to tuboidal domains (of $n$ copies) of the complexified covering space of the AdS spacetime. This continuation includes the Euclidean AdS spacetime and satisfies there Osterwalder-Schrader positivity. Euclidean AdS spacetime is a Riemannian manifold with constant negative curvature.\footnote{It is called 'hyperbolic space' in the mathematical literature.} Moreover, Birke and Fr\"ohlich [BiFr], establishing in an algebraic approach Wick rotation of quantum field theories at finite temperature, also presented a reconstruction of quantum field theories on specific curved spacetimes from corresponding imaginary-time formulations, using group-theoretical techniques.\\ Our work starts straight considering a Riemannian manifold as given "spacetime". This manifold is assumed to be geodesically complete and to have all its sectional curvatures confined by a negative lower and a positive upper bound. We then study perturbative renormalizability of massive $\vp_4^4\,$-theory defined on such a manifold by analy\-sing the generating functional $L^{\La,\Lao}$ of connected (free propagator) amputated Schwinger functions (CAS). From the physical point of view it seems justified to restrict to this class of manifolds, since in situations where curvature becomes large or where singularities appear the treatment of gravity as a classical background effect becomes questionable anyway. As there is no translation symmetry, the CAS and thus the system of flow equations relating them have to be dealt with in position space. Establishing bounds involving these CAS we heavily rely on global lower and upper bounds for the heat kernel on the manifold, found in the mathematical literature. Around the beginning of the eighties a considerable amount of work was carried out to formulate quantum field theory perturbatively on curved spacetime. Based on the intuition that ultraviolet divergences involve arbitrarily short wavelenghts, an approximating local momentum space representation of the Feynman propagator in curved spacetime was developed in [Bir],[BPP],[BuPn] for the $\phi^4$-theory and generalized in [BuPr],[Bun1]. Combined with dimensional regularization, the euclideanized $\phi^4$-theory was then shown to be renormalizable with local counterterms in one- and two-loop order. Furthermore, choosing the same general approach, Bunch [Bun2] has demonstrated the BPHZ renormalization of the $\phi^4$-theory on euclideanized curved spacetime, by taking into account the power counting singular contributions in the asymptotic expansion of the propagator around its euclidean form. A different kind of generally applicable dimensional regularization scheme has been given by L\"uscher [L\"u], who applies it to the $\phi^4$-theory on an arbitrary compact four-dimensional manifold with positive metric and to the Yang-Mills gauge theory on $ S^4$. He also shows the renormalizability of the $\phi^4$-theory by local counterterms at the one- and two-loop levels. Further references on work before 1982 can be found in the monograph [BiDa]. More recently, the perturbative construction of the $\phi^4$-theory has been performed in an algebraic setting by Brunetti, Fredenhagen, Hollands and Wald [BrFr],[HoWa1],[HoWa2]. These authors adapted the renormalization method of Epstein and Glaser to construct the algebra of local, covariant quantum fields of the $\phi^4$-model on a globally hyperbolic curved spacetime to any order of the perturbative expansion, making use of techniques from microlocal analysis. The crucial notion of a local and covariant quantum field introduced in [HoWa1,2] has been further formalized by Brunetti, Fredenhagen and Verch [BFV]. \\ This paper is organized as follows~: In Section 2 we collect and slightly adapt global bounds on the heat kernel found in the mathematical literature, which are pertinent to our treatment. The action considered and the system of perturbative flow equations satisfied by the CAS is set up in Section 3. To establish bounds on the CAS, being distributions, they have to be folded first with test functions. In Section 4 a suitable class of test functions is introduced, together with tree structures with the aid of which the bounds to be derived on the CAS are going to be expressed. In Section 5 we state the boundary and the renormalization conditions used to integrate the flow equations of the irrelevant and relevant terms, respectively. The flow equations permit to be quite general in this respect, englobing basically all situations of interest. Section 6 is the central one of this paper. We state and prove inductive bounds on the Schwinger functions which, being uniform in the cutoff, directly lead to renormalizability. Beyond they imply tree decay of the Schwinger functions between their external points. The last section is devoted to the proof that the bare action of the theory may be chosen minimally, i.e. with position independent counter terms apart from one (logarithmically divergent) term which is proportional to the scalar curvature of the manifold. Here we have to make the assumption that geometric quantities on the manifold have a smooth expansion (to lowest orders) w.r.t. contributions of curvature terms of increasing mass dimension. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ The heat kernel} We consider geodesically complete simply connected Riemannian manifolds $\cal M\,$ of dimension $n$ without boundary, whose sectional curvatures are bounded between two constants $-k^2$ and $\kappa^2\,$. The related heat kernel then has the following properties~: \eq K(t,x,y) \in C^{\infty}( (0,\infty) \times {\cal M} \times {\cal M})\ , \label{hk0} \eqe \eq 0 < \ K(t,x,y) < \infty \ , \label{hk1} \eqe \eq K(t,x,y) = \ K(t,y,x)\ , \label{hk2} \eqe \eq \int_{\cal M} K(t,x,y)\ dV(y)\ =\ 1\ , \label{hk3} \eqe \eq K(t_1+t_2,x,y) =\ \int_{\cal M} K(t_1,x,z) \ K(t_2,z,y)\ dV(z)\ . \label{hk4} \eqe Stochastic completeness (\ref{hk3}) holds due to the assumed bounded curvature, cf. [Tay, ch.6, Prop.2.3 ]. Mathematicians have established quite sharp pointwise bounds on the respective heat kernels of various classes of manifolds. We are going to explicit now some of these bounds, because we will rely on them in the subsequent construction. Some bounds are known to hold for $0 < t < T\,$, others for $0 < t \,$ or $ T < t $. We will write $t_{\de} =t(1+\de)\,$ where the parameter $\de\,$ satisfies $\,0 < \de <1\,$ and may be chosen arbitrarily small. Furthermore $c,\ C,$ collectively denote constants which depend on $\de, n\,$ and - if involved in the claim - on $k^2, T\,$.\\ On complete Riemannian manifolds of dimension $n$ with nonnegative Ricci curvature the heat kernel satifies the lower and upper bounds [LiYa, Dav1] \eq \frac{c}{\sqrt{|{\cal B}(x,t^{1/2})|\, |{\cal B}(y,t^{1/2})|}}\, e ^{-\frac{d^2(x,y)}{4 t(1-\de)}} \ \le \ K(t,x,y)\ \le \ \frac{C}{\sqrt{|{\cal B}(x,t^{1/2})| \, |{\cal B}(y,t^{1/2})|}}\, e ^{-\frac{d^2(x,y)}{4 t(1+\de)}} \label{hk5} \eqe valid for all $x,y \in \cal{M}$ and $ t >0\, $.\\ In the case of negative Ricci curvature bounded below let $ E \geq 0 $ denote the bottom of the spectrum of the operator $ - \Delta\, $. Then it holds, see [Dav2, Theorems 16 and 17 ]~:\\ If $ \delta > 0 $, there exists a constant $ c_{\delta } $ such that \begin{equation} \label{f20} K(t,x,y) \leq c_\delta \big (\,|{\cal B}(x,t^{1/2})| \, |{\cal B}(y,t^{1/2})|\big )^{-\frac{1}{2}} \exp \Big( - \frac{d^2(x,y)}{4t(1+\de)} \Big) \end{equation} for $ 0 < t < 1$ and for all $ x,y \in \mathcal{M} $, whereas \\ \begin{equation} \label{f21} K(t,x,y) \leq c_{\delta} \Big (|{\cal B}(x,1)| \, |{\cal B}(y,1)| \Big )^{- \frac{1}{2}} \exp \Big \{ ( \delta - E ) t - \frac{d^{\, 2}(x,y)}{4t(1+\de)} \Big \} \end{equation} for $ 1 \leq t < \infty $ and for all $x,y \in \mathcal{M} $. \\ Moreover, a lower bound of the form appearing in (\ref{hk5}) holds here, too, however restricted to $ 0 < t < T$, [Var].\\ In addition, given bounded sectional curvature $ - k^2 \leq Sec_{\cal{M}} \leq \kappa^2 $, there is the lower bound \eq K(t,x,y) \geq c\, \exp\bigg ( - \tilde{E} t- C\, \frac{ d^{\,2}(x,y)}{t} \bigg ) \label{F22} \eqe for all $ x,y \in \mathcal{M} $ and for $ t > T $, with constants $ c,C > 0 $ and $\tilde{E} > E $, possibly much larger, [Gri, ch. 7.5 ]. \\ On a Cartan-Hadamard manifold of dimension $n\,$, i.e. a geodesically complete simply connected noncompact Riemannian manifold with nonpositive sectional curvature, and assuming that the sectional curvature is bounded below by $- K^2 $, we also have for all $ x,y \in \mathcal{M} $, and for $0 0 $, fixed, then \\ \eq \frac{\pi^2}{2} t^2 \leq |{\cal B}(x,t^{1/2})| \leq \frac{\pi^2}{2} t^2\, h_4(k\, t^{1/2}) \label{comp} \eqe with the positive increasing function $$ h_4(r)=\ \frac{\cosh(3r)-9\cosh r +8}{3r^4}, \quad h_4(0) = 1 \, ,$$ ii) if all sectional curvatures of $\cal{M}$ have values in $ [0, \ka^2],\, \ka > 0 $, fixed, then \footnote{ The restriction on $ t $ accounts for the injectivity radius of the manifold.}\\ \eq \frac{\pi^2}{2} t^2 \, s_4(\ka \, t^{1/2}) \leq |{\cal B}(x,t^{1/2})| \leq \frac{\pi^2}{2} t^2 ,\quad \mbox{for}\quad \ka\, t^{1/2} < \pi, \label{comp'} \eqe with the positive decreasing function, $ 0 < r < \pi$, $$ s_4(r)=\ \frac{\cos(3r)-9\cos r +8}{3r^4}, \quad s_4(0) = 1 \, .$$ Taking (\ref{hk5}) together with (\ref{comp'}), as well as taking (\ref{hk8}) or (\ref{f20}) and the lower bound from (\ref{hk5}) \footnote{Remember the statement after (\ref{f21}).} together with (\ref{comp}), we obtain, restricting \footnote{The restriction is necessary both for the upper and lower bounds.} to $\, 0 < t < T $: \eq \frac{c}{t^2}\ \exp(-\frac{d^2(x,y)}{4 t(1-\de)}) \ \le \ K(t,x,y)\ \le \ \frac{C}{t^2} \ \exp(-\frac{d^2(x,y)}{4 t(1+\de)})\ . \label{hk10} \eqe The constants $c,\ C $ depend on $ k^2, \ka^2, \de, T $, but do not depend on $ t $. As a consequence of this lower and upper bound we obtain under the same conditions \eq d^s(x,y) \,K(t,x,y) \ \le \ c\,'\ t^{\, s/2}\ K(t_{\,\de\, '},x,y)\ \quad \mbox{for} \,\, t \leq T , \label{d} \eqe with $\de\,' > \de $. For $\,1\le s \le 3\,$ we also need the bound \eq |\nabla^s \,K(t,x,y)| \ \le \ C \ t^{-s/2}\ K(t_{\de},x,y)\ \label{D} \eqe based on [CLY], [Dav3] and valid for $ 0 < t < T $. Here $\,\nabla^s\,$ denotes a covariant derivative of order $s\,$ w.r.t. $x\,$ and the norm is that of (\ref{nor}). The constant $C$ here also depends on the norm of the covariant derivatives of the curvature tensor up to order $s-1\,$. From (\ref{D}) and the heat equation it follows directly that \eq |\pa_t \,K(t,x,y)| \ \le \ C \ t^{-1}\ K(t_{\de},x,y)\ . \label{pat} \eqe Finally we note the following recently proven bound on the logarithmic derivative of the heat kernel [SoZh] which holds for $Ric_{\cal M} \ge -k^2\,$ and for $\,0 < t < T\,$: \eq \frac{|\nabla \,K(t,x,y)|}{K(t,x,y)} \ \le \ O(1)\ \frac{1}{t^{1/2}} \ (1+\frac{d^2(x,y)}{t})\ . \label{logD} \eqe In closing this section we remark that the restriction to manifolds $\cal M$ of the kind considered is not dictated by the validity of our methods of proof. It rather seems to be a choice which is reasonable and interesting on physical grounds. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Action and the Flow Equations} The regularized (free) propagator is given in terms of the heat kernel by \eq {C}^{\vep,t}(x,y)\,=\, \int_{\vep}^{t} dt' \ e ^{-m^2\,t'}\ K(t',x,y)\ . \label{propa} \eqe Its derivative w.r.t. $t\,$ is denoted as \eq C_t(x,y):=\pa_{t}C^{\vep,t}(x,y)=\ e ^{-m^2\,t}\ K(t,x,y)\ . \label{dpropa} \eqe We assume $\,0 < \vep \le t < \infty\,$ so that the flow parameter $t$ takes the role of a long distance cutoff, whereas $\vep$ is a short distance regularization. The full propagator is recovered for $\vep=0$ and $t \to \infty\,$. For finite $\vep$ and in finite volume the positivity and regularity properties of ${C}^{\vep,t}\,$ permit to define the theory rigorously from the functional integral \eq e^{- {1\over \hbar} (L^{\vep,t}(\vp)+ I^{\vep,t})} \,=\, \int \, d\mu_{\vep,t}(\phi) \; e^{- {1\over\hbar} L^{\vep,\vep}(\phi\,+\,\vp)} \ , \quad L^{\vep,t}(0):=0\ , \label{funcin} \eqe where the factors of $\hbar\,$ have been introduced to allow for a consistent loop expansion in the sequel. In (\ref{funcin}) $\,d\mu_{\vep,t}(\phi) $ denotes the Gaussian measure with covariance $\hbar C^{\vep,t}(x,y)$. The normalization factor $\,e^{-{1\over\hbar} I^{\vep,t}} $ is due to vacuum contributions. It diverges in infinite volume so that we can take the infinite volume limit only when it has been eliminated. We do not make the finite volume explicit here since it plays no role in the sequel. The functional $\,L^{\vep}(\vp)~:=\,L^{\vep,\vep}(\vp)\,$ is the bare (inter)action including counterterms, viewed as a formal power series in $\,\hbar\,$. The superscript $\vep\,$ indicates the UV cutoff. For shortness we will pose in the following, with $x,y \in \cal{M}$, $\vec x = (x_1,\cdots, x_n)\in {\cal{M}}^{\times n} $, \[ \int_x\ :=\ \int_{\cal M} dV(x)\, , \qquad \int_{\vec x}\ :=\ \prod_{i=1}^{n}\int_{\cal M} dV(x_i)\, , \] and \[ \ti \de(x,y)\,:=\ |g|^{-1/2}(x) \ \de(x,y)\ . \] As is known from lowest order calculations [Bir],[BPP],[L\"u], in curved spacetime there will appear an additional counterterm of the type $\int_x R(x)\, \vp^2(x)\,$ which is proportional to the scalar curvature $\,R(x)\,$ of the spacetime manifold $\cal M\,$ considered. So the bare interaction for the symmetric $\vp_4^4$ theory would be \eq L^{\vep}(\vp) = {\lambda \over 4!} \int_x \vp^4(x) \, +\, {1 \over 2} \int_x\ \{ (a ^{\vep} +\xi ^{\vep} R(x))\vp^2(x) + b ^{\vep} g^{\mu \nu}(x)\ \partial_{\mu}\vp(x)\cdot\partial_{\nu} \vp(x) + {2 \over 4!}\, c ^{\vep} \vp^4(x)\} \label{nawi} \eqe where $\la >0$ is the renormalized coupling, and the cutoff dependent parameters $ a ^{\vep},\ \xi ^{\vep},\ b ^{\vep},\ c ^{\vep} $ - which remain to be fixed and which are directly related to the mass, curvature, wave function, and coupling constant counterterms \footnote{Since it is not necessary in the flow equation framework to introduce bare fields in distinction from renormalized ones (our field is the renormalized one in this language), there is a slight difference, which is to be kept in mind only when comparing to other schemes. For our purposes the functional arguments $\vp(x)$ may be assumed to live in the Schwartz space ${\cal S}(\cal M)$.} - will fulfill \eq a ^{\vep},\ \xi ^{\vep},\ b ^{\vep},\ c ^{\vep} =O(\hbar)\,. \label{coef} \eqe It seems to us that there is no a priori reason to restrict to bare interactions of this form. In fact, since there is no translation invariance in curved space time, {\it all} counter terms and even the coupling $\la$ itself may be position dependent. Quite generally the bare action is not a directly observable physical object, and the constraints on its form stem from the symmetry properties of the theory which are imposed, on its field content and on the form of the propagator. The symmetry properties depend in particular on the renormalization conditions which fix the physical (relevant) parameters of the theory. %In curved space time they should reflect intrinsic %geometrical properties. They might be position dependent~: e.g. a local scattering experiment performed at different places at the same external momenta might give different cross sections and, as a consequence of this, the renormalized coupling, fixed in terms of the cross section, would be position dependent. It is therefore natural to admit more general bare interactions \footnote{One could of course be even more general.} \[ L^{\vep}(\vp) = \int_x {\la(x) \over 4!} \,\vp^4(x) \ + \ \] \[ {1 \over 2} \int_x\ \{a^{\vep}(x) \,\vp^2(x) + u^{\,\mu,{\vep}}(x) \,\vp(x)\,\nabla_{\mu}\vp(x) + \hat b^{\vep}(x) g^{\mu \nu}(x)\ \partial_{\mu}\vp(x)\cdot\partial_{\nu} \vp(x) + {2 \over 4!}\, c ^{\vep}(x) \vp^4(x)\}\, . \] Here $\, \la(x),\ a ^{\vep}(x),\ \hat b ^{\vep}(x),\ c ^{\vep}(x)\,$ are general scalars and $u^{\,\mu,\vep}(x)$ is a general vector, all functions are supposed to be smooth, and $|\la(x)|\,$ (of course) uniformly bounded on $\cal M\,$. When calculating the two point function ${\cal L}^{\vep}(x_1,x_2)=\ \de/\de_{\vp(x_1)}\,\de/\de_{\vp(x_2 )}\, L^{\vep}(\vp)|_{\vp \equiv 0}\,$ from this bare action one obtains \begin{eqnarray} \mathcal{L}_{2}^{\,\epsilon}(x_1, x_2) &= & a^{\epsilon}(x_1) \, \tilde{\delta}(x_2,x_1) - \frac{1}{2}( \nabla_\mu u^{\,\mu, \epsilon})(x_1) \, \tilde{\delta}(x_2,x_1) \nonumber \\ &-& |g(x_2)|^{-{1 \over 2}}\, \pa_{\mu}^{(2)}\,\hat{b}^{\vep}(x_2) \, g^{\mu \nu}(x_2)\,|g(x_2)|^{1 \over 2}\,\pa_{\nu}^{(2)} \, \tilde{\delta}(x_2,x_1)\ . \label{gen2} \end{eqnarray} This means that $u^{\,\mu, \epsilon}\,$ only contributes via its divergence, and that the contribution $\sim \nabla_\mu u^{\,\mu,\epsilon} \,$ can be absorbed in $a^{\epsilon}\,$. On the other hand the particular tensor coupling $ \hat b^{\vep}(x)g^{\mu \nu}(x)\, $ can be generalized - without changing symmetry properties - into a smooth symmetric $ (2,0)$-tensor field $ b^{\,\mu \nu, \vep}(x)\,$. In (\ref{gen2}) the product $ \hat b^{\vep}(x)g^{\mu \nu}(x)$ is then replaced by $ b^{\,\mu \nu, \vep}(x)\,$, and we recognise that in (\ref{gen2}) the generalisation \eq \Delta^{(b)}\, := \, |g(x)|^{- {1 \over 2}}\,\pa_{\mu} \, b^{\,\mu \nu}(x)\,|g(x)|^{1 \over 2}\,\pa_{\nu} \label{gede} \eqe of the Laplace-Beltrami operator $\De $, (\ref{a4}), appears. We thus adopt a bare interaction of the form \eq L^{\vep}(\vp) = \int_x {\la(x) \over 4!} \, \vp^4(x) + {1 \over 2}\int_x\ \{\, {\tilde a}^{\vep}(x) \,\vp^2(x) + \, b^{\,\mu \nu, \vep}(x)\ \partial_{\mu}\vp(x)\cdot\partial_{\nu} \vp(x) + {2 \over 4!}\, c ^{\vep}(x) \vp^4(x)\} \label{nawig} \eqe with smooth scalar functions $\,{\tilde a}^{\vep}(x),\ c ^{\vep}(x)\,$ and a smooth symmetric tensor field $\ b ^{\mu \nu,\vep}(x)\,$ - which remain to be fixed and which are of (at least) order $\,\hbar\,$. The flow equation (FE) is obtained from (\ref{funcin}) on differentiating w.r.t. $t\,$. It is a differential equation for the functional $L^{\vep,t}\,$~: \eq \partial_{t}(L^{\vep,t} + I^{\vep,t} ) \,=\,\frac{\hbar}{2}\, \langle\frac{\delta}{\delta \vp}, C_t\, \frac{\delta}{\delta \vp}\rangle L^{\vep,t} \,-\, \frac{1}{2}\, \langle \frac{\delta}{\delta \vp}L^{\vep,t}, C_t\, \frac{\delta}{\delta \vp} L^{\vep,t}\rangle \;\,. \label{feq} \eqe By $\langle\ ,\ \rangle$ we denote the standard inner product in $L^2({\cal M}, dV( x))\,$. The FE can also be stated in integrated form \eq e^{- {1\over \hbar} (L^{\vep,t}(\vp)+ I^{\vep,t})} \,=\, e^{\hbar\De_{\cal F}(\vep,t)}\ e^{- {1\over\hbar} L^{\vep,\vep}(\vp)} \ . \label{intfe} \eqe The functional Laplace operator $\De_{\cal F}(\vep,t)\,$ is given by \[ \De_{\cal F}(\vep,t)=\,\frac12\ \langle \de_{\vp},\ C^{\vep,t} \ \de_{\vp} \rangle \] using the notation $\delta_{\vp(x)}=\delta/\delta\vp(x)$. We may expand $L^{\vep,t}(\vp)\,$ w.r.t. the number of fields $\vp\,$ setting \eq L^{\vep,t}_{n}(\vp)~:= \ \frac{1}{n!}\, {\pa ^n\over \pa \ka ^n}\ L^{\vep,t}(\ka \vp)|_{\ka =0}\ . \eqe The functional $L^{\vep,t}(\vp)\,$ can also be expanded in a formal powers series w.r.t. $\hbar$, and in a double series w.r.t. $\hbar$ and the number of fields \eq L^{\vep,t}(\vp)\,=\,\sum_{l=0}^{\infty} \hbar^l\,L^{\vep,t}_{l}(\vp)\,=\, \sum_{l=0}^{\infty}\hbar^l\,\sum_{n =2}^{\infty}L^{\vep,t}_{n,l} (\vp) \ ,\quad L^{\vep,t}_{2,0}(\vp)\equiv 0\ . \label{3.3} \eqe Corresponding expansions for $ \ti a ^{\vep}(x),\ b^{\,\mu \nu, \vep}(x),\ c ^{\vep}(x)\,$ are $ \ti a ^{\vep}(x)=\sum_{l \ge 1} \ti a_l ^{\vep}(x) \hbar^l$ etc. We can then rewrite (\ref{feq}) in loop order $l\ $ as \[ \partial_{t}L^{\vep,t}_{n,l}(\vp ) \,=\, \] \eq\frac{1}{2}\, \int_{x,y} C_{t}(x,y)\ \Bigl[\de_{\vp(x)}\ \de_{\vp(y)}\,L^{\vep,t}_{n+2,l-1}(\vp ) \,-\ \sum_{n_1+n_2 =n+2 \atop l_1+l_2=l } (\de_{\vp(x)}\,L^{\vep,t}_{n_1,l_1}(\vp ))\ \de_{\vp(y)}\,L^{\vep,t}_{n_2,l_2}(\vp ) \Bigr]\ . \label{nfeq} \eqe From $L^{\vep,t}_{l}(\vp)$ we obtain the connected amputated Schwinger functions of loop order $l$ as \eq {\cal L}^{\vep,t}_{n,l}(x_1,\ldots,x_{n}) \,:=\ \de_{\vp(x_1)} \ldots \de_{\vp(x_n)} L^{\vep,t}_l|_{\vp \equiv 0}\ . \label{CAS} \eqe It is straightforward to realize that the ${\cL}^{\vep,t}_{n,l}\,$ are distributions \\ 1) which are completely {\it symmetric} w.r.t. permutations of the arguments $(x_1 ,\ldots, x_{n})\,$\\ 2) and which {\it fall off rapidly} with the distances $d(x_i,x_j)\,$.\\ These facts follow from (\ref{nawig}), (\ref{intfe}) and the properties of the regularized propagator (\ref{hk5}), (\ref{hk10}). The distributional character of the ${\cL}^{\vep,t}_{n,l}\,$ is related to the fact that we consider {\it amputated} Schwinger functions. Thus there is associated a factor of $\ti\de(x_i,z)\,$ to the external line joining $x_i$ to an internal vertex at $z$ which is integrated over. Therefore the distributional character is different according to whether one or several external lines end in a given $z$-vertex. From the point of view of the FE the distributional character is a consequence of the boundary conditions, see (\ref{nawig}) and (\ref{bo2}), (\ref{bo3}). Note that by (\ref{hk0}) the propagators $C^{\vep,t}\,$ which join different vertices are smooth functions of their position arguments for $\vep >0\,$. The two point function is the most divergent object as regards its flow for small $t$. But the distributional structure of its regularized version is particularly simple. In fact it follows from (\ref{nawig}) and (\ref{intfe}) that $\,{\cal L}^{\vep,t}_{2,l} (x_1,x_2) \,$ can be written as a sum over regularized Feynman amplitudes with vertices from $\, L^{\vep} \,$ and with regularized propagators which satisfy (\ref{hk0}), (\ref{hk5}). Thus the only distributional singularities appearing are of the form ${\ti \de}(x_2,x_1)\,$ and ${\De^{(b_l^{\vep,\vep})}}_{x_2}\ {\ti \de}(x_2,x_1)\,$. Thus $\,{\cal L}^{\vep,t}_{2,l} (x_1,x_2) \,$ can be written as a linear combination of these two contributions and a smooth function $f(x_1,x_2)\,$ of rapid decrease in $\,d(x_1,x_2)\,$.\\ The FE for the Schwinger functions derived from (\ref{nfeq}) takes the following form~: \eq \pa_{t} {\cL}^{\vep,t}_{n,l}(x_1,\ldots,x_{n})=\, \frac12\,\int_{x,y} C_t(x,y)\ \Biggl\{ {\cL}^{\vep,t}_{n+2,l-1}(x_1,\ldots,x_{n},x,y)\ - \label{fequ} \eqe \[ \sum_{l_1+l_2=l,\atop n_1+n_2=n}\Bigl[ {\cal L}^{\vep,t}_{n_1+1,l_1}(x_1,\ldots,x_{n_1},x)\, \, {\cal L}^{\vep,t}_{n_2+1,l_2}(y,x_{n_1+1}, \ldots,x_{n})\Bigr]_{sym}\Biggr\}\, . \] Here $sym$ means symmetrization - i.e. summing over all permutations \footnote{by our choice of normalization there is no normalization factor to be divided out} of $(x_1,\ldots, x_{n})$ {\it modulo those} which only rearrange the arguments of a factor. \footnote{This may be implemented by counting only the configurations in which the permuted position variables appearing in ${\cL}^{\vep,t}_{n_1+1}\,$ and ${\cL}^{\vep,t}_{n_2+1}\,$ appear in lexicographic order.} \\ For the renormalization proof we also need the FE for the Schwinger functions derived w.r.t. the UV cutoff $\vep$. Integrating the FE over $t'\,$ between $\vep$ and $t$ and then deriving w.r.t. $\vep$ we obtain \[ \pa_{\vep} \,{\cL}^{\vep,t}_{n,l}(x_1,\ldots,x_{n})=\ \pa_{\vep}{\cL}^{\vep,\vep}_{n,l}(x_1,\ldots,x_{n})\, -\, \frac12\,\int_{x,y} C_{\vep}(x,y)\ \Biggl\{ {\cL}^{\vep,\vep}_{n+2,l-1}(x_1,\ldots,x_{n},x,y)\ - \] \eq \sum_{l_1+l_2=l,\atop n_1+n_2=n}\Bigl[ {\cal L}^{\vep,\vep}_{n_1+1,l_1}(x_1,\ldots,x_{n_1},x)\, \, {\cal L}^{\vep,\vep}_{n_2+1,l_2}(y,x_{n_1+1}, \ldots,x_{n})\Bigr]_{sym}\Biggr\}\ + \label{fequvep} \eqe \[ \frac12\,\int_{x,y} \int_{\vep}^t dt'\ C_{t'}(x,y)\ \Biggl\{ \pa_{\vep} {\cL}^{\vep,t'}_{n+2,l-1}(x_1,\ldots,x_{n},x,y)\ - \] \[ \sum_{l_1+l_2=l,\atop n_1+n_2=n}\Bigl[ \pa_{\vep} \Bigl({\cal L}^{\vep,t'}_{n_1+1,l_1}(x_1,\ldots,x_{n_1},x)\, \, {\cal L}^{\vep,t'}_{n_2+1,l_2}(y,x_{n_1+1}, \ldots,x_{n})\Bigr)\Bigr]_{sym}\Biggr\}\ . \] Integrating the FE instead from $t<1$ to $t=1$ and then deriving w.r.t. $\vep\,$ we get \eq \pa_{\vep} \,{\cL}^{\vep,t}_{n,l}(x_1,\ldots,x_{n})=\ \pa_{\vep}{\cL}^{\vep,1}_{n,l}(x_1,\ldots,x_{n})\ - \label{fequvep2}\eqe \[ \frac12\,\int_{x,y} \int_{t}^1 dt' \ C_{t'}(x,y)\ \Biggl\{ \pa_{\vep} {\cL}^{\vep,t'}_{n+2,l-1}(x_1,\ldots,x_{n},x,y)\ - \] \[ \sum_{l_1+l_2=l,\atop n_1+n_2=n}\Bigl[ \pa_{\vep} \Bigl({\cal L}^{\vep,t'}_{n_1+1,l_1}(x_1,\ldots,x_{n_1},x)\, \, {\cal L}^{\vep,t'}_{n_2+1,l_2}(y,x_{n_1+1}, \ldots,x_{n})\Bigr)\Bigr]_{sym}\Biggr\}\ . \] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Test functions and Tree structures} The distributional character of the ${\cal L}^{\vep,t}_{n,l}\, $ necessitates the introduction of test functions against which they will be integrated.\\[.1cm] {\it Definition 1}~: For $\, n \in \bbbn\,$ we set \[ \cH_n: =\{ \vp(\xv_n)= \vp_1(x_1)\ldots \vp_n(x_n)\ | \ \vp_i \in C^{\infty}({\cal M})\cap L^{\infty}({\cal M})\} \ . \] We wrote $\,\xv_n=\ (x_1,\ldots,x_n)\,$ and we shall write \footnote{By the bosonic symmetry of the ${\cal L}^{\vep,t}_{n,l}\,$ all bounds are independent of the particular role assigned to the coordinate $x_1\,$, which can be exchanged with any other coordinate.} $\,x_{2,n}=\, (x_2,\ldots,x_n) \in {\cal M}^{\times (n-1)} \,$.\\ For $\vp\, \in \cH_{n-1}\,$ we set \eq {\cal L}^{\vep,t}_{n,l}(x_1,\vp) :=\, \int_{x_{2,n}} {\cal L}^{\vep,t}_{n,l} (\xv_n)\ \vp(x_{2,n})\ . \label{lin} \eqe The regularized Schwinger functions are obviously linear w.r.t. the test functions: \eq {\cal L}^{\vep,t}_{n,l}(x_1,a \,\vp_1+ b\,\vp_2) =\, a\ {\cal L}^{\vep,t}_{n,l}(x_1,\vp_1) \,+\, b\ {\cal L}^{\vep,t}_{n,l}(x_1,\vp_2) \ ,\quad a,b \in \bbbc\ , \label{lin1} \eqe and it also follows from from (\ref{nawig}) and (\ref{intfe}) and the properties of the regularized propagator (\ref{hk0}), (\ref{hk8}) that they satisfy \[ {\cal L}^{\vep,t}_{n,l}(x_1,\vp)\in \cH_{1}\ . \] We will also consider Schwinger functions multiplied by products of factors $\si(x_j,x_1) ^{\mu}$, (\ref{sig}).\\ {\it Definition 2}~: We introduce a smooth (external) covector field $ \om_\mu (x)$ and form the bi-scalar insertions \eq E_{(i)}\equiv E(x_i, x_1~;\om) := \si(x_i, x_1)^\mu\, \om_{\mu}(x_1), \qquad i=2, \cdots, n\ , \label{ino} \eqe and, more generally, for $r\in \bbbn\,$, \eq E ^{(r)}_{(i)}\equiv E(x_i, x_1~;\om^{(r)}) := \si(x_i, x_1)^{\mu_1}\ldots \si(x_i, x_1)^{\mu_r}\, \om^{(r)}_{\mu_1 \ldots\mu_r }(x_1) \label{inor} \eqe with a smooth (external) symmetric covariant tensor field $\om^{(r)}_{\mu_1 \ldots\mu_r } (x)$ of rank $r$. We have, because of (\ref{nsi}), \eq | E(x_i, x_1~;\om^{(r)}) | \leq |\,\om^{(r)}(x_1)| \, d^{\,r}(x_i,x_1) \label{nino} \eqe with the norm $ |\,\om^{(r)}(x_1)| $ according to (\ref{nor}). For $r\,\in \bbbn\,$ we then pose \eq {\cal L}^{\vep,t}_{n,l}(x_1, E^{(r)}_{(i)}\, \vp):= \int_{x_{2,n}} E(x_{i},x_{1}~;\om^{(r)})\, {\cal L}^{\vep,t}_{n,l} (\xv_n)\ \vp(x_{2,n})\ . \label{ins} \eqe Mostly we will suppress $\om$ in the notation as we did in (\ref{ins}). Furthermore, for given $x_1, x_2\, \in \cal{ M}$ we consider the products \eq F^{(r)}_{(12)} {\cal L}^{\vep,t}_{n,l}(x_1, x_2, \vp) := d^{\,3-r}(x_1, x_2)E(x_{2},x_{1}~;\om^{(r)})\, \int_{x_{3,n}}{\cal L}^{\vep,t}_{n,l} (\xv_n)\ \vp(x_{3,n}) \label{2ins} \eqe for $ r = 0,1,2\, $, with $ E \equiv 1\mbox{ if}\,\, r = 0\, $, and with $\vp(x_{3,n})\equiv 1\,$ (and no integration) \mbox {if} $n <3\,$. \noindent In the subsequent definitions we introduce tree structures, which will appear in the bounds for the CAS (\ref{lin}) to be established later. These bounds are proven inductively, and the tree structures are fixed such that they are reproduced under the operations which are employed to conclude from the CAS appearing on the r.h.s. of the FE on the CAS appearing on the l.h.s. In this paper a tree is understood to be a set of lines and vertices satisfying the subsequent definitions. \noindent {\it Definition 3}~: For $ s \geq 2\, $ let $\{x_1\}\,$ and $Y=\{y_2,\ldots,y_s\}\,$ be sets of coordinates in $\cal M\,$. We call $x_1\,$ the root vertex and $y_i$ an external vertex of the tree\, \footnote{In mathematically straight notation a vertex should be viewed as the image of an element of a discrete set under a mapping from this set into $\cal M$.}. Likewise let $Z=\{ z_1,\ldots, z_r\}\,$ for $r \ge 0\,$ be a set of coordinates in $\cal M\,$, called internal vertices of the tree. Furthermore let $\cal P$ be a set of 'adjacent pairs' of distinct coordinates $\{a,b\}\,$, $a,\ b \in \{x_1\}\cup Y \cup Z\,$, called lines, with the following properties~:\\ i) For two distinct elements $a,\ b \in \{x_1\}\cup Y \cup Z\,$ there exists a {\it unique} path in $\cal P$ from $a$ to $b$. By this we mean that either $\{a,b\}\in \cal P\,$, or there exists a unique set of (mutually distinct) pairs $p_1, \ldots,p_{n+1}\,$; $p_i \in \cal P\,$, $n \ge 1$, such that $p_1=\{a, v_1\}\,$, $p_{\nu+1} =\{v_{\nu},v_{\nu+1}\}\,$, $1 \le \nu \le n-1\,$, and $p_{n+1}=\{v_n,b\}\,$.\\ ii) For $y_i \in Y\,$ there exists exactly one $p \in \cal P$ such that $y_i \in p\,$. For $x_1$ there exist $p_1,\ldots,p_{c_1}\in\cal P$ with $1 \le c_1 \le s-1$ such that $x_1 \in p_1,\ldots,x_1 \in p_{c_1}$. For $z_j \in Z$ there exist $p^{(z_j)}_1,\ldots,p^{(z_j)}_{c_j}\in\cal P$ with $2 \le c_j \le s \,$ such that $z_j \in p^{(z_j)}_1,\ldots, z_j \in p^{(z_j)}_{c_j}\,$. We call $c_1=c(x_1)$ the coordination number of the root vertex and $c(z_j)=c_j\,$ the coordination number of the internal vertex $z_j$ of the tree.\\ We call a line $p \in \cal P$ an external line of the tree if there exists $y_i$ such that $y_i \in p\,$. The set of external lines is denoted $\cal J$. The remaining lines are called internal lines of the tree and are denoted by $\cal I$, hence $\cal P = \cal J \cup \cal I $.\\ Denoting by $ v_c $ the number of vertices having coordination number $c$, it follows from the definition that $\sum_{c\ge 2} (c-2)\, v_c = s-3+ \delta_{c_1,1}\,$. We denote by $\mathcal{T}^s $ the set of all trees in the sense of the previous definition. By $ T_l^s $ we denote a tree $ T^s \in \mathcal{T}^s$ satisfying $ v_2+\de_{c_1,1} \leq \, 3l-2+s/2\, $ for $l\ge 1\,$ and satisfying $ v_2=0\,$ for $l =0\,$. Then $ \mathcal{T}_l^s $ denotes the set of all trees $ T_l^s $. We indicate the external vertices and internal vertices of the tree by writing $T_l^{s}(x_1, y_{2, \, s}, \vec z \,)$ with $y_{2, \, s}=(y_2,\ldots,y_s)\,,\ \vec z =(z_1,\ldots,z_r)\,$.\\ Finally we also define for $i \le s$ the set of {\it twice rooted trees} denoted as $\mathcal{T}^{s,(12)}_l\,$. The trees ${T}^{s,(12)}_l\,\in \mathcal{T}^{s,(12)}_l\,$ are defined exactly as the trees $\,T^{s}_l\, $ apart from the fact that they have two root vertices $x_1, \ x_2\,$ with the properties of Definition 3,ii) above, and $s-2$ external vertices. \\[.1cm] {\it Definition 4}~: For a tree $T_l^{s+2}(x_1, y_{2, \,s+2},\vec z \,)$ we define the {\it reduced tree}\\ $ T_{l,y_i,y_j}^{s}(x_1,y_2, \ldots, y_{i-1},y_{i+1}, \ldots,y_{j-1},y_{j+1},\ldots, y_{s+2}, \vec z_{ij})\,$ to be the unique tree to be obtained from $T_l^{s+2}(x_1, y_{2, \,s+2},\vec z \,)$ through the following procedure~:\\ i) By taking off the two external vertices $y_i, y_j\,$ together with the external lines attached to them.\\ ii) By taking off the internal vertices - if any - which have acquired coordination number $c=1\,$ through the previous process, and by also taking off the lines attached to them.\\ iii) If the process ii) has produced a new vertex of coordination number 1 go back to ii). \\[.1cm] In the sequel we shall bound the CAS folded with test functions. Here we restrict to test functions of the following form~: Let $1\le s \le n\,$ and $\,\tau= \tau_{2,s}= (\tau_2,\ldots,\tau_{s})\,$ with $\,0< \tau_i \,$ \footnote{The function $\mathbf{1}(x)\,$ is obtained on integrating $K(\tau,x,y)\, $ over $y\,$. This could be used to unify the notation.} \eq \vp_{\tau,y_{2,s} }( x_{2,n})~:=\ \prod_{i=2}^{s} K(\tau_{i}, x_i,y_i)\ \prod_{i=s+1}^n \mathbf{1}(x_i)\ . \label{phic} \eqe Here $\mathbf{1}(x)=1\ \ \, \forall x \in \cal M\,$. The pair $ \tau_j , y_j $ determines the width and the center of localisation of the test function. This definition can be generalized by choosing any other subset of $s$ coordinates among $x_2, \ldots, x_n\,$. We also define \footnote{Note that $\vp^{(j)}_{\tau,y_{2,s} }$ depends on $x_1$ which is not indicated.} for $2 \le j \le s\,$ \eq \vp^{(j)}_{\tau,y_{2,s} }( x_{2,n})~:=\ K^{(1)}(\tau_{j}, x_j,x_1;y_j)\ \prod_{i=2,i\neq j}^{s} \ K(\tau_{i}, x_i,y_i)\ \prod_{i=s+1}^n \mathbf{1}(x_i) \label{phid} \eqe with \eq K^{(1)}(\tau_{j}, x_j,x_1;y_j) =\ K(\tau_j, x_j,y_j)-\ K(\tau_j, x_1,y_j)\ . \label{k1} \eqe \noindent {\it Definition 5:} Given $\tau$, $y_{2,s}\,$, $\de >0\,$, and a set of internal vertices $\vec z =(z_1,\ldots, z_r)\,$, and attributing positive parameters $t_{{\cal I}}= \{ t_I| I \in {\cal I}\}\,$ to the internal lines, the {\it weight factor $ {\cal F}(t_{{\cal I}},\tau;T_l^{s}( x_1, y_{2,s},\vec z))$ of a tree $T_l^{s}( x_1, y_{2,s},\vec z)\,$ at scales $t_{{\cal I}}$ } is defined as a product of heat kernels associated with the internal and external lines of the tree. We set \eq {\cal F}(t_{{\cal I}},\tau;T_l^{s}(x_1, y_{2,s},\vec z)) :=\ \prod_{I\in {\cal I}}C_{t_{I,\de}}(I)\ \prod_{J \in {{\cal J}}}\ K(\tau_{J,\de}, J)\ . \label{f} \eqe Here we denote by $\tau_{J}\,$ the entry $\tau_i\,$ in $\tau\,$ carrying the index of the external coordinate $y_i$ in which the external line $J\,$ ends. For $I =\{a,b\}\,$ the notation $C_{t_I}(I)\,$ stands for $C_{t_I}(a,b)\,$. \\ We then also define the {\it integrated weight factor} of a tree by \eq {\cal F}(t,\tau;T_l^{s}~; x_1, y_{2,s})~:= \sup_{\{t_I| I \in {\cal I},\, \vep \le t_I \le t\}} \int_{\vec z} {\cal F}(t_{{\cal I}},\tau;T_l^{s}( x_1, y_{2,s},\vec z))\ . \label{ff} \eqe It depends on $\vep$, but note that its limit for $\vep \to 0\,$ exists, and that typically the $\sup\,$ is expected to be taken for the maximal values of $t\,$ admitted. Therefore we suppress the dependence on $\vep\,$ in the notation. Finally we introduce the shorthand notation for the {\it global weight factor $\,{\cal F}_{s,l}(t,\tau;x_1, y_{2,s})\,$ or more shortly $\,{\cal F}_{s,l}(t,\tau)\,$ } which is defined as follows \eq {\cal F}_{s,l}(t,\tau)\,\equiv\,{\cal F}_{s,l}(t,\tau;x_1, y_{2,s})\,:=\, \sum_{T_l^{s} \in {\cal T}_l^s} {\cal F}(t,\tau;T_l^{s}~;x_1, y_{2,s})\ . \label{ges} \eqe In complete analogy we define the weight factors and global weight factors for twice rooted trees which we denote as ${\cal F}(t,\tau;T_l^{s,(12)}~;x_1,x_2, y_{3,s})\,$ resp. ${\cal F}^{(12)}_{s,l}(t,\tau;x_1,x_2,y_{3,s})\,$ or $\,{\cal F}^{(12)}_{s,l}(t,\tau)\,$. Following the definitions (\ref{f})-(\ref{ges}) we also define for $t \ge 1\,$ \eq {\cal F}^{\,t}(\tau;T_l^{s}(x_1, y_{2,s},\vec z)) :=\ \sup_{\{t_I| I \in {\cal I},\, \vep \le t_I \le 1\}} \prod_{I\in {\cal I}}[(C_{t_{I,\de}}(I) +\int_{1}^{t}C_{t'}(I)\ dt')] \ \prod_{J \in {{\cal J}}}\ K(\tau_{J,\de}, J)\ , \label{fi} \eqe \eq {\cal F}^{\, t}(\tau;T_l^{s}~; x_1, y_{2,s})~:= \int_{\vec z} {\cal F}^{\,t}(\tau;T_l^{s}( x_1, y_{2,s},\vec z))\ , \label{iff} \eqe and \eq {\cal F}^{\,t}_{s,l}(\tau)\,:=\, \sum_{T_l^{s} \in {\cal T}_l^s} {\cal F}^{\,t}(\tau;T_l^{s}~;x_1, y_{2,s})\ . \label{iges} \eqe For $s=1\,$ we set $\, {\cal F}_{1,l}(t,\tau)\,\equiv\,1 \ .$\\ We give more explicitly the form of ${\cal F}_{2,l}(t,\tau;x,y)\,$. It is by definition given through \[ {\cal F}_{2,l}(t,\tau;x,y)\,=\, \sum_{T_l^2} {\cal F}_{2,l}(t,\tau;T_l^2~;x,y)\ = \ K(\tau_{\de}, x,y)\ +\ \] \[ \sum_{n = 1}^{3l-2}\, \, \sup_{\{t_{I_i}| \vep\, \leq\, t_{I_i}\, \leq\, t,\, i=1.\cdots,n\}} \, \,[\prod_{1 \le i \le n } \int_{z_i}]\ C_{t_{I_1,\de}}(x,z_1) \ldots C_{t_{I_n,\de}}(z_{n-1},z_n)\ K({\tau}_{\de}, z_n,y)\ . \] Using (\ref{hk4}) we get \eq {\cal F}_{2,l}(t,\tau;x,y)\ = \sum_{n = 0}^{3l-2}\, \, \sup_{\{t_{I_i}| \vep\, \leq\, t_{I_i}\, \leq\, t,\, i=1.\cdots,n\}} C_{\tau_{\de}+ \sum_1^n t_{I_i,\de}}(x,y) \ e ^{m^2\tau_{\de}}\ . \label{f2r} \eqe %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Boundary and renormalization conditions} Form the mathematical point of view the renormalization problem in the FE framework appears as a mixed boundary value problem. The relevant terms are fixed by renorma\-lization conditions at a large value \,$t_R$\, of the flow parameter $t$, all other boundary terms are fixed at the short-distance cutoff $ t=\vep $.\\[.2cm] To extract the relevant terms - contained in ${\cal L}^{\vep,t}_{2,l}(x_1,\vp)\,$ and ${\cal L}^{\vep,t}_{l,4}(x_1,\vp)\,$- a covariant Taylor expansion with remainder term (\ref{sloe}), (\ref{Rn}) of the test function $ \, \vp \,$ is used, $\vep \le t$~: \eq {\cal L}^{\vep,t}_{2,l}(x_1,\vp)\,=\, a^{\vep,t}_l(x_1)\ \vp(x_1) \, - \, f^{\mu,\vep,t}_l(x_1)\ (\nabla _{\mu} \vp)(x_1) \, - \, b^{\,\mu \nu,\vep,t}_l(x_1) (\nabla_\mu \nabla_\nu \vp)(x_1) \,+\, {\ell}^{\vep,t}_{2,l}\ (x_1, \vp)\ , \label{2l} \eqe \eq {\cal L}^{\vep,t}_{4,l}(x_1,\vp)=\ c^{\vep,t}_l (x_1)\ \vp_2(x_1)\ \vp_3(x_1) \vp_4(x_1) + \ {\ell}^{\vep,t}_{4,l} (x_1, \vp)\,. \label{4l} \eqe Then the relevant terms appear as $$ a^{\vep,t}_l(x_1) = \int_{x_2}\! {\cal L}^{\vep,t}_{2,l} (x_1,x_2)\ ,\quad f ^{\mu,\vep,t}_l(x_1) = \int_{x_2}\! \si(x_2,x_1)^{\mu} \ {\cal L}^{\vep,t}_{2,l} (x_1,x_2)\ , $$ \eq b^{\mu\nu,\vep,t}_l(x_1)=-\, \frac12 \, \int_{x_2}\! \si(x_2,x_1)^{\mu} \ \si(x_2,x_1)^{\nu} {\cal L}^{\vep,t}_{2,l} (x_1,x_2) \, , \label{coeff} \eqe \eq c^{\vep,t}_l(x_1)= \int_{x_2,x_3,x_4}\!\!\! {\cal L}^{\vep,t}_{4,l} (x_1, \ldots ,x_4)\, , \label{coeff4} \eqe and the `remainders' ${\ell}^{\vep,t}_{2,l}$ and ${\ell}^{\vep,t}_{4,l}$ have the respective forms \eq {\ell}^{\vep,t}_{2,l} (x_1, \vp)\ =\ \int_{x_2} {\cal L}^{\vep,t}_{2,l} (x_1,x_2) \int_{0}^{s} dr\, \frac{ (s-r)^2}{2!} \,\, {\dot x}_{12}^{\nu_{3}} (r)\, {\dot x}_{12}^{\nu_{2}} (r) \, {\dot x}_{12}^{\nu_1} (r) \big( \nabla_{\nu_{3}} \nabla_{\nu_{2}} \label{re2} \nabla_{\nu_{1}}\vp \big)(x_{12}(r)) \eqe where $ s = d(x_1, x_2)$\, and $x_{12}(r)\, $ is the point on the geodesic segment from $x_1$ to $x_2\,$ at arc length $r\,$; and \[ {\ell}^{\vep,t}_{4,l} (x_1, \vp)\ =\ \int_{x_2,x_3,x_4} {\cal L}^{\vep,t}_{4,l} (x_1,\ldots,x_4) \Bigl[ \int_{0}^{s_{12}} dr\, \,\, {\dot x}^{\nu}_{12} (r)\ \big( \nabla_{\nu} \vp_2\big)(x_{12}(r)) \ \vp_3(x_3)\vp_4(x_4) \ +\ \] \[ \vp_2(x_1) \int_{0}^{s_{13}} dr\, \,\, {\dot x}^{\nu}_{13} (r)\ \big( \nabla_{\nu} \vp_3\big)(x_{13}(r)) \ \vp_4(x_4) \ + \] \eq \vp_2(x_1) \ \vp_3(x_1) \int_{0}^{s_{14}} dr\, \,\, {\dot x}^{\nu}_{14} (r)\ \big( \nabla_{\nu} \vp_4\big)(x_{14}(r))\Bigr]\, . \label{re4} \eqe Reparametrizing the geodesic segment $ x_{12}(r) = X(\rho),\, r = d(x_1, x_2) \rho\, ,\, 0 \leq \rho \leq 1 $, we can rewrite the remainder (\ref{re2}) employing (\ref{Rna}) $$ {\ell}^{\vep,t}_{2,l} (x_1, \vp) = \int_{x_2}\!\!\! d^{\,3}(x_1, x_2) {\cal L}^{\vep,t}_{2,l} (x_1,x_2) \int_{0}^{1}\!\!\! d\rho \frac{ (1-\rho)^{\,2}}{2 !\,d^{3}(x_1, x_2)} {\dot X}^{\nu_{3}}(\rho)\, {\dot X}^{\nu_{2}}(\rho) \, {\dot X}^{\nu_1}(\rho) \, \om^{(3)}_{{\nu}_3 {\nu}_2 {\nu}_1}(X(\rho)) $$ \eq \mbox{where}\qquad \om^{(3)}_{{\nu}_3 {\nu}_2 {\nu}_1}(x) \, = \, \big( \nabla_{\nu_{3}} \nabla_{\nu_{2}} \nabla_{\nu_{1}}\vp \big)(x)\ . \label{re22} \eqe 1) {\bf Boundary conditions} at $\,t =\vep\,$~:\\ The bare interaction (\ref{nawig}) implies that at $t =\vep\ $ - with $ {\cal L}^{\vep}\equiv {\cal L}^{\vep,\vep}$ - \eq {\cal L}^{\vep}_{n,l} (x_1,\ldots x_{n}) \equiv 0\ \mbox{ for}\quad n >4\ , \quad {\cal L}^{\vep}_{2,0} \equiv 0 \label{bo1} \eqe \eq {\cal L}^{\vep}_{2,l} (x_1,x_2) = {\tilde a}_l^{\epsilon}(x_1)\, \tilde{\delta}(x_2,x_1) - \ \Delta^{(b)}_2 \, \tilde{\delta}(x_2,x_1)\,, \,b = b_l^{\,\mu \nu,\, \vep} (x_2) \,, \label{bo2} \eqe \eq {\cal L}^{\vep}_{4,l} (x_1,\ldots x_{4}) = ({ \de}_{l,0}\ \la(x_1) \,+\,(1-{ \de}_{l,0})\,c_l^{\vep}(x_1))\ {\ti \de}(x_2,x_1)\,{\ti \de}(x_3,x_1)\,{\ti \de}(x_4,x_1)\ . \label{bo3} \eqe To cope with the relevant part of the expansion (\ref{2l}) we consider a corresponding bare part \eq {\cal L}^{\,\vep}_{2,\,l}(x_1,\vp)\,=\, a^{\,\vep}_l(x_1)\ \vp(x_1) \, - \, f^{\mu,\,\vep}_l(x_1)\ (\nabla _{\mu} \vp)(x_1) \, - \, b^{\,\mu \nu,\,\vep}_l(x_1) (\nabla_\mu \nabla_\nu \vp)(x_1)\,. \label{baex} \eqe The identity \eq -\, b^{\,\mu \nu}_l(x) (\nabla_\mu \nabla_\nu \vp)(x) \,= \, \nabla_\nu\, b^{\,\mu \nu}_l (x)\cdot \big (\nabla_\mu \vp\big )(x)\, -\, \Delta ^{(b)} \vp (x)\,, \quad b = b^{\,\mu \nu}_l(x) \label{cov2} \eqe suggests to decompose the bare vector coefficient appearing in (\ref{baex}) as \eq f^{\mu,\,\vep}_l(x_1)\, = \, {\tilde f}^{\mu,\,\vep}_l(x_1)\, + \, \nabla_\nu\, b^{\,\mu \nu,\,\vep}_l (x_1)\, . \label{bave} \eqe By folding (\ref{baex}) with the test function $ \vp $ we obtain after partial integration \footnote{\, If $ \cal M $ is noncompact, $ \vp \in {\cal S} (\cal M) $ is assumed. } \eq \int_{x} \,\vp(x)\,{\cal L}^{\,\vep}_{2,\,l}(x,\vp)\,=\, \int_{x}\Big \{ \big ( a^{\vep}_l(x)+ \frac{1}{2}\, \nabla_{\mu} {\tilde f}^{\mu,\,\vep}_l(x)\big )\, \vp^2(x) + \, b^{\,\mu \nu, \vep}_l(x)\ \partial_{\mu}\vp(x)\cdot\partial_{\nu} \vp(x) \Big \} \,. \label{baac} \eqe This agrees in form with the corresponding content of the bare action (\ref{nawig}). From the boundary conditions (\ref{bo1})-(\ref{bo3}) we deduce \eq {\ell}^{\vep,\vep}_{2,l}\ (x_1,\,\vp)\ =\ 0\ ,\quad {\ell}^{\vep,\vep}_{4,l}\ (x_1,\,\vp)\ =\ 0\ . \label{bell} \eqe The renormalization problem is related to the behaviour of the heat kernel at small values of $t$. Therefore this problem is essentially solved if we can integrate the flow equations up to some finite value $t_R$ of $t$. For shortness we choose units such that $t_R =1$. We will come to the limit $t\to \infty\,$ later, see Proposition 3. The positive mass $m>0\,$ only plays a role when this limit is taken. We pose\\ 2) {\bf Renormalization conditions} at $t = t_R := 1\,$ ~: \footnote{The scale $t_R\,$ is related to the scale $T$ appearing in the bounds on the heat kernel (\ref{hk10}), (\ref{d}), (\ref{D}) .} \eq a ^{\vep, 1}_l(x_1):=\, a ^R_l(x_1) ,\quad f^{\mu,\vep, 1}_l(x_1):=\, f^{\mu,R}_l(x_1),\quad b^{\mu\nu, \vep, 1}_l(x_1):=\, b^{\mu\nu, R}_l(x_1)\, , \label{renbed} \eqe \eq c ^{\vep, 1}_l(x_1) :=\, c ^R_l(x_1)\, ,\qquad \label{renbed4} \eqe where $b^{\mu\nu, R}_l(x)\,$ is a smooth symmetric tensor of type $(2,0)\,$, $f^{\mu, R}_l(x)\,$ is a smooth vector and $a^R_l(x)\,,\ c ^R_l(x)\,$ are smooth scalars on $\cal M\,$, all uniformly bounded in the norm (\ref{nor}). Typically the renormalization conditions are assumed to be cutoff-independent. To be able to analyse the relation between the bare (inter)action and the renormalization conditions in more detail later on, we shall be more general in also admitting weakly $\vep$-dependent renormalization functions satisfying \eq |\pa_{\vep}\,a_l^R(x)| \,< \, O(\vep^{-\eta}) \ ,\quad |a_l^R(x)| \,< \, const\,+\, O(\vep^{1-\eta})\ , \quad \eta \le 1/2\ , \label{vepdep} \eqe with analogous expressions for the other renormalization functions. In the particular case of $\, {\cal M}\,$ having constant curvature, i.e. where all sectional curvatures of $\, {\cal M}\,$ have a constant value $\, \rho \, $, a transitive isometry group $ \, G \,$ acts on $\, {\cal M}\,$. There are three types of such manifolds: The sphere $\, {\cal S}^4 \,$ with $\, \rho = k^2 $ \, and \,$G = SO(5)$\,, the flat space $ {\mathbf R}^4\,$ with $\, \rho = 0 $ \,and \,$G = SO(4)\otimes_s {\mathbf R}^4 $\,, the hyperbolic space \,$ {\cal H}^4$\, with $\, \rho = - k^2 $\, and \, $ G = SO_0(4,1)$,\, the subscript denoting the component connected to the identity. \\ Requiring the Schwinger functions to show this symmetry $\,G\,$, results in the following restrictions on the relevant terms: \\ i)$\quad a^{\vep,t}_l(x), \, c^{\vep,t}_l (x)\quad $ do not depend on\, $ x \in {\cal M} $, \\ ii)$\quad f^{\mu,\vep,t}_l(x)\equiv 0\, , \quad b^{\,\mu \nu,\vep,t}_l(x) = g^{\,\mu \nu}(x)\, b^{\,\vep,t}_l $\,,\,\, hence \, $\nabla_{\nu}\, b^{\,\mu \nu,\vep,t}_l(x)\equiv 0 $. \\ However, there is a further (dimensionless) parameter \, $ \zeta = k^2/ m^2 $\, on which $a^{\vep,t}_l,\ b^{\vep,t}_l,\ c^{\vep,t}_l\, $ may depend, in general. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Renormalizability} The subsequent proposition is proven for test functions of the form $\,\vp_{\tau_{2,s},y_{2,s}}( x_{2,n})$, (\ref{phic})\,. Bounds for more general test functions then follow by linearity (\ref{lin}) and through extension by continuity arguments. By Bose symmetry the bounds stay unaltered if any permuted subset of external coordinates (and not $\,x_2, \ldots, x_s\,$) is folded with test functions. \\ \noindent {\bf Proposition 1}: \\ {\it We consider $0 < \vep \le t \le 1\,$ and $\vep < \tau_i \,$, furthermore $1 \le s \le n\,$, $2 \le i \le n\,$, $2 \le j \le s \,$ and $0 \le r \le 3\,$. We consider test functions either of the form $\vp_{\tau_{2,s},y_{2,s}}( x_{2,n})\,$ or $\vp^{(j)}_{\tau_{2,s},y_{2,s}}( x_{2,n})\,$, which are also denoted in shorthand as $\vp_s \,$ resp. $\vp^{(j)}_s\,$.\\ In all subsequent bounds we understand ${\cal P}_{l}$ to denote a polynomial of degree $\le \sup(l,0)\,$ - each time it appears possibly a new one - with nonnegative coefficients which may depend on $l,n,\de$ \footnote{We suppose that $\de>0$ may be chosen arbitrarily small in the definition of $\cal F$. The constants in ${\cal P}_{l}$ then depend on the choice of $\de$.}, on $\sup_{\cal M}|\la(x)|\,$, as well as on $k^2\,$, $\ka ^2\,$ and the bounds on the first and second covariant derivatives of the curvature tensor (see (\ref{hk10}) - (\ref{D})), {\it but not on $\,\vep,\,t,\,m\, $} and $\tau$. Constants $O(1)\,$ in the subsequent proof are to be understood in the same way. By $(t,\tau)\,$ we denote $\inf \{\tau_2,\ldots,\tau_s, t\}$, by $(t,\tau)_{i}\,$ we denote $\inf \{\tau_2,\ldots, \not\!\! \tau_i,\ldots, \tau_s, t\}$. \noindent Then we claim the following bounds - using the shorthand (\ref{ges}) - \eq |\,{\cal L}^{\vep,t}_{n,l} (x_1,\vp_{\tau,y_{2,s}})|\,\le\ t^{\frac{n-4}{2}}\ {\cal P}_l\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau) \ , \quad n \ge 4 \label{prop1} \eqe \eq |\, {\cal L}^{\vep,t}_{n,l} (x_1, E_{(i)}^{(r)} \,\vp_{\tau,y_{2,s}})|\, \le\ |\,\om^{(r)}(x_1) | \ t^{\frac{n+r-4}{2}}\ {\cal P}_l\log(t,\tau)_{i}^{-1}\ {\cal F}_{s,l}(t,\tau) \ , \quad n > 4, \ r>0 \label{prop1r} \eqe \eq |\, {\cal L}^{\vep,t}_{n,l} (x_1, E_{(i)}^{(r)} \vp_{\tau,y_{2,s}})|\, \le\ |\,\om^{(r)}(x_1) | \ t^{\frac{n+r-4}{2}}\ {\cal P}_{l-1}\log(t,\tau)_{i}^{-1}\ {\cal F}_{s,l}(t,\tau) \label{prop3} \eqe \centerline{$\, n=2, r = 3\,$ or $\, n=4,\ r>0\,$ } \eq |\, {\cal L}^{\vep,t}_{2,l} (x_1,\vp_{{\tau},y_{2}})|\,\le\ (t,\tau)^{-1}\ {\cal P}_{l-1}\log (t,\tau)^{-1}\ {\cal F}_{2,l}(t,\tau) \label{prop20} \eqe \eq |\, {\cal L}^{\vep,t}_{2,l} (x_1,E_{(2)}\, \vp_{{\tau},y_{2}})|\, \le\ \nom \ (t , \tau)^{-1/2}\ {\cal P}_{l-1}\log (t,\tau)^{-1}\ {\cal F}_{2,l}(t,\tau) \label{prop21} \eqe \eq |\, {\cal L}^{\vep,t}_{2,l} (x_1,E_{(2)}^{(2)}\, \vp_{{\tau},y_{2}})|\, \le\ |\,\om^{(2)}(x_1) | \ {\cal P}_{l-1}\log (t,\tau)^{-1}\ {\cal F}_{2,l}(t,\tau) \label{prop22} \eqe \eq |\,{\cal L}^{\vep,t}_{n,l} (x_1,\vp^{(j)}_{\tau,y_{2,s}})|\,\le\ t^{\frac{n-4}{2}}\ \bigg(\frac{t}{\tau_j}\bigg)^{1/2}\ {\cal P}_l\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau) \ ,\quad n>2 \label{prop5} \eqe \eq |\, {\cal L}^{\vep,t}_{2,l} (x_1,\vp^{(2)}_{{\tau}, y_{2}})|\,\le\ (\frac{t}{\tau})^{1/2} (t, \tau)^{-1}\ {\cal P}_{l-1}\log(t,\tau)^{-1}\ {\cal F}_{2,l}(t,\tau) \label{prop2j} \eqe \eq |\,F^{(r)}_{(12)}{\cal L}^{\vep,t}_{n,l}(x_1, x_2, \vp)| \,\le\ |\,\om^{(r)}(x_1) | \ t^{\frac{n-1}{2}}\ {\cal P}_{l-1}\log(t,\tau)^{-1}\ {\cal F}^{(12)}_{s,l}(t,\tau)\ . \label{prop2r} \eqe $$ r = 0,1,2 \quad and \, \, |\,\om^{(0)}(x_1) | \equiv 1 \, .$$ } {\sl Remark~:} The full series of the previous bounds is needed to close the inductive argument in the subsequent proof. The reader who only wants to know what the bounds are can restrict to (\ref{prop1}), (\ref{prop20}). \noindent {\it Proof}~: The bounds stated in the proposition are proven inductively using the (standard) inductive scheme which proceeds upwards in $n+2l$, and for given $n+2l$ upwards in $l$.\\ Thus the induction starts with the pair $(4,0)\,$. For this term the r.h.s. of the FE vanishes so that ${\cL}^{\vep,t}_{4,0}(x_1,\ldots,x_4) = \la(x_1)\,{\ti \de}(x_2,x_1)\, {\ti \de}(x_3,x_1)\,{\ti \de}(x_4,x_1)\, $ which is compatible with our bounds (after folding with suitable $\vp\,$). Generally it is important to note that the boundary conditions are compatible with the bounds of the proposition.\\ We will first derive bounds for the derivatives $\pa_t\, {\cal L}^{\vep,t}_{n,l} (x_1, E_{(i)}^{(r)} \vp_s)\,$, where $\, E_{(i)}^{(0)}\equiv 1 $, and $\pa_{t}\, {\cal L}^{\vep,t}_{n,l} (x_1,\vp^{(j)}_s)\,$. Afterwards these bounds are integrated over w.r.t. $t$. \\ A) We start considering the case $r=0\,$ and test functions $\vp_s\,$.\\ A1) Here we first treat the first term on the r.h.s. of (\ref{fequ}) \[ R_1 := \, \int_{x_{2,n}, x,y} \vp_s( x_{2,n})\ C_{t}(x,y)\ {\cL}^{\vep,t}_{n+2,l-1}(\xv_n,x,y) \ . \] We may rewrite this expression as \[ R_1 = \int_v \int_{x_2,\ldots,x_n, x,y} \vp_s(x_{2,n})\ C_{t/2}(x,v)\ C_{t/2}(v,y) \ {\cL}^{\vep,t}_{n+2,l-1}(\xv_n,x,y)\ =\ \] \[ \int_v {\cL}^{\vep,t}_{n+2,l-1}(x_1,\vp_s \times C_{t/2}(\cdot,v)\times C_{t/2}(v,\cdot))\ . \] Applying the induction hypothesis to ${\cL}^{\vep,t}_{n+2,l-1}(x_1,\vp_s \times C_{t/2}(\cdot,v)\times C_{t/2}(v,\cdot))\,$ we thus obtain the bound \eq |R_1| \le t^{\frac{n+2-4}{2}} \ {\cal P}_{l-1}\log(t,\tau)^{-1} \int_v \int_{\vec z} \sum_{T_{l-1}^{s+2}(x_1,y_{2,s},v,v)} {\cal F}(t,\tau,\frac t2,\frac t2; T_{l-1}^{s+2}(x_1,y_{2,s},v,v,\vec z))\ . \label{1st1} \eqe For any contribution to (\ref{1st1}) we denote by $z',\ z''\,$ the vertices in the respective tree $\,T_{l-1}^{s+2}( x_1, y_{2,s},v,v,\vec z)$ to which the test functions $ C_{t/2}(\cdot,v),\ C_{t/2}(v,\cdot)\,$ are attached. Interchanging $\int_{\vec z}$ (see (\ref{ff})) and $\int_v\,$ and performing the integral over $v\,$ using (\ref{hk4}), (\ref{dpropa}), we get a contribution \eq \int_{v} C_{t/2}(z',v)\ C_{t/2}(v,z'')\ = \ \ C_t (z',z'')\ \le \ O(1)\ t^{-2}\ . \label{rom} \eqe Using this bound we can majorize $\,\int_v {\cal F}(t,\tau,\frac t2,\frac t2; T_{l-1}^{s+2}~;x_1,y_{2,s},v,v)\,$ by \[ O(1)\ t^{-2}\ {\cal F}(t,\tau;T_{l}^{s}~;x_1,y_{2,s}) \] where the tree $\,T_{l}^{s}\,$ is the {\it reduced tree} obtained from $\,T_{l-1}^{s+2}\,$ by taking away the two external lines ending in $v\,$, see Definition 4. The reduction process for each tree fixes uniquely the set of internal vertices of $\,T_{l}^{s}\,$ in terms of those of $\,T_{l-1}^{s+2}\,$. Note that the elimination of vertices of coordination number 1 together with their adjacent line is justified by the fact that $\int_{z'} C_{t_I,\de_l}(z,z') \le 1\,$. Note also that in the tree $T^{s}_l\,$ the number $v_2\,$ of vertices of coordination number $2$ may have increased by at most $2\,$, as compared to $T_{l-1}^{s+2}\,$, so that $T^{s}_l\,$ is indeed an element from $\,{\cal T}_l^{s}\,$. We keep track of this lower index $l$ in the tree basically to show that the number of vertices always stays finite (in fact does not grow faster than linearly in $l\,$ for $n$ fixed).\\ The final bound for the first term on the r.h.s. of the FE is thus \eq |R_1| \le t^{\frac{n-6}{2}}\ \ {\cal P}_{l-1}\log(t,\tau)^{-1}\ \sum_{T_{l}^{s}(x_1,y_{2,s})} {\cal F}(t,\tau;T_{l}^{s}~;x_1,y_{2,s}) \label{1st} \eqe where constants have been absorbed in $\,{\cal P}_{l-1}\log\,$.\\ \noindent A2) We now consider the second term in (\ref{fequ}) for\\ i) $n > 4 $\\ Picking a generic term from the symmetrized sum and arguing as in A1) we have to bound \[ R_2 := \int_{x_{2,n}, x,y} \vp_s(x_{2,n})\ C_{t}(x,y)\ {\cL}^{\vep,t}_{n_1+1,l_1}(x_1,\ldots,x_{n_1},x) \ {\cL}^{\vep,t}_{n_2+1,l_2}(y,x_{n_1+1},\ldots,x_{n}) \] which we rewrite similarly as in A1) \eq R_2 = \int_v \int_{x_2,\ldots,x_n, x,y} \vp_s(x_{2,n})\ C_{t/2}(x,v)\ C_{t/2}(v,y)\ {\cL}^{\vep,t}_{n_1+1,l_1}(x_1,\ldots,x) \ {\cL}^{\vep,t}_{n_2+1,l_2}(y,\ldots,x_{n})\ . \label{nuj} \eqe Denoting \[ \vp'_{s_1}(x_{2,n_1})= \prod^{n_1}_{r=2}\vp_r(x_r)\ , \quad \vp_{s_2}''(x_{n_1+1,n-1})= \prod_{r= n_1+1}^{n-1} \vp_r(x_{r}) \] we identify the two terms \[ \int_{x_2,\ldots,x_{n_1}, x} \vp'_{s_1}(x_{2,n_1})\ C_{t/2}(x,v)\ {\cL}^{\vep,t}_{n_1+1,l_1}(x_1,\ldots,x) \] and \[ \int_{x_{n_1+1},\ldots,x_n, y} \vp_{s_2}''(x_{n_1+1,n-1})\ C_{t/2}(v,y)\ {\cL}^{\vep,t}_{n_2+1,l_2}(y,\ldots,x_{n}) \] and thus write (\ref{nuj}) as \eq R_2 = \int_{x_n} \int_{v} {\cL}^{\vep,t}_{n_1+1,l_1}(x_1,\vp'_{s_1}\times C_{t/2}(\cdot ,v)) \ {\cL}^{\vep,t}_{n_2+1,l_2}(x_n,\,C_{t/2}(v,\cdot)\times \vp_{s_2}'' ) \ \, \vp_n(x_n)\ . \label{bose2} \eqe On applying the induction hypothesis to both terms in (\ref{bose2}), restricting first to $ n_1, n_2 > 1,$ we obtain the bound \[ |R_2| \le \, t^{\frac{n+2-8 }{2}}\ {\cal P}_{l_1}\log(t,\tau)^{-1}\ \int_{x_n}\int_v \sum_{T_{l_1}^{s_1+1},\ T_{l_2}^{s_2+1}} {\cal F}(t,\tau',t/2; T_{l_1}^{s_1+1}~;x_1,y_2,\ldots,y_{s_1},v)\, \cdot \] \eq \cdot \ {\cal P}_{l_2}\log(t,\tau)^{-1}\ {\cal F}(t,\tau'',t/2;T_{l_2}^{s_2+1}~;x_n, v,y_{s_1+1},\ldots \ldots, y_{s(n)}) \ \, \vp_n(x_n) \ . \label{gensi} \eqe Here $s(n)=s$ if $s 4 \ . \label{formeln1} \eqe \noindent ii) $n\le 4$ \\ In this case we have $\,n_1+1 =2\,$ and/or $\,n_2+1 =2\,$. Thus at least one of the polynomials $\, {\cal P}_{l_i}\log(t,\tau)^{-1}\,$ appearing in the bounds (\ref{gensi}) can by induction be replaced by $\, {\cal P}_{l_i-1}\log t^{-1}\,$. Therefore proceeding exactly as in the previous case we obtain the bounds \eq |\pa_t {\cal L}_{n,l}^{\vep,t}(x_1,\vp_s)| \ \le \ t^{\frac{n-6}{2}}\ {\cal P}_{l-1}\log (t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau) \ , \quad n \le 4 \ . \label{formeln2} \eqe \\ \noindent B) $\,r >0\,, \,$ cf.(\ref{ins})\\ For the first term on the r.h.s. of the flow equation resulting from (\ref{fequ}) the bounds for $1 \le r \le 3\,$ are proven exactly as in A1). For the second term we proceed similarly as in A2). We pick a generic term on the r.h.s. \[ \int_{x_{2,n}, x,y} \vp_s(x_{2,n})\ C_{t}(x,y)\ E(x_k,x_1~; \om^{(r)}) {\cL}^{\vep,t}_{n_1+1,l_1}(x_1,\ldots,x_{n_1},x) \ {\cL}^{\vep,t}_{n_2+1,l_2}(y,x_{n_1+1},\ldots,x_{n})\ . \] In the case where $k \le n_1\,$ the proof is the same as for $r=0$, up to inserting the modified induction hypothesis for \[ \int_{x_{2,n_1}, x} \vp_{s_1}(x_{2,n_1})\ C_{t/2}(x,v)\ E(x_k,x_1~; \om^{(r)}) {\cL}^{\vep,t}_{n_1+1,l_1}(x_1,\ldots,x_{n_1},x)\ . \] If $k > n_1\,$ we assume without restriction $k = n\,$ and proceed again as in A2) to obtain the bound \[ t^{\frac{n-6 }{2}}\ {\cal P}_{l_1}\log(t,\tau)_{n}^{-1}\ \int_{x_n}\int_v | E(x_n,x_1~; \om^{(r)})| \ \sum_{T_{l_1}^{s_1+1},\ T_{l_2}^{s_2+1}} {\cal F}(t,\tau,t/2; T_{l_1}^{s_1+1}~;x_1,y_2,\ldots,y_{s_1},v)\ \] \eq \cdot \ {\cal P}_{l_2}\log(t,\tau)_{n}^{-1}\ {\cal F}(t,\tau,t/2;T_{l_2}^{s_2+1}~;x_n, v,y_{s_1+1},\ldots \ldots, y_{s(n)}) \ \vp_n(x_n)\ . \label{gensis} \eqe Observing the inequality (\ref{nino}) together with \eq d(x_n,x_1) \le \ \sum_{a=1}^q d(v_a,v_{a-1}) \label{sid} \eqe where $\{v_a\}\,$ are the positions of the internal vertices in the tree $T_l^{s}(T_{l_1}^{s_1+1},\, T_{l_2}^{s_2+1})$ defined as in A2), on the path joining $x_1=v_0$ and $x_n = v_q\,$, we then use the bound (\ref{d}). The cases $s=n$ and $s 4 \label{bdprop2} \eqe \eq |\pa_t \ {\cal L}^{\vep,t}_{4,l}(x_1, E_{(i)}^{(r)}\, \vp_s)|\,\le\ | \om^{(r)}(x_1) | \, t^{\frac{r-2}{2}}\ {\cal P}_{l-1}\log(t,\tau)_{i}^{-1}\ {\cal F}_{s,l }(t,\tau) \label{bdprop3} \eqe \eq |\pa_t \ {\cal L}^{\vep,t}_{2,l}(x_1, E_{(2)}^{(r)}\ \vp_2)|\,\le\ \, |\,\om^{(r)}(x_1) | \ t^{\frac{r-4}{2}}\ {\cal P}_{l-2}\log t^{-1}\ {\cal F}_{2,l }(t,\tau) \label{bdprop22} \eqe \eq |\,\pa_t \ F^{(r)}_{(12)} {\cal L}^{\vep,t}_{n,l}(x_1, x_2,\vp)| \,\le\ |\,\om^{(r)}(x_1) | \ t^{\frac{n-3}{2}}\ {\cal P}_{l-1}\log(t,\tau)^{-1}\ {\cal F}^{(12)}_{s,l}(t,\tau)\ . \label{prop2rr} \eqe In (\ref{prop2rr}) $\tau\,$ stands for $\,(\tau_3,\ldots,\tau_s)\,$, furthermore $\, r = 0,1,2 \, \mbox{ and } \, |\,\om^{(0)}(x_1) | \equiv 1\, $.\\[.2cm] The bounds for (\ref{coeff})-(\ref{coeff4}), \eq |\,\pa_{t}\, c^{\vep,t}_{l}(x_1)\,| \le \ {1 \over t}\ {\cal P}_{l-1}\log \frac{1}{ t} \ ,\quad |\ \pa_{t}\,a^{\vep,t}_{l}(x_1)\,| \le \ t^{-2} \ {\cal P}_{l-1}\log \frac{1}{ t}\ , \label{ca} \eqe \begin{eqnarray} |\,\pa_{t}\, f^{\mu,\vep,t}_{l}(x_1) \,\om_\mu (x_1) | & \le & \nom \,\ t^{-3/2} \ {\cal P}_{l-2}\log \frac{1}{ t}\ , \label{b2pt1}\\ |\,\pa_{t}\, \, b^{\mu\nu,\vep,t}_{l}(x_1) \,\om_{\mu \nu}^{(2)} (x_1)\,| & \le & \ |\, \om^{(2)}(x_1) | \, {1 \over t}\ {\cal P}_{l-2}\log \frac{1}{ t} \label{b2pt} \end{eqnarray} are obtained on restricting the previous considerations to the case $s=1\,$, in which all external coordinates are integrated over, e.g. \[ \pa_{t}\,a^{\vep,t}_{l}(x_1)\,=\, \frac12\,\int_{x_2,x,y} C_{t}(x,y) \Biggl\{ {\cL}^{\vep,t}_{4,l-1}(x_1,x_{2},x,y)\, - \sum_{l_1+l_2=l} \Bigl[ {\cal L}^{\vep, t}_{2,l_1}(x_1,x)\, {\cal L}^{\vep, t}_{2,l_2}(y,x_{2})\Bigr]_{sym}\Biggr\} \ . \] The polynomials appearing in (\ref{b2pt1}), (\ref{b2pt}) are of degree $\le \,l-2\,$, corresponding to the fact that on the r.h.s. of the FE (\ref{fequ}) for these terms, there appear $ {\cal L}^{\vep,t}_{l-1,4}$ and $ {\cal L}^{\vep,t}_{l_1,2}\, {\cal L}^{\vep,t}_{l_2,2}\,$ with insertions $ E_{(2)}^{(r)}\, , \,r = 1, 2\,$. Both are bounded inductively by polynomials of total degree $\le \sup (l-2,0)\,$.\\ \noindent C) We come to the bound on $\pa_t {\cal L}^{\vep,t}_{n,l} (x_1,\vp^{(j)}_{s})\,$, cf. (\ref{prop5}). As compared to B) the only case which requires new analysis is the bound on the second term from the r.h.s. of the FE (\ref{fequ}), in the case $j >s_1\,$. Then we assume without restriction, similarly as in B), that $j=s\,$. The term to be bounded corresponding to (\ref{gensi}) is then \[ t^{\frac{n-6 }{2}}\ {\cal P}_{l_1}\log(t,\tau)^{-1}\ \int_v \sum_{T_{l_1}^{s_1+1},\ T_{l_2}^{s_2+1}} {\cal F}(t,\tau',t/2; T_{l_1}^{s_1+1}~;x_1,y_2,\ldots,y_{s_1},v)\ \cdot \] \eq \ \cdot\ {\cal P}_{l_2}\log(t,\tau)^{-1}\ \int_{x_s} {\cal F}(t,\tau'',t/2;T_{l_2}^{s_2+1}~;x_s, v,y_{s_1+1},\ldots \ldots, y_{s-1}) \ |K^{(1)}(\tau_s, x_s,x_1;y_s)| \ . \label{gensik} \eqe To bound this expression we telescope the difference $\,K^{(1)}(\tau_s, x_s,x_1;y_s)$, cf.(\ref{k1}), along the tree $T_l^s( T_{l_1}^{s_1+1}, T_{l_2}^{s_2+1})\,$ obtained from the two initial trees by joining them via $v$ as in A2) and proceeding similarly as in (\ref{sid}). We then have to bound expressions of the type \[ C_{t_{I,\de}}(v_{a-1},v_a)\ |K(\tau_{s}, v_a,y_s)\ -\ K(\tau_{s}, v_{a-1},y_s)| \] where $v_{a-1},v_a\,$ are adjacent internal vertices in $T_l^s( T_{l_1}^{s_1+1}, T_{l_2}^{s_2+1})\,$ on the unique path from $\,x_1\,$ to $\,y_s\,$. Making use of the covariant Schl\"omilch formula (\ref{sloe})-(\ref{Rnb}) for the difference $\,K^{(1)}(\tau_s, v_a,v_{a-1};y_s)$, we obtain $$ |K(\tau_{s}, v_a,y_s)\ -\ K(\tau_{s}, v_{a-1},y_s)| \leq \int_{0}^{s} d r \, |\, \nabla_{(1)} K(\tau_s, z(r),y_s)\,| $$ $$ = d(v_{a-1},v_a) \int_{0}^{1} d \rho \, |\, \nabla_{(1)} K(\tau_s, v_{a-1, a}(\rho),y_s)\,| $$ \eq \leq O(1)\, \frac{ d(v_{a-1},v_{a})}{\sqrt{\tau_s}}\, \int_{0}^{1} d \rho \, K(\tau_{s,\de '}, v_{a-1, a}(\rho),y_s) \label{tel} \eqe where $ z(r)=v_{a-1, a}(\rho)$ lies at distance $ r = \rho\, d(v_{a-1},v_{a}), \, 0 \leq \rho \leq 1, $ from $v_{a-1}\,$ on the reparametrized geodesic segment from $ v_{a-1}$ to $v_a$. The last inequality results from (\ref{D}). Introducing for \eq 3\, \de < 1 :\qquad b = 2\,\frac{1+3\de}{1-3\de}\ , \label{tb} \eqe we then bound, with $ \de' > 0$ to be fixed later, \begin{eqnarray} C_{t_{I,\de}}(v_{a-1},v_{a})\ |\,K(\tau_{s}, v_a,y_s)\ -\ K(\tau_{s}, v_{a-1},y_s)| \ \le{\qquad\qquad\qquad\qquad\qquad } \nonumber \\ \le \ \left\{ \begin{array}{r@{\quad \quad}l} C_{t_{I,\de}}(v_{a-1},v_{a})\,K(\tau_{s}, v_a,y_s)\ + C_{t_{I,\de}}(v_{a-1},v_{a})\, K(\tau_{s}, v_{a-1},y_s) \ ,\quad\,b \,t \ge \de '\tau_s \\ O(1)\ C_{t_{I,2\de}}(v_{a-1},v_{a})\,(\frac{t_I}{\tau_s})^{1/2}\ \int_{0}^{1} d\rho \ K(\tau_{s,\de '}, v_{a-1,a}(\rho),y_s)\, \ ,\quad b\,t < \de '\tau_s \,\, . \end{array}\right . \label{arr} \end{eqnarray} The last line follows using (\ref{tel}) and absorbing the factor $d(v_{a-1},v_a)$ in $ C_{t_{I,\de}}(v_{a-1},v_a)\,$ with the aid of (\ref{d}), by changing $\de$ to $ 2\de$ .\\ The last line in (\ref{arr}) has to be bounded in such a way as to reproduce a contribution compatible with the induction hypothesis. To this end we use the (upper) bound (\ref{hk10}) \[ C_{t_{2\de}}(v_{1},v_{2})\ K(\tau_{\de '}, v_{1,2}(\rho),y)\ \le \ O(1)\, \frac{1}{t^2}\ \frac{1}{\tau^2}\ \exp \bigg(-\frac{d^2(v_1,v_2)}{4t(1+3 \de )}\ -\frac{d^2(v_{1,2}(\rho),y)}{4\tau(1+{\de '})^2}\,\bigg)\ . \] Noting that $d(v_{1},v_2)=d(v_1,v_{1,2}(\rho)) + d(v_{1,2}(\rho),v_2)\,$ we deduce \[ \frac{1}{\de '}\ d^2(v_{1},v_2)+d^2(v_{1,2}(\rho),y) \ge \frac{1}{\de '}\ d^2(v_1,v_{1,2}(\rho)) + d^2(v_{1,2}(\rho),y) \ge \] \[ \frac{1}{1+\de '}\ \left(d (v_1,v_{1,2}(\rho)) + d(v_{1,2}(\rho),y)\right)^2 \ge \frac{1}{1+\de '}\ d^2(v_{1},y)\ . \] Hence, observing (\ref{tb}), we find for $ b\,t < \de '\tau \,$ \begin{eqnarray} \frac{d^2(v_1,v_2)}{4 t(1+3\de)} + \frac{d^2(v_{1,2}(\rho),y)}{4 \tau (1+\de')^2} & = & \frac{d^2(v_1,v_2)}{8 t} + \frac{d^2(v_1,v_2)}{4 b t} + \frac{d^2(v_{1,2}(\rho),y)}{4 \tau(1+\de ')^2 } \nonumber \\ & \geq & \frac{d^2(v_1,v_2)}{8 t} \ +\ \frac{d^2(v_1,y)}{4\tau(1+\de ')^3 } \ . \nonumber \end{eqnarray} With the aid of the lower bound (\ref{hk10}) we then arrive at \eq C_{t_{I,2\de}}(v_{1},v_{2})\! \int_{0}^{1}\!\!\! d\rho \ K(\tau_{s,\de '}, v_{1,2}(\rho),y_s) \le O(1)\, C_{2 t_{I},\de}(v_{1},v_{2})\, K(\tau_s (1+\de ')^4, v_{1},y)\ . \label{abs} \eqe Taking into account (\ref{hk4}) and choosing $\de ' $ such that $(1+\de ')^4 = 1+\de\,$, i.e. $\de ' =\de/4+O(\de ^2)\,$, we may thus bound the last line in (\ref{arr}) by \eq O(1)\ (\frac{t}{\tau_s})^{1/2}\ K(\tau_{s,\de}, v_{a-1},y_s) \ \int_v C_{t_{I},\,\de}(v_{a-1},v)\, C_{t_{I},\,\de}(v,v_{a})\ , \quad b\,t < \de ' \ \tau\ . \label{arr2} \eqe Note that the addition of a new internal vertex $v\,$ of coordination number 2 in (\ref{arr2}) is compatible with the inequality $v_2+\de_{c_1,1} \le 3l - 2 +s/2\,$. \\ Using these bounds and going back to (\ref{gensik}) we realize that the two terms in the first line of (\ref{arr}) - case $b\,t \geq \de' \tau_s $ - correspond to two new trees of type ${\cal T}_l^{s}\,$, where in comparison to $T_l^s( T_{l_1}^{s_1+1}, T_{l_2}^{s_2+1})\,$ the coordination number of the vertex $v_{a-1}$ or $v_{a}$ has increased by one unit. Similarly (\ref{arr2}) - case $b\,t < \de ' \tau_s\,$ - corresponds to a new tree where the coordination number of the vertex $v_{a-1}$ has increased by one unit. In (\ref{gensik}) an integral over $x_s\,$ is performed. If in the new tree \\ i) $x_s$ has $c(x_s)>1\,$\footnote{remember that the vertex $x_s$ in $T_{l_2}^{s_2+1})\,$ is a root vertex by construction}, then $x_s$ takes the role of an internal vertex of the new tree,\\ ii) $x_s$ has $c(x_s)=1\,$ we integrate over $x_s$ using (\ref{hk3}) so that the vertex $x_s$ disappears. As a consequence of the previous bounds, on replacing again $s \to j\,$ in (\ref{arr}),(\ref{arr2}), we thus obtain for $ n > 4$ - see also A2) ii) for (\ref{bd6}), (\ref{bd2}) - \begin{eqnarray} |\pa_t \,{\cal L}^{\vep,t}_{n,l} (x_1,\vp^{(j)}_{\tau_{2,s}, y_{2,s}})|\,\le\ \left\{ \begin{array}{r@{\quad \quad}l} t^{\frac{n-6}{2}}\ {\cal P}_l\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau) \ ,\quad \, b\,t \ge \de '\tau_j \\ t^{\frac{n-6}{2}}\ (\frac{t}{\tau_j})^{1/2}\ {\cal P}_l\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau) \ ,\quad b\,t < \de '\tau_j \end{array}\right . \label{bd5} \end{eqnarray} \begin{eqnarray} |\pa_t \,{\cal L}^{\vep,t}_{4,l} (x_1,\vp^{(j)}_{\tau_{2,s}, y_{2,s}})|\,\le\ \left\{ \begin{array}{r@{\quad \quad}l} t^{-1}\ {\cal P}_{l-1}\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau) \ ,\quad b\,t \ge \de ' \tau_j \\ t^{-1}\ (\frac{t}{\tau_j})^{1/2}\ {\cal P}_{l-1}\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau) \ ,\quad b\, t < \de '\tau_j \end{array}\right . \label{bd6} \end{eqnarray} \begin{eqnarray} |\pa_t \,{\cal L}^{\vep,t}_{2,l} (x_1,\vp^{(2)}_{\tau,y_{2}})|\,\le\ \left\{ \begin{array}{r@{\quad \quad}l} t^{-2}\ {\cal P}_{l-1}\log t^{-1}\ {\cal F}_{2,l}(t,\tau) \ ,\quad b\,t \ge \de '\tau_2 \ \ \\ t^{-2}\ (\frac{t}{\tau})^{1/2}\ {\cal P}_{l-1}\log t^{-1}\ {\cal F}_{2,l}(t,\tau) \ ,\quad b\,t < \de '\tau_2 \ . \end{array}\right . \label{bd2} \end{eqnarray} \\ \noindent D) From the bounds on the derivatives $\,\pa_t \,{\cal L}^{\vep,t}_{n,l} \,$ we then verify the induction hypothesis on integrating over $t\,$. In all cases we need the bound \eq {\cal F}_{s,l}(t',\tau)\ \le \ {\cal F}_{s,l}(t,\tau) \quad \mbox{for }\ t' \le t\ , \label{bdF} \eqe which follows directly from the definition (\ref{ff}). \noindent a) In the cases \underline{$n+r >4\,$} we have, due to the boundary conditions encoded in the form of (\ref{nawig}) \[ {\cL}^{\vep,t}_{n,l}(x_1,\vp)=\, \int_{\vep}^{t} dt'\ \pa_{t'} {\cL}^{\vep,t'}_{n,l}(x_1,\vp)\ . \] Then we get from (\ref{formeln1}), (\ref{bdprop2})-(\ref{bdprop22}), due to (\ref{bdF}), \eq |\, {\cal L}^{\vep,t}_{n,l} (x_1,\vp_s)|\,\le\ t^{\frac{n-4}{2}}\ {\cal P}_l\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau)\ \label{res1} \eqe \eq | {\cal L}^{\vep,t}_{n,l} (x_1, E_{(k)}^{(r)}\, \vp_s)|\,\le\ |\, \om^{(r)}(x_1) | \, t^{\frac{n+r-4}{2}}\ {\cal P}_l\log(t,\tau)_k^{-1}\ {\cal F}_{s,l}(t,\tau)\ ,\quad n > 4 \label{res2} \eqe \eq | {\cal L}^{\vep,t}_{4,l} (x_1, E_{(k)}^{(r)}\, \vp_s)|\,\le\ |\, \om^{(r)}(x_1) | \, t^{\frac{r}{2}}\ {\cal P}_{l-1}\log (t,\tau)_k^{-1}\ {\cal F}_{s,l }(t,\tau) \ ,\quad r > 0 \label{res3} \eqe \eq | {\cal L}^{\vep,t}_{2,l} (x_1, E_{(2)}^{(3)} \vp_s)|\,\le\ |\,\om^{(3)}(x_1) | \, t^{\frac{1}{2}}\ {\cal P}_{l-1}\log t^{-1}\ {\cal F}_{s,l}(t,\tau) \ , \label{res4} \eqe which proves the proposition for these cases.\\ b) Similarly, for \underline{$n \ge 4\,$} the boundary conditions (\ref{nawig}) imply that \[ {\cL}^{\vep,t}_{n,l}(x_1,\vp^{(j)}_s)=\, \int_{\vep}^{t} dt'\ \pa_{t'} {\cL}^{\vep,t'}_{n,l}(x_1,\vp^{(j)}_s)\ , \] and we then obtain from (\ref{bd5}), (\ref{bd6}) together with (\ref{bdF}) \eq |{\cal L}^{\vep,t}_{n,l} (x_1,\vp^{(j)}_s)|\,\le\ t^{\frac{n-4}{2}}\ \bigg (\frac{t}{\tau_j}\bigg)^{1/2}\ {\cal P}_l\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau)\ . \label{res5} \eqe We note that for $b\,t \ge \de '\tau_j\,$ the integral $\int_{\vep}^{t} dt'\,$ has to be split into $\int_{\vep}^{ \de '\tau_j/b} dt'+ \int_{ \de ' \tau_j/b}^t dt'\,$, and that in the case $n=4$ the polynomial in logarithms may increase in degree by one unit due to the logarithmically divergent $t$-integral, see (\ref{d242})-(\ref{acht}) below for more details.\\[.1cm] c) In the case \underline{$n=4\,, \ r=0$} we start from the decomposition (\ref{4l}) \[ {\cal L}^{\vep,t}_{4,l} (x_1,\vp)\,=\ c_l^{\vep,t}(x_1)\ \vp(x_1,x_1,x_1)\,+\ {\ell}^{\vep,t}_{4,l} (x_1,\vp)\ . \] On integrating the bound for $c_l^{\vep,t}(x_1)$, (\ref{ca}), from $t\,$ to $1\,$ and using the boundary condition (\ref{renbed4}) we get \eq |\, c^{\vep,t}_{l}(x_1)\,| \le \ {\cal P}_{l}\log t^{-1} \ . \label{4ptt} \eqe Taking together (\ref{ca}) and (\ref{formeln2}) we verify \[ | \pa_{t} {\ell}^{\vep,t}_{4,l} (x_1,\vp_s)|\ \le \ t^{-1}\ {\cal P}_{l-1}\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau)\ . \] A sharper bound for $\, \pa_{t} {\ell}^{\vep,t}_{4,l} (x_1,\vp_s) \,$, which when integrated over $t \ge \vep\,$ stays uniformly bounded in $\vep$, is obtained as follows. In the case $s=4\,$\footnote{In this case $\vp_i(x_i)=\, K(\tau_i, x_i, y_i)\,,\ 1 \le i \le 4\,$. If $\vp_i = \mathbf{1}\,$ for some $i\,$, the corresponding contribution to the subsequent sum over $i\,$ vanishes.} we decompose the test function $$ \vp_4(x_2, x_3, x_4) := \prod_{i=2}^{4} K(\tau_i, x_i, y_i) = \vp_4(x_1, x_1, x_1) + \psi(x_2, x_3, x_4) \,, $$ $$ \psi(x_2, x_3, x_4) = \sum_{i=2}^{4} \, \prod_{f=2}^{i-1} \, K(\tau_f, x_1, y_f)\, K^{(1)}(\tau_i, x_i, x_1; y_i) \prod_{j=i+1}^{4} \, K(\tau_j, x_j, y_j)\ . $$ Then ${\ell}^{\vep,t}_{4,l} (x_1,\vp_4) = \mathcal{L}_{4,l}^{\vep, t}(x_1, \psi)\, $, and hence the FE (\ref{fequ}) provides \eq \pa_{t}\,{\ell}^{\vep,t}_{4,l} (x_1,\vp_4) \,=\ \frac12 \int_{x_2, x_3, x_4, x,y} \psi(x_2, x_3, x_4)\, C_t(x,y)\ \Bigl\{{\cL}^{\vep,t}_{6,l-1}(x_1,\ldots,x_4, x,y) \label{s4t} \eqe \[ -\ \sum_{l_1+l_2=l} \Bigl[ {\cal L}^{\vep,t}_{4,l_1}(x_1, x_2 ,x_{3},x)\, {\cal L}^{\vep,t}_{2,l_2}(y,x_{4}) \ + {\cal L}^{\vep,t}_{2,l_1}(x_1,x)\, \, {\cal L}^{\vep,t}_{4,l_2}(y,x_2,x_3,x_{4})\Bigr]_{sym}\,\Bigr\}\ . \] The r.h.s. is a sum over expressions of the same form as the one for $\pa_t\mathcal{L}_{4,l}^{\vep, t}(x_1, \vp^{(j)}_{\tau_{2,s}, y_{2,s}})\,$ in part C. Setting $\tau =\inf_j\{ \tau_j\}$ we obtain, in the same way as there, the bound \begin{eqnarray} | \pa_{t} {\ell}^{\vep,t}_{4,l} (x_1,\vp_s)| \,\le\ \left\{ \begin{array}{r@{\quad \quad}l} t^{-1}\ {\cal P}_{l-1}\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau) \ ,\quad b\,t \ge \de ' \tau\ , \\ t^{-1}\ (\frac{t}{\tau})^{1/2}\ {\cal P}_{l-1}\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau) \ ,\quad b\,t < \de '\tau \ . \end{array}\right . \label{d242} \end{eqnarray} Using the boundary condition (\ref{bell}) we integrate (\ref{d242}) over $t\,$. This gives for $b\,t < \de ' \tau$ (and $\vep\,$ sufficiently small) \eq \Big |\int_{\vep}^{t} dt'\ \pa_{t'}\, {\ell}^{\vep,t'}_{4,l} (x_1,\vp_s)\Big |\ \le \ (\frac{t}{\tau})^{1/2} \ {\cal P}_{l-1}\log{t}^{-1}\ {\cal F}_{s,l}(t,\tau)\, , \label{sieben} \eqe and for $b\,t > \de '\tau \,$, in which case we may have $t > \tau$ or $ t < \tau\,$, $$ \Big |\int_{\vep}^{t} dt'\ \pa_{t'}\, {\ell}^{\vep,t'}_{4,l} (x_1,\vp_s)\Big|\ \le \ \Big |\int_{\vep}^{ \de '\tau/b} dt'\pa_{t'}\, {\ell}^{\vep,t'}_{4,l} (x_1,\vp_s)\Big |+ \Big |\int_{ \de '\tau/b}^t dt'\pa_{t'}\, {\ell}^{\vep,t'}_{4,l} (x_1,\vp_s)\Big | $$ \eq \le \ \bigg(\Big( \frac{\de '}{b}\Big )^{1/2} \, {\cal P}_{l-1}\log \frac{b}{\de '\tau} +\ {\cal P}_l\log \frac{b}{\de '\tau} \bigg)\ {\cal F}_{s,l}(t,\tau ) \ . \label{acht} \eqe Hence, absorbing powers of $\log( \de '/b)\,$ in the coefficients of ${\cal P}_{l}\log\,$ as usual, \eq | {\ell}^{\vep,t}_{4,l} (x_1,\vp_s)|\, \le \ {\cal P}_{l}\log{\tau}^{-1} \ {\cal F}_{s,l}(t,\tau) \ . \label{df} \eqe From (\ref{4l}), (\ref{4ptt}), (\ref{sieben}) and (\ref{df}) we obtain \[ | {\cL}^{\vep,t}_{4,l} (x_1,\vp_s)|\ \le\ {\cal P}_{l}\log(t,\tau)^{-1}\ {\cal F}_{s,l}(t,\tau)\ . \] \noindent d) In the case \underline{$n=2\,$} we have the decomposition (\ref{2l}). In addition to the bounds (\ref{ca})-(\ref{b2pt}) to be integrated from 1 to $ t \leq 1 $, we need for $\, \pa_t\, {\ell}^{\vep,t}_{2,l}\ (x_1, \vp) $,\ (\ref{re2}), a bound, which upon integration from $\vep $ to $t\,$ becomes a uniformly bounded function on $ \vep \geq 0 $. To this end we use the form (\ref{re22}) and choose the test function $\vp(x_2) = K(\tau, x_2,y_2) \,$. Taking into account the bound (\ref{prop2rr}) for $ n = 2, r=0 $ together with (\ref{Rnb}) yields $$ |\, \pa_t\, {\ell}^{\vep,t}_{2,l}\ (x_1, \vp)| \leq \int_{x_2} \, t^{- \frac{1}{2}}\ {\cal P}_{l-1}\log t^{-1}\ {\cal F}_{2,l}^{(12)}(t) \int_{0}^{1}d\rho \, \frac{ (1-\rho)^2}{2!}\, | \, \nabla_{(1)}^3 K(\tau, X(\rho),y_2) | $$ \[ \qquad \,\, \leq \, t^{- \frac{1}{2}}\, \tau^{- \frac{3}{2}} \, {\cal P}_{l-1}\log t^{-1}\int_{x_2} {\cal F}_{2,l}^{(12)}(t) \int_{0}^{1} d\rho \, K(\tau_{ \de '}, X(\rho),y_2) \] where (\ref{D}) has been used. By definition we have \[ {\cal F}^{(12)}_{2,l}(t)\ =\ {\cal F}_{2,l}(t;x_1,x_2)\,=\, \sum_{T_l^{2,(12)}} {\cal F}_{2,l}(t;T_l^{2,(12)}~;x_1,x_2) \] \[ =\ \sum_{n = 1}^{3l-2}\, \, \sup_{\{t_{I_{\nu}}| \vep\, \leq\, t_{I_i}\, \leq\, t,\, i=1.\cdots,n\}} \, \,[\prod_{1 \le \nu \le n } \int_{z_{\nu}}]\ C_{t_{I_1,\de}}(x_1,z_1) \ldots C_{t_{I_n,\de}}(z_{n},x_2) \] \[ =\ \sum_{n = 1}^{3l-2}\, \, \sup_{\{t_{I_{\nu}}| \vep\, \leq\, t_{I_{\nu}}\, \leq\, t,\, \nu=1.\cdots,n\}} C_{\sum_1^n t_{I_{\nu},\de}}(x_1,x_2) \] where we used (\ref{hk4}) and (\ref{f2r}). We then proceed similarly as in and after (\ref{arr}). Setting $N=3l-2\,$ we bound for $Nb\,t < \de ' \tau \,$ and for $n \le N\,$ as in (\ref{abs}) \eq C_{\sum_1^n t_{I_{\nu},\de}}(x_1,x_2)\ \int_{0}^{1} d\rho \ K(\tau_{\de '}, X(\rho),y_2)\, \ \le O(1) \,\, C_{2\sum_1^n t_{I_{\nu},\de}}(x_{1},x_{2})\ K(\tau_{\de}, x_{1},y_2) \label{rr} \eqe so that, observing (\ref{hk3}), \eq |\, \pa_t\, {\ell}^{\vep,t}_{2,l}\ (x_1, \vp)| \ \le \ \tau^{-3/2}\ t^{-1/2}\ {\cal P}_{l-1}\log t^{-1}\ K(\tau_{\de}, x_{1},y_2)\ ,\quad N b\,t < \de ' \tau \ . \label{rl2} \eqe To verify the induction hypothesis (\ref{prop20}) we resort to the decomposition (\ref{2l}) and denote the sum of the first, second and third terms there by $ {\cal L}^{\vep,t}_{2,l}(x_1,\, \vp)_{rel}$. Integrating the corresponding bounds (\ref{ca})-(\ref{b2pt}) from $1$ to $t$, and using again (\ref{D}) gives \eq \label{rel} |\, {\cal L}^{\vep,t}_{2,l}(x_1,\, \vp)_{rel} | < \Bigl(\, \frac{1}{t}\ {\cal P}_{l-1}\log t^{-1}\ + \frac{1}{(t \tau)^{\frac{1}{2}}}{\cal P}_{l-2}\log t^{-1} + \frac{1}{\tau} {\cal P}_{l-1}\log t^{-1}\Bigr)\ K(\tau_{\de}, x_1,y_2)\ . \eqe Integrating the remainder (\ref{rl2}) from (small) $\vep$ with vanishing initial condition (\ref{bell}) to $ t < \de '\tau /(bN)$ leads to \eq |\,{\ell}^{\vep,t}_{2,l}\ (x_1, \vp)| \ \le \ \tau^{-3/2}\ t^{1/2}\ {\cal P}_{l-1}\log t^{-1}\ K(\tau_{\de}, x_{1},y_2)\ . \label{bn2} \eqe By way of (\ref{2l}) we obtain from the bounds (\ref{formeln2}) and (\ref{ca})-(\ref{b2pt}) the bound for $N b\,t \ge \de ' \tau \,$ \eq |\, \pa_t\, {\ell}^{\vep,t}_{2,l}\ (x_1, \vp)| \leq \ t^{-2}\ {\cal P}_{l-1}\log (t,\tau)^{-1}\ {\cal F}_{2,l}(t,\tau)\ +\ \label{bn22} \eqe \[ \Bigl(t^{-2}\ {\cal P}_{l-1}\log t^{-1}\ +\,\sum_{j=1}^2 t^{\frac{j-4}{2}}\ \tau ^{-j/2}\ {\cal P}_{l-2}\log t^{-1}\Bigr)\ K(\tau_{\de}, x_{1},y_2)\ . \] Hence, integration and majorization, again observing both $\tau > t $ and $t < \tau $, gives for $ t > \de '\tau /(bN)$ \eq |\,{\ell}^{\vep,t}_{2,l}\ (x_1, \vp)|\, \leq \, \bigg(\, \frac{bN}{\de '\tau}\, {\cal P}_{l-1}\log \frac{bN}{\de '\tau} + \theta (t - \tau)\frac{1}{\tau} {\cal P}_{l-1}\log \tau^{-1} \bigg )\,{\cal F}_{2,l}(t,\tau)\ +\ \label{tgta} \eqe $$ \bigg (\, \frac{bN}{\de '\tau}\, {\cal P}_{l-1}\log \frac{bN}{\de '\tau} + \bigg(\frac{bN}{\de '\tau^2}\bigg)^{1/2} {\cal P}_{l-2}\log \frac{bN}{\de '\tau}\, + \frac{1}{\tau} {\cal P}_{l-1}\log \frac{bN}{\de '\tau} \bigg )\, K(\tau_{\de}, x_1,y_2)\ . $$ From (\ref{rel}), (\ref{bn2}) and (\ref{tgta}), absorbing constants as usual in $ {\cal P}\log\,$, we then get \footnote{The bound (\ref{tgta}) diverges linearly with $\de '$, whereas in (\ref{acht}) the divergence was only logarithmic. This indicates rapid growth since the bounds then behave as $(\de ') ^{-l}\,$, a factor of $(\de ')^{-1}\,$ being produced per loop order. Without trying at all to optimize constants, we still note that it is possible to choose for this case $\de =2\,$ in $K(\tau_{\de}, x_{1},y_2)\,$ and bound the two point function inductively by ${\cal F}_{2,l}(t,3\tau)$ without changing the bounds on the other functions. The only place in the proof where there is a modification due to this factor is in part A2). But here the value of $\tau$ appearing is $t/2$, see (\ref{falt}), and $3t/2$ can be accommodated for in the proof by introducing a new vertex of coordination number 2 while respecting the bound on the number of those vertices. A value $\de =2\,$ then gives for suitable choice of $b$ the value $b/\de ' \simeq 6\,$.} \eq |\, {\cal L}^{\vep,t}_{2,l}(x_1,\, \vp)\,|\ \le \ (t,\tau)^{-1}\ {\cal P}_{l-1}\log (t,\tau)^{-1}\ {\cal F}_{2,l}(t,\tau) \eqe in accord with (\ref{prop20}). To establish the bounds on $\, {\cal L}^{\vep,t}_{2,l} (x_1,E_{(2)}^{(r)} \vp) ,\, r=1,2\, , $ we expand the respective test functions as follows, employing (\ref{Rna}) and using the notations (\ref{coeff}), (\ref{2ins}) $$ {\cal L}^{\vep,t}_{2,l} (x_1,E_{(2)}^{(1)} \vp) = \vp(x_1) f^{\mu,\vep,t}_l(x_1)\ \om_{\mu}(x_1) + 2\, b^{\,\mu \nu,\vep,t}_l(x_1)\, \om_{\mu}(x_1) \, (\nabla _{\nu} \vp)(x_1)$$ \eq + \int_{x_2} F^{(1)}_{(12)} {\cal L}^{\vep,t}_{2,l} (x_1,x_2) \int_{0}^{1} d\rho\, \frac{ (1-\rho)}{d^{\,2}(x_1, x_2)}\, {\dot X}^{\mu}(\rho)\, {\dot X}^{\nu}(\rho) (\nabla_\mu \nabla_\nu \vp)(X(\rho))\, , \label{ein1} \eqe $$ {\cal L}^{\vep,t}_{2,l} (x_1,E_{(2)}^{(2)} \vp) = - 2 \,\vp(x_1)\, b^{\,\mu \nu,\vep,t}_l(x_1)\, \om^{(2)}_{\mu \nu}(x_1) $$ \eq + \int_{x_2} F^{(2)}_{(12)} {\cal L}^{\vep,t}_{2,l} (x_1,x_2) \int_{0}^{1} d\rho\, \frac{1}{d(x_1, x_2)} \, {\dot X}^{\mu}(\rho)\, (\nabla_\mu \vp)(X(\rho))\, . \label{ein2} \eqe The local, i.e. relevant terms have already been dealt with in (\ref{b2pt1}), (\ref{b2pt}), and the remainders are treated as $\, {\ell}^{\vep,t}_{2,l}\ (x_1, \vp)$ ; one obtains \[ |\, {\cal L}^{\vep,t}_{2,l} (x_1,E_{(2)}^{(1)} \vp_2)| \ \le\ |\,\om^{(1)}(x_1)|\ (t, \tau)^{-1/2}\ {\cal P}_{l-1}\log(t,\tau)^{-1}\ {\cal F}_{2,l}(t,\tau)\ , \] \[ |\, {\cal L}^{\vep,t}_{2,l} (x_1,E_{(2)}^{(2)}\vp_2)| \ \le\ |\om^{(2)}(x_1)|\ {\cal P}_{l-1}\log (t,\tau)^{-1}\ {\cal F}_{2,l}(t,\tau)\ . \] Finally, we realize that ${\cal L}^{\vep,t}_{2,l}(x_1,\vp^{(2)}_2)$ equals the r.h.s. of (\ref{2l}) without its first term. Proceeding again similarly as before - see (\ref{rel}), (\ref{bn2}) and (\ref{tgta}) - provides \[ |{\cal L}^{\vep,t}_{2,l}(x_1,\vp^{(2)}_2)| \ \le\ (\frac{t}{\tau})^{1/2}\ (t,\tau)^{-1}\ {\cal P}_{l-1}\log (t,\tau)^{-1}\ {\cal F}_{2,l}(t,\tau) \ . \] This ends the proof of Proposition 1. \qed \\ \noindent The behaviour of the CAS upon removing the UV cutoff, i.e. $ \vep \searrow 0 $, follows from\\ \noindent {\bf Proposition 2}:\\ {\it Let $\vep $ be (sufficiently) small. With the notations, conventions and the same class of renormalization conditions as in Proposition 1 we have the bounds} \begin{eqnarray} |\,\pa_{\vep}\, {\cal L}^{\vep,t}_{n,l} (x_1,\vp_{\tau_{2,s},y_{2,s}})| & \le & \vep^{-\frac{1}{2}} \,\, {\cal P}_l\log{\vep}^{-1}\,\, \, t^{\frac{n-5}{2}} \,\, {\cal F}_{s,l}(t,\tau) \label{uv1} \\ |\,\pa_{\vep}\, {\cal L}^{\vep,t}_{n,l} (x_1, E_{(i)}^{(r)}\vp_{\tau_{2,s},y_{2,s}})| & \le & \vep^{-\frac{1}{2}}\,\, {\cal P}_l\log{\vep}^{-1}\,\,|\,\om^{(r)}(x_1) |\, t^{\frac{n+r-5}{2}}\,\, {\cal F}_{s,l}(t,\tau) \label{uv3} \\ |\,\pa_{\vep}\, {\cal L}^{\vep,t}_{n,l} (x_1,\vp^{(j)}_{\tau_{2,s}, y_{2,s}})|\, & \le & \vep^{-\frac{1}{2}} \,\, {\cal P}_l\log{\vep}^{-1}\,\, t^{\frac{n-4}{2}}\,\, \tau_j^{-\frac{1}{2}}\,\, {\cal F}_{s,l}(t,\tau) \label{uv4}\\ |\,\pa_{\vep}\, F_{(12)}^{(0)} {\cal L}^{\vep,t}_{n,l} (x_1, x_2, \vp_{\tau_{2,s},y_{2,s}})| & \le &\vep^{-\frac{1}{2}} \,\, {\cal P}_l\log{\vep}^{-1}\,\, t^{\frac{n-2}{2}}\,\, {\cal F}_{s,l}^{(12)}(t,\tau) \label{uv5} \\ |\,\pa_{\vep}\, {\cal L}^{\vep,t}_{2,l}(x_1,\vp_{\tau, y})| & \le & \vep^{-\frac{1}{2}}\,\, {\cal P}_{l-1}\log{\vep}^{-1}\,\, (t,\tau)^{-\frac{3}{2}}\,\, {\cal F}_{2,l}(t,\tau) \label{uv2} \\ |\,\pa_{\vep}\, F_{(12)}^{(0)} {\cal L}^{\vep,t}_{2,l} (x_1, x_2)| & \le & \vep^{-\frac{1}{2}} \,\, {\cal P}_{l-1}\log{\vep}^{-1}\,\, \, {\cal F}_{2,l}^{(12)}(t)\ . \label{uv6} \end{eqnarray} {\it Proof:} We apply the method developed in the previous proof. The bound (\ref{uv1}) obviously holds in the starting case\, $n=4, l=0\,$. Because of the bare interaction (\ref{nawig}) the FE (\ref{fequvep}) is used if $n+r>4,$ where the difference test function in (\ref{uv4}) and the modified insertion in (\ref{uv5}),(\ref{uv6}) count as $r=1$ and $r=3$, respectively. Regarding the r.h.s. of (\ref{fequvep}) we note that the first and second term do not contribute to the cases considered, and the third one only if $n=2,4,6.$\\ Proceeding inductively as in A, B) and C) of the previous proof, and using the bounds of Proposition 1, reproduces (\ref{uv1}) for $n>4$ and (\ref{uv3})-(\ref{uv5}) and (\ref{uv6}).\\ The FE (\ref{fequvep2}) provides bounds on the relevant parts of the cases $n+r \leq 4$. As the renormalization conditions (\ref{renbed}), (\ref{renbed4}), (\ref{vepdep}) depend at most weakly on $\vep$, we obtain inductively \begin{eqnarray} %\eq |\,\pa_{\vep}\, c^{\vep,t}_{l}(x_1)\,| \le \, \vep^{-\frac{1}{2}}\,\, {\cal P}_{l-1}\log {\vep}^{-1} \,\cdot t^{- \frac{1}{2}} & , & |\,\pa_{\vep}\, a^{\vep,t}_{l}(x_1)\,| \le \, \vep^{-\frac{1}{2}}\, {\cal P}_{l-1}\log {\vep}^{-1} \cdot t^{- \frac{3}{2}} \label{uvr1}\\ %\eqe |\,\pa_{\vep}\, f^{\mu,\vep,t}_{l}(x_1)\,\om_\mu (x_1) | & \le & \nom \,\,\vep^{-\frac{1}{2}}\,\, {\cal P}_{l-1}\log {\vep}^{-1} \,\cdot t^{-1} \label{uvr2}\\ |\,\pa_{\vep}\, b^{\mu\nu,\vep,t}_{l}(x_1)\,\om_{\mu \nu}^{(2)} (x_1)\ | & \le & |\, \om^{(2)}(x_1)\,| \,\, \vep^{-\frac{1}{2}}\,\, {\cal P}_{l-1}\log {\vep}^{-1} \,\cdot t^{- \frac{1}{2}} \ . \label{uvr3} \end{eqnarray} With the aid of the decomposition (\ref{4l}), the bound (\ref{uv1}) for $n=4$ follows from (\ref{uvr1}) and (\ref{uv4}). It remains to show (\ref{uv2}). We use the decomposition (\ref{2l}) and perform similar steps as in D)d). From (\ref{re22}) and (\ref{uv6}) we obtain \begin{eqnarray} |\, \pa_\vep\, {\ell}^{\vep,t}_{2,l}\ (x_1, \vp_{\tau, y})| & \leq & \int_{x_2} \,| \,\pa_{\vep}\, F_{(12)}^{(0)} {\cal L}^{\vep,t}_{2,l} (x_1, x_2)\,| \ \int_{0}^{1}d\rho \, \frac{ (1-\rho)^2}{2!}\, | \,( \nabla^{\,3} \vp_{\tau, y})( X(\rho))\, | \nonumber \\ & \leq & \vep^{-\frac{1}{2}}\,\, {\cal P}_{l-1}\log {\vep}^{-1} \,\, \,\tau^{- \frac{3}{2}} \,\int_{x_2} {\cal F}_{2,l}^{(12)}(t) \int_{0}^{1} d\rho \, K(\tau_{ \de '}, X(\rho),y)\, \nonumber \end{eqnarray} and herefrom, cf. (\ref{f2r}), (\ref{rr}) for $N b\,t <\de '\tau$ $(N=3l-2)\,$, \eq |\, \pa_\vep\, {\ell}^{\vep,t}_{2,l}\ (x_1, \vp_{\tau, y})| \ \le \ \vep^{-\frac{1}{2}}\,\, {\cal P}_{l-1}\log {\vep}^{-1} \,\, \tau^{-3/2}\ K(\tau_{\de}, x_{1},y)\ . \label{p2s} \eqe From (\ref{uvr1})-(\ref{uvr3}) follows \eq \label{p2rel} |\,\pa_\vep \, {\cal L}^{\vep,t}_{2,l}(x_1,\, \vp_{\tau, y})_{rel} | < \vep^{-\frac{1}{2}}\,\, {\cal P}_{l-1}\log {\vep}^{-1} \,\cdot \frac{1}{t^{\frac{1}{2}}} \bigg(\, \frac{1}{t} + \frac{1}{(t \tau)^{\frac{1}{2}}} + \frac{1}{\tau}\, \bigg)\ K(\tau_{\de}, x_1,y)\, . \eqe On account of (\ref{2l}) the bounds (\ref{p2s}), (\ref{p2rel}) establish (\ref{uv2}) for $ N b\,t < \de ' \tau $.\\ To obtain an extension of the bound (\ref{p2s}) to $N b\,t \geq \de '\tau$ we again resort to the decomposition (\ref{2l}), yielding $$ \pa_t\, \pa_\vep\, {\ell}^{\vep,t}_{2,\,l}(x_1, \vp) \, = \,\pa_t \,\pa_\vep \, {\cal L}^{\vep,t}_{2,\,l}(x_1,\, \vp) $$ \eq \label{p2d} -\, \pa_t \,\pa_{\vep}\, a^{\vep,t}_{l}(x_1)\vp (x_1) +\, \pa_t \,\pa_{\vep}\, f^{\mu,\vep,t}_{l}(x_1)\,\om_\mu (x_1) +\, \pa_t\,\pa_{\vep}\, b^{\mu\nu,\vep,t}_{l}(x_1)\,\om_{\mu \nu}^{(2)} (x_1)\, , \eqe with $\, \om_{\mu}(x) = \nabla_{\mu} \vp(x), \, \, \om^{(2)}_{\mu \nu}(x) = \nabla_{\mu} \nabla_{\nu} \vp(x), \, \, \vp (x) = K(\tau, x, y) $. Employing on the r.h.s. of (\ref{p2d}) in the various terms the corresponding FE (\ref{fequ}) derived w.r.t. $\vep$ and then making use of bounds of Proposition 1 and of Proposition 2 already established inductively, leads with now familiar steps to $$ |\,\pa_t\, \pa_\vep\, {\ell}^{\vep,t}_{2,\,l}(x_1, \vp) | \leq \ \vep^{-\frac{1}{2}}\,\, {\cal P}_{l-1}\log {\vep}^{-1} \,\cdot \Big [ \, ( t,\tau)^{-\frac{5}{2}} \,\,{\cal F}_{2,\,l}(t,\tau) $$ \eq \label{dbp2} +\, \Big ( \, t^{-\frac{5}{2}} + \tau^{-\frac{1}{2}}\, t^{-2} + \tau^{-1}\, t^{-\frac{3}{2}} \, \Big ) \, K(\tau_{\de}, x_1,y)\,\Big ] \, . \eqe On integrating $\,\pa_t\, \pa_\vep\, {\ell}^{\vep,t}_{2,l}\ (x_1, \vp)\,$ from $ t = \vep $ (small) with vanishing initial condition up to $ t \geq \de '\tau/bN $ the integral has to be split at $ t = \de '\tau/bN $. A bound on the lower part of the integral is given by (\ref{p2s}). The upper part of the integral can be bounded using (\ref{dbp2}) observing both $ \tau > t $ and $ \tau < t$, and majorizing constants. Combining both contributions yields for $ t \geq \de '\tau/bN $ $$ |\, \pa_\vep\, {\ell}^{\vep,t}_{2,\,l}(x_1, \vp) | \leq \ \vep^{-\frac{1}{2}}\,\, {\cal P}_{l-1}\log {\vep}^{-1} \,\cdot \,\bigg ( \frac{b N}{\de ' \tau}\bigg )^{\frac{3}{2}} $$ \eq \label{bp2l} \cdot\, \bigg [ \,{\cal F}_{2,\,l}(t,\tau) +\, \bigg ( \,1 + \Big (\,\frac{\de '}{b N}\,\Big )^{\frac{1}{2}} + \frac{\de '}{b N} \, \bigg ) \, K(\tau_{\de}, x_1,y)\,\bigg ] \, . \eqe Taking into account once more the decomposition (\ref{2l}), the bound (\ref{p2rel}) on the relevant part together with the bounds (\ref{p2s}), (\ref{bp2l}) on the remainder reproduce (\ref{uv2}). Thus the proof of Proposition 2 is complete. \qed \\ \noindent From (\ref{uv1}), (\ref{uv2}) follows the integrability at $\vep = 0$ and hence the existence of finite limits $$ \lim_{\vep \searrow 0} \, {\cal L}^{\vep,t}_{n,\,l}\, (x_1,\vp_{\tau_{2,s},y_{2,s}})\, , \quad n \geq 2 \ .$$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent {\bf Proposition 3}:\\ {\it With the notations, conventions and the same class of renormalization conditions as in Proposition 1 - up to the fact that the constants in ${\cal P}_l\log\,$ may now also depend on the mass $m\,$ - we claim the following bounds for the CAS in the interval $ \,1 \le t \le \infty$~: \eq |\,{\cal L}^{\vep,t}_{n,l} (x_1,\vp_{\tau,y_{2,s}})|\,\le\ {\cal P}_l\log \tau^{-1}\ {\cal F}^{\,t}_{s,l}(\tau) \ , \quad n \ge 4 \label{ip1} \eqe \eq |\, {\cal L}^{\vep,t}_{2,l} (x_1,\vp_{{\tau},y})|\,\le\ (1,\tau)^{-1}\ {\cal P}_{l-1}\log (1,\tau)^{-1}\ {\cal F}^{\,t}_{2,l}(\tau)\ . \label{ip2} \eqe The definition of ${\cal F}^{\,t}_{s,l}(\tau)\,$ is given in (\ref{iges}).} \noindent {\it Proof}~: The bounds stated in the proposition are proven inductively using again the standard scheme. The boundary conditions are the bounds from Proposition 1 taken at $t=1\,$. They obviously satisfy the bounds (\ref{ip1}), (\ref{ip2}). The FE is treated in the same way as in parts A1) and A2) of the proof of Proposition 1. The integration w.r.t $t$ is performed using the fact that ${\cal F}^{\,t}_{s,l}(\tau)\,$ is montonically increasing with $t\,$. As regards part A1) we now use for $\,t \geq 1\,$ instead of (\ref{rom}) now $\,C_t(z, z') \leq O(1)\, \exp( - ( m^2 -\de)\,t\, )\,$, which results from the upper bounds (\ref{hk5}), (\ref{f21}) on the heat kernel, and obtain upon integration \[ \int_1^{\,t} dt'\ {\cal F}^{\,t'}_{s,l}(\tau)\ e ^{-(m^2 - \de)\,t'}\ \le \, O(1/m^2) \,{\cal F}^{\,t}_{s,l}(\tau)\ . \] As regards A2) the internal line generated, which connects the two (partial) trees, see (\ref{gensi}), (\ref{2nd}), has the weight (\ref{falt}). Integrating, we majorize the weights of the other internal lines by their values at $\, t \, $ and use for (\ref{falt}) $$ \int_1^{\,t} dt'\,C_{t'}(z', z'') \,\leq \,C_{\underline{t}}\,(z', z'') + \int_1^{\,t} dt'\,C_{t'}(z', z'') $$ valid for any $\, 0 < \underline{t} \leq 1 \, $, thus reproducing the weight factor $\,{\cal F}^{\,t}_{s,l}(\tau)\,$, (\ref{iges}), in this case, too. \qed Note that the renormalization conditions at $\,t=1\,$ are in one to one relation with the values of the corresponding relevant terms at $\,t=\infty\,$, which have been shown to be finite for $m^2 >0\,$ in according to Proposition 3. Therefore renormalization conditions at $\,t=1\,$ are tantamount to renormalization conditions at $\,t=\infty\,$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Scaling transformations and the minimal form of the bare action } In this section we want to show that the theory can be renormalized starting from a bare (inter)action of the form (\ref{nawi}). This requires that we do not introduce any position dependent quantity in the theory which is not intrinsic to $\, (\mathcal{M}, g ) \, $. Thus we only consider position independent coupling $\la$, and renormalization conditions in terms of intrinsic geometric quantities. We then introduce scaling tranformations of the following kind~:\\ \noindent For a four-dimensional Riemannian manifold $\, (\mathcal{M}, g ) \, $ we scale its metric by a constant conformal factor, [NePa], \begin{equation} \label{sc1} \rho \in \mathbf{R}_+ : \qquad g_{\mu \nu}(x) \rightarrow \rho ^2 \,g_{\mu \nu}(x)\,,\quad \mbox{shortly }\ g \rightarrow \rho ^2 \,g\ . \end{equation} This leads to corresponding changes of geometrical quantities $$ g^{\mu \nu} \rightarrow \rho^{ - 2}g^{\mu \nu},\quad \Delta \rightarrow \rho^{ - 2}\Delta, \quad |\,g|^{1/2} \rightarrow \rho ^4 |\,g|^{1/2},\quad \ti \de \rightarrow \rho ^{-4} \, \ti \de$$ \begin{equation} \label{sc2} d(x,y) \rightarrow \rho\, d(x,y) , \quad \sigma(x,y)^\mu \rightarrow \sigma(x,y)^\mu \end{equation} $$\Gamma_{\mu \nu}^{\lambda} \rightarrow \Gamma_{\mu \nu}^{\lambda}\,, \quad \nabla_{\mu} \rightarrow \nabla_{\mu} \,, \quad R^{\lambda}_{\, \,\mu \nu \sigma} \rightarrow R^{\lambda}_{\, \,\mu \nu \sigma}\,, \quad R_{\mu \nu} \rightarrow R_{\mu \nu}\,, \quad R \rightarrow \rho^{ - 2} R\,. $$ Moreover, the heat kernel $\, K (t, x, y\,; g) \, $ satisfies the scaling relation \begin{equation} \label{sc3} K(t, x, y \,;\, g)\, = \, \rho^4\, K ( \,\rho^2 t, x, y\, ;\,\rho^2 g) \, , \end{equation} which follows from its evolution equation $ \,( \partial_t - \Delta_g ) K (t, x, y\, ;\, g) = 0 \, $ together with stochastic completeness (\ref{hk3}). As a consequence the regularized free propagator (\ref{propa}), $0 < \varepsilon < t \leq \infty $, $$ C^{\,\varepsilon,\, t}(x, y;\,m^2 \,,g)\, = \,\int_{\varepsilon}^{\,t}dt '\, e^{\,- m^2 t'}\, K(t', x, y;\,g) \, ,$$ satisfies \begin{equation}\label{sc4} C^{\,\varepsilon,\, t}(x, y;m^2 \,,g)\, = \rho^2 \,C^{\,\rho^2\varepsilon,\,\rho^2 t} (x, y;\frac{m^2}{\rho^2} \,,\rho^2 g)\,. \end{equation} Regarding for a moment the action of the \emph{classical scalar field theory}, \begin{equation} S(\varphi, m^2, \xi, \lambda; g) \, =\frac{1}{2}\, \int_x\, \Big (\, \varphi (-\Delta)\varphi + m^2 \varphi^2 + \xi\, R(x)\,\varphi^2 + 2 \frac{\lambda}{4!}\, \varphi^4 \Big )\, , \end{equation} we observe, that it is invariant if we supplement the scaling (\ref{sc1}) of the metric by the transformations \begin{equation} \varphi(x) \rightarrow \rho^{-1}\varphi(x)\,,\quad m^2 \rightarrow \rho^{-2} m^2 \, ,\quad \xi \rightarrow \xi \, , \quad \lambda \rightarrow \lambda \ . \end{equation} We now consider the perturbative expansion of a regularized $\lambda \phi^4 $- theory without counter terms, i.e. in (\ref{funcin}) we have \, $ L^{\vep, \vep}( \phi) = \lambda \int dV(x)\, \phi ^4 (x) \,$. A Feynman diagram contributing to an $n$-point CAS having\, $v$\, four-vertices and $\, I\, $ internal lines obeys the topological relation $ 4 v = n + 2 I .$ \, This together with the scaling property (\ref{sc4}) of the propagator implies for an $n$-point function folded with a test function $ \varphi = \varphi(x_2, \dots, x_n)$ \begin{equation} \label{sc5} \mathcal{L}^{\,\varepsilon,\,t}_{n, \,l} (x_1, \varphi\, ; m^2, \lambda, g) = \rho^{4-n} \,\mathcal{L}^{\,\rho^2\varepsilon,\,\rho^2 t }_{n, \,l} (x_1,\varphi \,; \frac{m^2}{\rho^2}\,, \lambda \,,\rho^2 g)\ . \end{equation} In the renormalization proof the CAS were constructed by imposing renormalization conditions for the relevant terms, see (\ref{renbed}), (\ref{renbed4}), and by requiring the irrelevant terms to vanish at scale $\vep$, see (\ref{bo1})-(\ref{bo3}). As noted the renormalization conditions will now be supposed to be expressed in terms of intrinsic quantities, and they will be supposed to satisfy scaling (both statements are true for vanishing renormalization conditions). Because of the behaviour of $\, \sigma(x,y)^\mu \, $ under scaling, (\ref{sc2}), this means \begin{eqnarray} a^{\varepsilon, \,\infty}_l ( x ; m^2, g) & = & \rho^2 \, a^{\,\rho^2 \varepsilon,\, \infty}_l ( x ; \rho^{-2}\, m^2 , \rho^2 \,g) \label{sc7} \\ f^{ \,\mu, \,\varepsilon, \,\infty}_l ( x ; m^2, g) & = & \rho^2 \, f^{\,\mu,\,\rho^2 \varepsilon,\, \infty}_l ( x ; \rho^{-2}\, m^2, \rho^2 \,g) \label{sc8} \\ b^{ \,\mu \nu, \, \varepsilon, \,\infty}_l ( x ; m^2, g) & = & \rho^2 \, b^{\,\mu \nu ,\,\rho^2 \varepsilon,\, \infty}_l ( x ; \rho^{-2}\, m^2, \rho^2 \,g) \label{sc9} \\ c^{\varepsilon, \,\infty}_l ( x ; m^2, g) & = & c^{\,\rho^2 \varepsilon,\, \infty}_l ( x ; \rho^{-2}\, m^2, \rho^2 \,g) \ . \label{sc10} \end{eqnarray} For the standard case of $\vep$-independent renormalization conditions the scaling of $\vep$ can of course be ignored. At the tree level the relation (\ref{sc5}) holds as shown above. Using the FE with the standard inductive scheme it then follows that\\ \centerline{{\it (\ref{sc5}) holds in the case of renormalization conditions satisfying (\ref{sc7})-(\ref{sc10}).}} Renormalization conditions imposed at some scale $t_R < \infty\,$ are in one to one relation to those imposed at $t= \infty\,$, and the local terms $ a^{\,\varepsilon,\, t_R}_l\,$ etc. can be viewed either as renormalization conditions imposed at this scale or as resulting from integrating the FE over $[t_R,\infty)\,$ with renormalization conditions imposed at $\infty\,$. From this fact and (\ref{sc5}) one deduces that the relations corresponding to (\ref{sc7})-(\ref{sc10}) for renormalization conditions imposed at finite $t_R\,$ are \eq a^{\varepsilon, \,t_R}_l ( x ; m^2, g)\, = \, \rho^2 \, a^{\,\rho^2 \varepsilon,\, \rho^2 t_R}_l ( x ; \rho^{-2}\, m^2 , \rho^2 \,g) \quad \mbox{etc.} \label{ss11} \eqe \noindent In the subsequent analysis of the counter terms it will be helpful to first analyse the {\it massless} theory for $t$ in the interval $[\vep,T]\,$ to eliminate one of the parameters subject to scaling. While restricting to $[\vep,T]\,$, the less singular corrections stemming from the massiveness (see (\ref{dev}) below) can be dealt with afterwards. The same can then be done (trivially) for the finite contributions coming from integrating the FE of the massive theory over $[T,\infty)\,$.\\ For the massless theory we introduce the following notation~: we denote \[ a_l^{\vep,t}(x;g) \to a_{l,t_R}^{\vep,t}(x;g)\,,\quad \mbox{ etc.} \] to explicitly introduce all parameters subject to scaling, including the scale of the renormalization point $t_R\,$. Furthermore we will introduce the sequence of scales \[ t_n~:= \ka ^{-n}\, t_R,\ \ \kappa>1\,,\quad 1 \le n \le N\,, \ \mbox{ such that } \ \vep =t_N\ . \] Then we use the shorthands \eq a_{l,t_R}^{n}(x;g)~:= a_{l,t_R}^{\vep,t_n}(x;g) \quad \mbox{ etc.}\ , \label{n1} \eqe and for the renormalization constants at $t=t_R$ \eq a_l^{t_R}(x;g)~:= a_{l,t_R}^{\vep,t_R}(x;g) \quad \mbox{ etc.} \label{n2} \eqe As a consequence of the properties of the heat kernel, the terms $ a_{l,t_R}^{n}(x;g)\,$ etc. are smooth scalars on the manifold. For the manifolds considered (of sectional curvature bounded above and below, as defined in Sect.2\,), we have proven bounds which are uniform in the curvature since our bounds on the heat kernel are uniform in this case. The same holds for their (low order) derivatives $(t \De)^s a_{l,t_R}^{n}(x;g)\,$ etc., since we obtain the same bounds for these derivatives due to (\ref{D}). We can therefore decompose these terms according to their tensorial character into individual contributions from curvature, respecting the scaling property, such that in this decomposition there will only appear terms depending smoothly on the geometric quantities. This gives \begin{align} a_{l,t_R}^{n}(x;g) = & \,\, \alpha_{l,t_R}^{n} + R (x) \,\xi_{l,t_R}^{n} + \de a_{l,t_R}^{n}( x ; g) \label{sc11} \\ f^{ \,\mu,\, n}_{l,t_R} ( x ; g) = & \,\, 0 + \de f^{ \,\mu, \,n}_{l,t_R} ( x ; g) \label{sc12} \\ b^{ \,\mu \nu, \,n}_{l,t_R} ( x ; g) = & \,\, g^{\mu \nu}(x) \,\beta_{l,t_R}^{n} + \de b^{ \,\mu \nu,\,n}_{l,t_R}(x;g)\label{sc13}\\ c_{l,t_R}^{n}(x;g) = & \,\, \gamma_{l,t_R}^{n} + \de c_{l,t_R}^{n}(x;g)\ . \label{sc14} \end{align} The zero written in (\ref{sc12}) reminds us that this term vanishes identically in the case of constant curvature. The remainder terms in this decomposition may be analysed further \begin{align} \de a_{l,t_R}^{n}( x ; g) = & \,\, t_R\, \Big( \Delta R(x)\, h^{(1,n)}_l + R^{\,2} (x)\, h^{(1',n)}_l + R^{\,\mu \nu}(x) R_{\,\mu \nu}(x)\, h^{(1'',n)}_l \nonumber\\ & \,\, + R^{\,\mu \nu\la \si }(x) R_{\,\mu \nu\la \si}(x)\, h^{(1''',n)}_l \Big ) \,\, + \cdots \label{dea} \\ \de f^{ \,\mu, \,n}_{l,t_R} ( x ;g) = & \,\, t_R\, g^{\,\mu \nu}(x) \, R_{ , \, \nu}(x) \, h^{(2,n)}_l + \cdots \label{def} \\ \de b^{ \,\mu \nu, \, n}_{l,t_R} ( x ; g) = & \,\, t_R\left( R^{\,\mu \nu}(x) \, h^{(3,n)}_l + g^{\,\mu \nu}(x) R(x) h^{(3',n)}_l \right)\,+ \cdots \label{deb}\\ \de c^{n}_{l,t_R} (x ; g) = & \,\, t_R\, R (x) \, h^{(4,n)}_l + \cdots \ \, .\label{dec} \end{align} All the $h$-functions in this decomposition have mass dimension zero and are therefore independent of $t_R$ which is the only scale. The dots indicate terms of higher scaling dimension in the expansion w.r.t. curvature terms. We then\\ {\it assume that these expansions are asymptotic \footnote{asymptoticity is obviously required up to second order in $\rho ^2\,$ only.}, in the sense that the remainders satisfy} \eq |\,\de a^{n}_{l,t_R} ( x ;\rho ^2 \,g)|\,,\ |\,\om_{\mu}(x) \,\de f^{ \,\mu, \,n}_{l,t_R} ( x ; \rho ^2 g)|\,,\ |\,\om^{(2)}_{\mu\nu}(x)\, \de b^{ \,\mu \nu, \, n}_{l,t_R} ( x ;\rho ^2 g)| \ \le\ O(\rho^{-4})\ , \label{desc1} \eqe \eq |\, \de c^{n}_{l,t_R} ( x ; \rho ^2 g) | \ \le \ O(\rho ^{-2})\ . \label{desc2} \eqe Here $ n\,$ and $t_R\, $ are (of course) kept fixed and furthermore, the rank 1 resp. rank 2 cotensor fields $\,\om_{\mu}(x)\,,\ \om^{(2)}_{\mu\nu}(x)\,$ are assumed to stay invariant under scaling $g\to \rho ^2 \,g\,$. The bounds are in agreement with the leading terms written in (\ref{dea})-(\ref{dec}). This assumption appears plausible and is often taken for granted, see e.g. [HoWa3]. Its proof requires a more thorough analysis of the heat kernel and its convolutions than is given here. \noindent {\bf Proposition 4}~:\\ {\it Assuming (\ref{desc1}),(\ref{desc2}), then for position independent coupling $\la$ there exist renormalization conditions of the form (\ref{renbed}, \ref{renbed4}) such that the bare action takes the simple form (\ref{nawi}), this means that for $l \ge 1$ \eq L_l^{\vep}(\vp) = \ {1 \over 2}\,\int_x\ \{ (\,\al_l^{\vep} +\,\xi_l^{\vep}\ R(x))\,\vp^2(x)\, -\, b_l^{\vep}\,\vp (x) \De\vp (x)\, +\, {2 \over 4!}\, c_l ^{\vep}\, \vp^4(x)\} \label{nawi1} \eqe with the following bounds \eq |\,\al^{\vep}_{l}\,| \le \ \frac{1}{ \vep}\ {\cal P}_{l-1}\log \frac{1}{ \vep}\ , \quad |\,\xi_l^{\vep}\,| \le \ {\cal P}_{l}\log \frac{1}{ \vep}\ , \quad |\,b_l^{\vep}\,| \le \ {\cal P}_{l-1}\log \frac{1}{ \vep}\ , \quad |\,c_l^{\vep}\,| \le \ {\cal P}_{l}\log \frac{1}{ \vep}\ . \label{cac} \eqe } \noindent {\it Proof~:}\\ We first note that Proposition 1 can be proven in complete analogy when imposing renormalization conditions of the form (\ref{renbed}), (\ref{renbed4}) at scale $t_R = T>0\,$ for the massless theory. The scale $T\,$ is the one up to which we have precise control on the heat kernel, cf. (\ref{hk10}), and it is thus related to the geometry of $\cal M\,$. Furthermore we can expand for $\vep \le t \le T\,$ \eq \,{\cal L}^{\vep,t}_{n,l} (m^2;x_1,\vp_{\tau,y_{2,s}})\ =\ \,{\cal L}^{\vep,t}_{n,l} (0;x_1,\vp_{\tau,y_{2,s}})\ +\ m^2\,\pa_{m^2}{\cal L}^{\vep,t}_{n,l} (0;x_1,\vp_{\tau,y_{2,s}}) \label{dev} \eqe \[ +\ m^4\,\int_{0}^{1} d\la \ (1-\la)\,\pa_{m^2}^2{\cal L}^{\vep,t}_{n,l} (\la m^2;x_1,\vp_{\tau,y_{2,s}})\ . \] We first analyse the massless theory and then comment on the derivative terms. We use the notation (\ref{n1}), (\ref{n2}). The theory is specified through renormalization conditions of the form (\ref{renbed}), (\ref{renbed4}) imposed at scale $t_R=T$~: \eq a^{T}_l(x;g)=0\, ,\ f^{\mu, T}_l(x;g)=0\, ,\ b^{\mu\nu,T }_l(x;g)=0\, ,\ c ^T_l(x;g)=0\ , \label{renscal} \eqe together with boundary conditions of the type (\ref{bo1})-(\ref{bo3}) at scale $\vep =\ka ^{-N} T$ for $l'\le l\,$. Our aim is to analyse the bare action. From Proposition 1 we obtain for $\, l > 0\,$ the bounds \begin{align} |\, a_{l, T}^{n}(x;g)| \leq & \,\, O(1)\ \ka ^{n}\,\,n^{l-1} \label{pp1} \\ |\,f^{ \,\mu,\, n}_{l,T} ( x ; g)\,\om_{\mu}(x)| \leq & \,\, O(1)\ |\,\om(x)|\,\, \ka ^{\frac{n}{2}}\,\, n^{l-1} \label{pp2} \\ |\, b^{ \,\mu \nu, \,n}_{l,T}( x ; g)\, \om^{(2)}_{\mu \nu} (x)| \leq & \,\, O(1) \ |\, \om^{(2)}(x)|\,\, n^{l-1} \label{pp3}\\ |\, c_{l,T}^{n}(x;g) | \leq & \,\, O(1) \ n^l \, . \label{pp4} \end{align} In the sequel we present the detailed argument for the relevant term $a(x;g)\,$, whereas the analogous treatment of the other ones is stated in shortened form. In view of the decomposition (\ref{sc11}) we want to prove inductively in $n$ \footnote{More precisely induction is in $(l,n)\,$ in the order $(l,1),(l,2),\ldots,(l,N),(l+1,1),\ldots$, but the step $(l,N)\to (l+1,1)$ is trivial.} \eq |\al_{l,T}^n| \le O(1) \sum_{n '=1}^{n} \ka ^{n'}\ {n'} ^{\,l-1}\,,\ |\xi_{l,T}^n| \le O(1) \sum_{n '=1}^{n} {n'}^{\,l-1} \,,\ |\de a_{l,T}^n(x;g)| \le O(1) \sum_{n '=1}^{n} \ka^{-n'}{n'}^{l-1}\ . \label{ind} \eqe First note that the uniqueness of the solutions of the FE implies that the relevant term $\,a_{l,T}^{n+1}(x,g)\,$ satisfies \eq a_{l,T}^{n+1}(x;g)= {\hat a}_{l,\kappa ^{-n} T}^1(x;g) \label{hut} \eqe where $ {\hat a}_{l,\kappa ^{-n} T}^1(x;g)\,$ is defined to be the corresponding relevant term at scale $\kappa ^{-(n+1)} T\,$ for the theory renormalized at scale $\kappa ^{-n} T$, with renormalization conditions of the following form \eq {\hat a}_{l}^{\kappa ^{-n} T}(x;g)= a_{l,T}^{n}(x;g) \ \ \mbox{ (analogously for the $f\,,\ b \,,\ c$-terms)}\ . \label{rhut} \eqe This just means that we take renormalization conditions at scale $T$, integrate down to $\kappa ^{-n} T\,$, and take the values we arrive at for the local terms, as renormalization conditions at the scale $\kappa ^{-n} T\,$. By the uniqueness statement we obtain the same Schwinger functions as when imposing $ a_{l}^{T}(x;g)\,$ etc. at scale $T\,$.\\ From the scaling relations, cf. (\ref{ss11}), we have \eq a_{l,T}^n(x; g)\ =\ \ka^{n}\ a_{l,\kappa ^{n}T}^n(x;\ka^{n}g) \ =\ \ka^{n}\ {\hat a}_{l}^{T}(x;\ka^{n}g)\, , \label{shutr} \eqe \eq a_{l,T}^{n+1}(x;g)\ =\ \ka^{n}\ a_{l,\kappa ^{n}T}^{n+1}(x;\ka^{n}g) \ =\ \ka^{n}\ {\hat a}_{l, T}^1(x;\ka^{n}g) \ . \label{shut} \eqe In the case of $ c_{l,T}^{n} (x;g)\ $ such relations hold without the external factor $\, \ka^n \,$. Moreover, \begin{align} b^{ \,\mu \nu, \,n}_{l,T}( x ; g)\, \om^{\,(2)}_{\mu \nu} (x)\, = & \,\,\ka^{n}\ {\hat b}^{\,\mu\nu,T }_{l}(x;\ka^n g )\, \om^{\,(2)}_{\mu \nu} (x)\, , \label{shutrb} \\ b^{ \,\mu \nu, \,n+1}_{l,T}( x ; g)\, \om^{\,(2)}_{\mu \nu} (x)\, = & \,\,\ka^n \,\, {\hat b}^{\,\mu\nu,1 }_{l, T}(x;\ka^n g )\, \om^{\,(2)}_{\mu \nu} (x)\ , \label{shutb} \end{align} and the analogue for $\,f^\mu \,$ is obtained replacing $\, b^{\mu \nu}\,$ by $\, f^\mu \,$ and $\, \om^{\,(2)}_{\mu \nu} (x)\,$ by $\, \om_{\mu} (x)$. Using (\ref{pp1})-(\ref{pp4}) and (\ref{shutr})-(\ref{shutb}), we then obtain \eq |\,{\hat a}^{T}_{l}(x;\kappa ^{n}g) |\ \le\ O(1)\ n^{l-1}\, ,\quad |\,{\hat f}^{\,\mu, T}_{l}(x;\ka ^{n} g)\, \om_{ \mu}(x) | \ \le \ O(1)\ |\, \om(x)|_{\,\kappa ^{n}g} \,\, n^{l-1} \,, \label{ren0} \eqe \eq |\,{\hat b}^{\mu\nu,T }_{l}(x;\ka ^{n} g)\, \om^{\,(2)}_{\mu \nu}(x)| \ \le\ O(1)\ |\, \om^{\,(2)}(x)|_{\,\kappa ^{n}g} \,\,n^{l-1}\, , \quad |\,{\hat c}^T_l(x;\kappa ^{n}g) | \ \le\ O(1)\ n^{l} \label{ren1} \eqe where we denoted by $\, | \cdot |_{\,\kappa ^{n}g} \,$ the norm (\ref{nor}) generated by $\, \kappa ^{n}g\, $. We now consider more general massless Schwinger functions $\,\hat {\cal L}^{\kappa ^{- 1}T,t}_{p,l}(x_1,\vp_{{\tau},y_{2,s}};\ti g)\,$ resulting from a metric $\ti g\,$ of the class defined in Section 2 \footnote{$\ti g =\kappa ^n g\,$ certainly belongs to this class if $g$ does} and satisfying renormalization conditions of the form (\ref{ren0}), (\ref{ren1}) at loop orders $l'