^sf||_{\C{G}}$; see page 343 in [ABG] for details). We consider only a rather restricted class of perturbations $V$ (see [Sh1] for the case of non-local singular perturbations satisfying natural extensions of the conditions (7.6.19), (7.6.20) in [ABG]). Let $\gs$ be a real number such that $0<\gs\leq m-1$ and let $V=\sum V_k$, where the sum runs over all the integers $k$ satisfying $0\leq k<\gs+1$. We assume that each $V_k$ is a bounded real function on $\D{R}^n$ which tends to zero at infinity and which satisfies $\sum_{|\ga|=k}|V_k^{(\ga)}(x)|\leq C\vv^{-1-\gs}$. Now let $s=\gs+1/2$ and denote by $\gk(H)$ the union of the set $\gk(h)$ of critical values of the function $h$ and of the set of eigenvalues of $H$; $\gk(H)$ is a closed real set and its accumulation points belongs to $\gk(h)$. Then for each $\gl\in\D{R}\setminus\gk(H)$ the limit $\lim_{\gm\to+0}(H-\gl \mp i\gm)^{-1}\equiv R(\gl\pm i0)$ exists in norm in $B(\C{G}_{s,\infty}^*;\C{G}_{-s,1})$, locally uniformly in $\gl$, and the maps $\gl\mapsto R(\gl\pm i0)\in B(\C{G}_{s,\infty}^*;\C{G}_{-s,1})$ are locally of class $\gL^\gs$ on $\D{R}\setminus \gk(H)$. These assertions follow easily from part (a) of Theorem A, as explained in \S 7.6.3 from [ABG]. Similar results hold for $N$-body hamiltonians. For the case of Dirac operators, Stark effect hamiltonians and simply characteristic operators, see [Sh1]. \subsection{} \label{s:1.5} The paper is organized as follows. In Section 2 we introduce two Besov type scales associated to a self-adjoint operator $A$: one consisting of vectors from $\C{H}$ (these are the spaces $\C{H}_{s,p}$ with $s>0$) and one consisting of bounded operators on $\C{H}$. We briefly recall those properties that will be needed later on (a complete treatment in a general setting may be found in [ABG], but see also [BB] and [BGSh2]) and we prove two estimates (Theorems 2.1 and 2.2) which will be important for the proofs of the main theorems. In \S 2.4 we recall a regularization procedure introduced in [BG1] and a result from [BG3]: these are the main technical tools of our paper. In Section 3 one may find a description of the regularity classes of arbitrary self-adjoint operators and of the Mourre estimate in the context of general resolvent families: the main result of this section is Proposition 3.1, which improves a result from [BGS]. Section 4 is the most technical one. We consider there a symmetric bounded $A$-regular operator $H\in B(\C{H})$, we regularize it by considering the operator $H(\ge)=\gq(\ge\C{A})H$ (in the commutative situation of \S 1.4 this means that we approximate the function $h$ by an entire function of exponential type; see the comment after Theorem 2.3), and then we ``twist'' $H(\ge)$ in the spirit of the theory of dilation analytic Schr\"odinger operators, i.e.\ we introduce the operator $H_\ge=e^{-\ge A}H(\ge)e^{\ge A}=e^{\ge\C{A}}\gq(\ge\C{A})H \equiv\gx(\ge\C{A})H$. All Section 4 is devoted to estimating the resolvent of the non-self-adjoint operator $H_\ge$. Finally, the main results of the paper are proved in Section 5. Note that the spectral gap hypothesis allows a rather straightforward reduction to the case when $H$ is a bounded (everywhere defined) operator. \section{Regularity Classes Associated to a Selfadjoint Operator $A$} \label{s:2} \setcounter{equation}{0} \subsection{} \label{s:2.1} Let $\mathbf{E}$ be a Banach space and $\phi:\D{R}\to\mathbf{E}$ a bounded continuous function. We recall that the modulus of continuity (or smoothness) of order $m$ (integer $\geq 1$) of $\phi$ is given by $\gw_m(\ge)=\sup_{x\in\D{R}}|| \sum_{k=0}^m(-1)^kC_m^k\phi(x+k\ge)||_{\mathbf{E}}$ for $\ge>0$. Let $s>0$ be a real number and $p\in[1,\infty]$. Then $\phi$ {\em is of class\/} $\gL^{s,p}$ if there is an integer $m>s$ such that $[\int_0^1(\ge^{-s}\gw_m(\ge))^p\ge^{-1}d\ge]^{1/p}<\infty$ (if $p=\infty$ this means $\gw_m(\ge)\leq c\ge^s$ for a finite constant $c$). One has $\gL^{s,p}\subset\gL^{t,q}$ if and only if $s>t$ or $s=t$ but $p\leq q$. The classes $\gL^{s,\infty}\equiv\gL^s$ are called {\em H\"older-Zygmund classes\/}. If $k\geq 1$ is an integer the classes $BC^k$ and $\Lip^{(k)}$ are defined as follows: $\phi\in BC^k$ means that the derivatives of order $\leq k$ of $\phi$ exist and are bounded and norm continuous; $\phi\in\Lip^{(k)}$ means that $\gw_k(\ge)\leq c\ge^k$ for a constant $c$ and all $\ge>0$ (this is a $k$-th order Lipschitz condition). Then we have $\gL^{k,1}\subset BC^k\subset\Lip^{(k)}\subset\gL^k$, all embeddings being strict and optimal on the scale $\gL^{s,p}$ (the spaces $\gL^{k,p}$ are not comparable with $BC^k$ and $\Lip^{(k)}$ if $1 0$ is a real number. Then $\phi\in\gL^{s,p}$ if and only if $\phi\in BC^k$ and $\phi^{(k)}\in\gL^{\gs,p}$. In particular, if $0<\gs<1$ then $\phi\in\gL^s$ means $\phi\in BC^k$ and $||\phi^{(k)}(x+\ge)-\phi^{(k)}(x)||_{\mathbf{E}} \leq C|\ge|^\gs$; if $\gs =1$ then $\phi\in\gL^s$ means that $\phi\in BC^k$ and $||\phi^{(k)}(x+\ge)+\phi^{(k)}(x-\ge)-2\phi^{(k)}(x)||_{\mathbf{E}} \leq C|\ge|$, i.e.\ $\phi^{(k)}$ has to verify a Zygmund condition. There are natural local classes associated to the preceding ones. For example, if $\gW$ is an open real set and $\phi:\gW\to \mathbf{E}$ is a continuous function, then $\phi$ is locally of class $\gL^{s,p}$ if $\gq \phi \in\gL^{s,p}$ for each $\gq\in C_0^\infty(\gW)$. \subsection{} \label{s:2.2} Let $\C{H}$ be a complex Hilbert space and $A$ a densely defined self-adjoint operator in $\C{H}$ (these objects are fixed from now on). We denote by $E_A$ the spectral measure of $A$ and we set $W_\gs =e^{iA\gs}$ for $\gs\in\D{R}$. We identify $\C{H}$ with its adjoint space $\C{H}^*$ (space of anti-linear continuous functionals on $\C{H}$) with the help of the Riesz isomorphism and we denote by $||\cdot||$ the norm in $\C{H}$ and in $B(\C{H})$. The scalar product $\vv<\cdot,\cdot>$ in $\C{H}$ is linear in the second variable. We denote $\C{H}_\infty$ the vector space $\cap_{k\in\D{N}}D(A^k)$ equipped with the natural Fr\'echet space topology and then we define $\C{H}_{-\infty}$ as the adjoint space $[\C{H}_\infty]^*$ equipped with the strong topology. Since the embedding $\C{H}_\infty\subset\C{H}$ is continuous and dense, we obtain by transposition a continuous dense embedding $\C{H}=\C{H}^*\subset\C{H}_{-\infty}$. The Besov spaces $\C{H}_{s,p}$ (with $s\in\D{R}$ and $p\in[1,\infty]$) associated to $A$ are Banach spaces continuously embedded in $\C{H}_{-\infty}$. To define them, note first that for each compact real set $K$ the spectral projection $E_A(K)$ extends to a continuous linear map $E_A(K):\C{H}_{-\infty}\to\C{H}_\infty$. For real $\gt$ we set $E_A(\gt)=E_A([-2\gt,-\gt]\cup[\gt,2\gt])$ and for each $f\in\C{H}_{-\infty}$, $s\in\D{R}$ and $1\leq p\leq\infty$ we define the number $||f||_{s,p}\in[0,\infty]$ by: \begin{equation} \label{2.1} ||f||_{s,p}\equiv ||E_A([-2,2])f||+\Bigl[\int_1^{\infty} ||\gt^sE_A(\gt)f||^p\gt^{-1}d\gt\Bigr]^{1/p}. \end{equation} If $p=\infty$ then the second term on the r.h.s.\ should be read $\sup_{\gt \geq 1}||\gt^sE_A(\gt)f||$. Finally, we may define $\C{H}_{s,p}$ as the space of vectors $f\in\C{H}_{-\infty}$ such that $||f||_{s,p}<\infty$, equipped with the Banach space structure associated to (2.1). We have $\C{H}_\infty\subset\C{H}_{s,p}$ and we denote by $\C{H}_{s,p}^{\circ}$ the closure of $\C{H}_\infty$ in $\C{H}_{s,p}$. In fact $\C{H}_{s,p}^{\circ}=\C{H}_{s,p}$ if $1\leq p<\infty$. We recall that $\C{H}_{s,p}\subset\C{H}_{t,q}$ (continuous embedding) if $s>t$ or if $s=t$ but $p\leq q$. If $1\leq p<\infty$ we have a continuous dense embedding $\C{H}_\infty\subset\C{H}_{s,p}$, hence one can realize $[\C{H}_{s,p}]^*$ as a subspace of $\C{H}_{-\infty}$; indeed, one has $[\C{H}_{s,p}]^*=\C{H}_{-s,p'}$ if $p^{-1}+{p'}^{-1}=1$. Similarly $[\C{H}_{s,\infty}^\circ]^*=\C{H}_{-s,1}$. If $s>0$ then $\C{H}_{s,p}$ is the set of vectors $f\in\C{H}$ such that the map $\gt\mapsto W_\gt f\in\C{H}$ is of class $\gL^{s,p}$. Another equivalent description is as follows: assume that $0

~~0$ and $c$ such that $|\gf(x)|\leq c|x|^\ga\min(|x|^\gn,|x|^{-\gn})$ for all $x\in\D{R}$. Then there is a constant $C<\infty$ such that for all $s\in\D{R}$, all $p\in[1,\infty]$ and all $f\in\C{H}_{-\infty}$: \begin{equation} \label{2.2} \Bigl[\int_0^1||\ge^{-\ga}\gf(\ge A)f||_{s,1}^p \ge^{-1}d\ge\Bigr]^{1/p}\leq C||f||_{s+\ga,p}. \end{equation} In particular $||\gf(\ge A)f||_{s,1}\leq C\ge^\ga||f||_{s+\ga,\infty}$ for all $\ge\in(0,1)$ and $f\in\C{H}_{-\infty}$. \end{thm} \begin{proof} (i) Set $\gr(u)=\min(u^\gn,u^{-\gn})$ for $u>0$ and observe that for $00$, $\gt>0$ and $f\in\C{H}_{-\infty}$ one has \begin{equation} \label{2.3} ||E_A(\gt)\gf(\ge A)f||\leq c2^\gn\max(1,2^\ga)(\ge\gt)^\ga \gr(\ge\gt)||E_A(\gt)f||. \end{equation} Indeed, we have by hypothesis $|\gf(x)|\leq c|x|^\ga\gr(|x|)$ and, if we denote by $\gc$ the characteristic function of the set $[-2,-1]\cup[1,2]$, then the l.h.s.\ of (2.3) can be estimated as follows \begin{align*} ||\gc(A/\gt)\gf(\ge A)f||&\leq c||\gc(A/\gt)|\ge A|^\ga\gr(\ge|A|)f||\\ &\leq c\max(1,2^\ga)(\ge\gt)^\ga||\gc(A/\gt)\gr (\ge|A|)f||\\ &\leq c\max(1,2^\ga)(\ge\gt)^\ga\sup_{\gt~~t$ or if $s=t$ but $p\leq q$. Now let $k\geq 1$ be an integer. We say that $S\in B(\C{H})$ is of class $C^k(A)$ if the map $S(\,\cdot\,):\D{R}\to B(\C{H})$ (defined above) is of class $\Lip^{(k)}$. This is equivalent with asking that $S(\,\cdot\,)$ be strongly of class $C^k$; if this map is norm $C^k$ we say that $S$ is of class $C_{\text{\rm u}}^k(A)$ (this makes sense and is not trivial even if $k=0$; on the other hand $C^0(A)=B(\C{H}))$. We have $\C{C}^{k,1}(A)\subset C_{\text{\rm u}}^k(A)\subset C^k(A)\subset\C{C}^k(A)$ and, if $A$ is not bounded, these embeddings are strict and optimal on the scale $\C{C}^{s,p}(A)$ (i.e.\ the spaces $\C{C}^{k,p}$ with $1 = \sum_{i+j=k}\frac{k!(-1)^j}{i!j!}\vv. \end{equation} We have $S\in C^k(A)$ if and only if the sesquilinear form $\ad_A^kS$ is continuous for the topology induced by $\C{H}$ on $\C{H}_k$. In this case, and if we denote by $\C{A}^k[S]\equiv \C{A}^kS$ the unique operator in $B(\C{H})$ such that $\vv

=(-1)^k\vv $ for all $f,g\in\C{H}_k$, then we have $\C{A}^kS=(-id/d\gt)^kS(\gt)|_{\gt=0}$ (the rather pedantic notation $\C{A}^kS$ is convenient for later purposes). Now assume that $s=k+\gs$ with $k\geq 1$ integer and $\gs>0$. Then one has $S\in\C{C}^{s,p}(A)$ if and only if $S\in C^k(A)$ and $\C{A}^kS\in \C{C}^{\gs,p}(A)$. Let $S\in\C{C}^{\ga,p}(A)$ for some $\ga >0$, $p\in[1,\infty]$. Then $S$ leaves $\C{H}_{\ga,p}$ invariant and has a canonical extension to a continuous operator $S:\C{H}_{-\ga,p'}\to\C{H}_{-\ga,p'}$ (if $1 0$ and some $p\in[1,\infty]$, then $\gP_\mp S\gP_\pm\C{H}\subset\C{H}_{\ga,p}$. In particular, if $S\in\C{C}^{\ga,2}(A)$ then $\gP_\mp S\gP_\pm\in B(\C{H}_s;\C{H}_{s+\ga})$ for all real $s$ such that $-\ga\leq s\leq 0$. \end{thm} \begin{proof} (i) We first prove a weak-type estimate, namely we show that $S_0\equiv\gP_-S\gP_+$ sends $\C{H}$ into $\C{H}_{m,\infty}$ if $S\in C^m(A)$ for some integer $m\geq 1$. Let $\gc$ be the characteristic function of the real set defined by $1\leq|x|\leq 2$. Then it suffices to show that $||\gc(\ge A)S_0||\leq C\ge^m$ for some constant $C$ and all $0<\ge<1$. Set $S_\gt=\exp(\gt A)S_0\exp(-\gt A)$ for $\gt\geq 0$. Then $\gt\mapsto S_\gt$ is strongly of class $C^m$ on $[0,\infty)$ and its $k$-th order derivative ($0\leq k\leq m$) is equal to $\ad_A^kS_\gt=\exp(\gt A)\gP_-(\ad_A^kS)\gP_+\exp(-\gt A)$. By making a Taylor expansion up to order $m$ we get : \begin{equation*} S_0=\sum_{k=0}^{m-1}\frac{(-1)^k}{k!}\ad_A^kS_1+ \frac{(-1)^m}{(m-1)!} \int_0^1\ad_A^m S_\gt\cdot\gt^{m-1}d\gt. \end{equation*} The operators $\ad_A^k S_1$ clearly send $\C{H}_{-\infty}$ into $\C{H}_{+\infty}$, so it suffices to consider the contribution of the integral term. We have: \begin{align*} &\int_0^1 ||\gc(\ge A)\ad_A^mS_\gt||\gt^{m-1}d\gt \leq ||\ad_A^mS_0||\int_0^1||\gc(\ge A)\gP_-e^{\gt A}||\gt^{m-1} d\gt\leq \\ &\leq||\ad_A^mS_0||\int_0^1 \sup_{x>0}\gc(\ge x)e^{-\gt x} \gt^{m-1}d\gt = ||\ad_A^mS_0|| \int_0^1e^{-\gt/\ge} \gt^{m-1}d\gt \leq C\ge^m, \end{align*} which is the desired estimate. (ii) Let $\C{P}:B(\C{H})\to B(\C{H})$ be the linear continuous operator given by $\C{P}S=\gP_-S\gP_+$. Then $||\C{P}||=1$ and $\C{P}C^m(A)\subset B(\C{H};\C{H}_{m,\infty})$ (by what we have shown above and the closed graph theorem). On the space $C^m(A)$ there is a natural Banach space structure such that the embedding $C^m(A)\subset B(\C{H})$ be continuous. Then one can obtain the spaces $\C{C}^{\ga,p}(A)$ by real interpolation: $\C{C}^{\ga,p}=(C^m(A),B(\C{H}))_{\gq,p}$ with $\gq=1-\ga/m$ if $0<\ga

-\vv $ (with domain $D(A)$) is continuous for the topology induced by $\C{H}$; and then $\C{A}[S]\equiv \C{A}S$ is the unique element of $B(\C{H})$ such that $\vv =\vv -\vv $ for all $f,g\in D(A)$. Clearly $C^k(A)$ coincides with the domain of the power $\C{A}^k$ of $\C{A}$, for each $k\in\D{N}$. Moreover, the following identity holds in $B(\C{H}_k;\C{H}_{-k})$: \begin{equation} \label{2.5} \C{A}^k[S]\equiv\C{A}^kS=(-1)^k\ad_A^kS=\sum_{i+j=k} \frac{k!}{i!j!} (-1)^iA^iSA^j \end{equation} The operator $\C{A}$ can be interpreted as the infinitesimal generator of a one-parameter group of automorphisms of $B(\C{H})$. For each real $\gt$ we define $\C{W}_\gt: B(\C{H})\to B(\C{H})$ by $\C{W}_\gt[S]\equiv\C{W}_\gt S=W_\gt^*SW_\gt$. Then $\C{W}_0=1$, $\C{W}_{\gt+\gs}=\C{W}_\gt\C{W}_\gs$ for all $\gt,\gs\in\D{R}$, and the function $\gt\mapsto\C{W}_\gt S\in B(\C{H})$ is strongly continuous (but not norm continuous in general). For $S\in B(\C{H})$ we have $S\in C^1(A)$ ($=$ domain of $\C{A}$) if and only if $\lim_{\ge\to 0}(i\ge)^{-1}(\C{W}_\ge-1)S$ exists in $B(\C{H})$ in the ultraweak (or weak, or strong) operator topology, and then the limit is just $\C{A}S$ and one has $\C{W}_\gt S=S+i\int_0^\gt\C{W}_\gs\C{A}Sd\gs$ for all $\gt\in\D{R}$. So $\C{A}$ is the infinitesimal generator of the ``weak'' one-parameter group $\{\C{W}_\gt\}_{\gt\in\D{R}}$ in the Banach space $B(\C{H})$ and this justifies the notation $\C{W}_\gt =\exp(i\gt\C{A}$) (the notion of weak semigroup is introduced in [BB]; note that $B(\C{H})$ is identified with the adjoint of the space of trace class operators). We shall define a functional calculus for the operator $\C{A}$ with the help of the group $\{\C{W}_\gt\}_{\gt\in\D{R}}$. Let $\C{M}=\C{M}(\D{R})$ be the unital subalgebra of the algebra $BC(\D{R})$ (bounded continuous complex functions on $\D{R}$) consisting of Fourier transforms of bounded Borel measures. The algebraic operations in $\C{M}$ are those inherited from the embedding $\C{M}\subset BC(\D{R})$, but we take as norm in $\C{M}$ of $\gf(t)=\int_{\D{R}}e^{it\gt}\gm(d\gt)$ the total variation of the measure $\gm$. This makes $\C{M}$ an abelian Banach algebra with unit. If $\gf$ is given by the preceding expression we define a linear continuous operator $\gf(\C{A}):B(\C{H})\to B(\C{H})$ by setting $\gf(\C{A})S\equiv\gf(\C{A})[S]\equiv \int_{\D{R}}W_\gt^*SW_\gt\gm(d\gt)$; the integral exists in the strong operator topology. Observe that the notation $\C{W}_\gt=e^{i\C{A}\gt}$ is consistent with the functional calculus, i.e.\ $\gf(\C{A})=\C{W}_\gt$ if $\gf$ is the function $\gf(t)=e^{it\gt}$. It is easily checked that the map $\gf\mapsto\gf(\C{A})$ is a unital homomorphism such that $(\gf(\C{A})[S])^*=\gf^+(\C{A})[S^*]$ if $\gf^+(t)=\overline{\gf(-t)}$, and that the norm of the operator $\gf(\C{A})$ (acting in the Banach space $B(\C{H})$) is $\leq||\gf||_{\C{M}}$. It is clear that $\gf(\C{A})$ commutes with $\C{A}$, in fact if $k\in\D{N}$ then $\gf(\C{A})C^k(A)\subset C^k(A)$ and $\C{A}^k\gf(\C{A})S=\gf(\C{A})\C{A}^kS$ for $S\in C^k(A)$. On the other hand, if $\gf$ decays at infinity then it improves regularity with respect to $\C{A}$. For $m\geq 1$ integer we define $\gf_{(m)}$ by $\gf_{(m)}(x)=x^m\gf(x)$. Then if $\gf\in\C{M}$ and $\gf_{(m)}\in\C{M}$, we have $\gf(\C{A})C^k(A)\subset C^{k+m}(A)$ for all $k\in\D{N}$ and $\C{A}^m\gf(\C{A})=\gf_{(m)}(\C{A})$. In particular, if $\gf\in C_0^\infty(\D{R})$ then $\gf(\C{A})B(\C{H})\subset C^\infty(A)$. The main purpose of the functional calculus introduced above is to allow us to construct operators of class $C^\infty(A)$ which approximate a given operator $S\in B(\C{H})$ rapidly enough, in a sense that we shall make precise below. Note first that if $\gf\in\C{M}$ and $\ge\in\D{R}$ then the function $x\mapsto\gf(\ge x)$, denoted $\gf^\ge$, belongs to $\C{M}$ and $||\gf^\ge||_{\C{M}}\leq||\gf||_{\C{M}}$ (the equality holds if $\ge\neq 0$). We set $\gf(\ge\C{A})=\gf^\ge(\C{A})$, in other terms $\gf(\ge\C{A})S=\int_{\D{R}}W_{\ge\gt}^*SW_{\ge\gt}\gm(d\gt)$. So for each $S\in B(\C{H})$ the map $\ge\mapsto\gf(\ge\C{A})S\in B(\C{H})$ is strongly continuous and $\gf(0\C{A})S=\gf(0)S$. In particular, if $\gf(0)=1$ and $\gf_{(m)}\in\C{M}$ for some integer $m\geq 1$, then by what we have seen before we have $\gf(\ge\C{A})S\in C^m(A)$ for $\ge\neq 0$ and $\gf(\ge\C{A})S\to S$ in the strong operator topology as $\ge\to 0$. Operators of the form $S_\ge=\gf(\ge\C{A})S$ with $\gf\in C_0^\infty(\D{R})$ and $\gf(0)=1$ will be called {\em regularizations of\/} $S$ ($\ge\neq 0$). The rapidity of the convergence of $S_\ge$ to $S$ is determined by the degree of regularity of $S$ with respect to $A$. We explain this fact in rather rough terms. Assume that $S\in C^k(A)$ for some integer $k\geq 1$ and let $\gf\in\C{M}$ be of the form $\gf(x)=1+x^k\gh(x)$ for some $\gh\in\C{M}$. Then $\gf(\ge\C{A})S=S+\ge^k\C{A}^k\gh(\ge\C{A})S=S+\ge^k\gh(\ge\C{A})\C{A}^kS$ for $\ge\neq 0$ hence $||\gf(\ge\C{A})S-S||\leq|\ge|^k||\gh||_{\C{M}}||\C{A}^kS||$. So if $\gf\in C_0^\infty(\D{R})$ and $\gf(x)-1=O(x^k)$ as $x\to 0$, then for each $S\in C^k(A)$ we have $||\gf(\ge\C{A})S-S||=O(\ge^k)$ as $\ge\to 0$. The case $p=\infty$ of the next theorem says that this behaviour characterizes the class of operators $\C{C}^k(A)$, which is slightly larger than $C^k(A)$. \begin{thm} Let $S\in B(\C{H})$, $s$ a strictly positive real number, and $p\in[1,\infty]$. If there is a function $\gq\in \C{M}(\D{R})$, which is not identically zero on $(0,\infty)$ and on $(-\infty,0)$, such that \begin{equation} \label{2.6} \biggl[\int_0^1 ||\ge^{-s}\gq(\ge\C{A})S||^p \ge^{-1}d\ge\biggr]^{1/p}<\infty \end{equation} then $S\in\C{C}^{s,p}(A)$. Reciprocally, if $S\in\C{C}^{s,p}$ and if $m>s$ is an integer, then $(2.6)$ holds for each $\gq$ such that $\gq^{(k)}\in\C{M}(\D{R})$ for $0\leq k\leq m$ and $\gq^{(k)}(0)=0$ for $0\leq k\leq m-1$. \end{thm} For the proof, see [BG 3]. This theorem characterizes the property $S\in\C{C}^{s,p}(A)$ in terms of the rapidity of the convergence of the regularizations $S_\ge$ of $S$. Let us say that an operator $T\in B(\C{H})$ has $A$-{\em exponential type less than\/} $r$ if there is a holomorphic function $T(\,\cdot\,):\D{C}\to B(\C{H})$ such that $T(\gt)=W_\gt^*TW_\gt$ for $\gt\in\D{R}$ and, moreover, there is a constant $C$ such that $||T(\gz)||\leq C\exp(r|\gz|)$ for all $\gz\in\D{C}$. One may show that $T$ has this property if and only if $\gf(\C{A})T=T$ for all $\gf\in\C{M}$ such that $\gf(x)=1$ on a neighbourhood of the interval $|x|\leq r$. Hence Theorem 2.3 is an extension of classical results of Jackson and Zygmund concerning the best approximation of H\"older-Zygmund functions by trigonometric polynomials. Note also that by taking $\gq(x)=(e^{ix}-1)^m$ we get that $S\in\C{C}^{s,p}(A)$ if and only if $[\int_0^1||\ge^{-s}(\C{W}_{\ge}-1)^mS||^p \ge^{-1}d\ge]^{1/p}<\infty$. \section{Resolvent Families and Mourre Estimates} \label{s:3} \setcounter{equation}{0} \subsection{} \label{s:3.1} A family $\{R(z)\mid z\in\D{C}\setminus\D{R}\}$ of bounded operators in the Hilbert space $\C{H}$ will be called a {\em (self-adjoint) resolvent family} if the following two conditions are satisfied: $R(z_1)-R(z_2)=(z_1-z_2)R(z_1)R(z_2)$ (first resolvent identity) and $R(z)^*=R(\overline{z})$ for all $z_1,z_2,z\in\D{C}\setminus\D{R}$. It follows easily from these relations that the map $R(\,\cdot\,):\D{C}\setminus\D{R}\to B(\C{H})$ is holomorphic and $(d/dz)^kR(z)=k!R(z)^{k+1}$. The {\em spectrum\/} of the resolvent family is the set of real numbers $\gl$ such that the function $R(\cdot)$ has no holomorphic extension to any neighbourhood of $\gl$. Clearly, the first resolvent identity holds for all complex numbers $z_1,z_2$ not in the spectrum of the resolvent family. Note also that we have $||R(z)||\leq|\Im z|^{-1}$ as a consequence of this identity; moreover, one has $\lim_{\ge\to 0}||R(\gl+i\ge)||=\infty$ if and only if $\gl$ belongs to the spectrum of $\{R(z)\}$. It is most convenient to think of resolvent families in terms of possibly non-densely defined self-adjoint operators in $\C{H}$. To be precise, we shall work with a slight extension of the standard notion of self-adjoint operator: for us a {\em self-adjoint operator in $\C{H}$} is a linear operator $H$ defined on a linear subspace $D(H)$ of $\C{H}$ with values in $\C{H}$, such that $HD(H)\subset\overline{D(H)}$ ($=$ closure of $D(H)$ in $\C{H}$) and such that, when considered as operator in the Hilbert space $\overline{D(H)}$, $H$ is self-adjoint in the usual sense (so a {\em densely defined\/} self-adjoint operator is a ``usual'' self-adjoint operator). Note that $H-z:D(H)\to\overline{D(H)}$ is bijective if $\Im z\neq 0$. The resolvent family associated to such an operator is defined by $R(z)f=(H-z)^{-1}f$ if $f\in\overline{D(H)}$ and $R(z)f=0$ if $f$ is orthogonal to $D(H)$. Reciprocally, if $\{R(z)\}$ is a resolvent family then there is a unique self-adjoint operator $H$ such that $R(z)$ be of the above form for $z\in\D{C}\setminus\D{R}$. Note that the spectrum of the densely defined self-adjoint operator $H$ in the Hilbert space $\overline{D(H)}$ coincides with the spectrum of the resolvent family $\{R(z)\}$. If $\gf$ is a complex continuous function on $\D{R}$ which tends to zero at infinity, then $\gf(H)$ is a well-defined bounded operator in $\overline{D(H)}$ (by the functional calculus associated to the densely defined selfadjoint operator $H$ in $\overline{D(H)}$). We extend $\gf(H)$ to a bounded operator on $\C{H}$ by setting $\gf(H)f=0$ if $f$ is orthogonal to $D(H)$. Then clearly we have \begin{equation} \label{3.1} \gf(H)=\wlim_{\ge\to+0}\frac{1}{\gp} \int_{\D{R}}\gf(\gl)\Im R(\gl+i\ge)d\gl \end{equation} where the integral exists in the weak topology. This formula expresses $\gf(H)$ directly in terms of the resolvent family but is not convenient for our purposes here. A more useful representation for $\gf(H)$ can, however, be easily deduced from (3.1). Let $r$ be a strictly positive number. We shall use Taylor's formula for the function $\gm\mapsto R(\gl+i\gm)$ on the interval $[\ge,r]$ with $0<\ge -\vv |\leq C||f||_H||g||_H \quad\forall f,g\in D(A)\cap D(H). \end{equation} \end{prop} \begin{proof} (i) Let $\C{D}$ be the set of $f\in D(A)\cap D(H)$ such that $Hf\in D(A)$. Each bounded operator of class $C^1(A)$ leaves $D(A)$ invariant, hence for any $z\in\D{C}\setminus\gs(H)$ one has $R(z)^2D(A)\subset\C{D}\subset R(z)D(A)$ (note that $\C{D}=R(z)D(A)$ if $H$ is densely defined; moreover, it follows from the first resolvent identity that the spaces $R(z)^2D(A)$ and $R(z)D(A)$ are independent of the choice of $z$). The operator $R(z)^2$ is a continuous surjective map of $\C{H}$ onto $D(H^2)$ (equipped with its graph topology), so it sends a dense subspace (e.g.\ $D(A)$) of $\C{H}$ onto a dense subspace of $D(H^2)$. Hence $R(z)^2D(A)$ is a dense subspace of $D(H^2)$, and so of $D(H)$. In particular $\C{D}$ is dense in $D(H)$. (ii) Let $f,g\in\C{D}$. Then for $z\in\D{C}\setminus\gs(H)$ one has \begin{align} \label{3.4} \vv -\vv &=\vv\\ &\qquad -\vv<(H-\overline{z})f,R(z)A(H-z)g>\notag\\ &=-\vv .\notag \end{align} By taking $z=i$ we get \begin{equation} \label{3.5} |\vv -\vv |\leq ||\C{A}[R(i)]|| \cdot||f||_H||g||_H. \end{equation} Our purpose is to show that this remains true for $f,g\in D(A)\cap D(H)$. (iii) We now point out two relations that will be needed below. If $z_1,z_2\in\D{C}\setminus\gs(H)$ then by applying $\C{A}$ to the first resolvent identity we get \begin{equation} \label{3.6} \C{A}[R(z_1)]\{1+(z_2-z_1)R(z_2)\} =\{1+(z_1-z_2)R(z_1)\}\C{A}[R(z_2)]. \end{equation} On the other hand, we clearly have $\{1+(z_2-z_1)R(z_2)\}\cdot\{1+(z_1-z_2)R(z_1)\}=1$. Hence \begin{equation} \label{3.7} \C{A}[R(z_2)]=\{1+(z_2-z_1)R(z_2)\} \C{A}[R(z_1)]\{1+(z_2-z_1)R(z_2)\}. \end{equation} (iv) For real $\ge\neq 0$ we set $R_\ge=(i\ge)^{-1}R(i/\ge)=(1+i\ge H)^{-1}P$, where $P$ is the orthogonal projection of $\C{H}$ onto $\overline{D(H)}$. By using (3.6) and (3.7) we get \begin{align*} &\C{A}[R_\ge]R_1=\{1+(\ge-1)R_\ge\}\C{A}[R_1]R_\ge,\\ &\ge\C{A}[R_\ge]=\{1+(\ge-1)R_\ge\}\C{A}[R_1]\{1+(\ge-1)R_\ge\}. \end{align*} When $\ge\to 0$ the operator $R_\ge$ converges strongly to $P$. Hence \begin{equation} \label{3.8} \slim_{\ge\to 0}\C{A}[R_\ge]R_1=P^\perp\C{A}[R_1]P, \end{equation} \begin{equation} \label{3.9} \slim_{\ge\to 0}\C{A}[R_\ge] =P^\perp\C{A}[R_1]P^\perp. \end{equation} Now set $S_\ge=(1+i\ge A)^{-1}$ for $\ge\in\D{R}$. Then for each $f\in\C{H}$ one has \begin{equation*} \vv =\vv +\vv , \end{equation*} and this clearly implies \begin{equation} \label{3.10} [S_\ge,R_1]=i\ge S_\ge\C{A}[R_1]S_\ge. \end{equation} (v) Finally, set $J_\ge=R_\ge^2S_\ge$ for $\ge\neq 0$, let $f\in D(A)\cap D(H)$ and denote $f_1=i(H-i)f\in\overline{D(H)}$, so that $f=R_1f_1$. We have $\slim_{\ge\to 0}J_\ge=P$ and (3.10) gives \begin{align*} (H-i)J_\ge f&=(H-i)J_\ge R_1f_1\\ &= (H-i)R_1J_\ge f_1+i\ge(H-i)R_\ge\cdot R_\ge S_\ge\C{A}[R_\ge]S_\ge f_1. \end{align*} When $\ge\to 0$ the operator $i\ge(H-i)R_\ge=\{1+(\ge-1)R_\ge\}P$ converges to zero and $J_\ge f_1\to f_1$, hence $(H-i)J_\ge f\to(H-i)R_1f_1=(H-i)f$. So $J_\ge f\to f$ in $D(H)$. On the other hand the bounded operator $J_\ge$ is clearly of class $C^1(A)$ and $\C{A}[J_\ge]=\C{A}[R_\ge^2]S_\ge$, hence $AJ_\ge f=J_\ge Af-\C{A}[R_\ge^2]S_\ge f$. By using (3.10) again we obtain \begin{align*} \C{A}[R_\ge^2]S_\ge R_1&=\C{A}[R_\ge^2]R_1S_\ge +\C{A}[R_\ge^2]i\ge S_\ge\C{A}[R_1]S_\ge \\ & =\C{A}[R_\ge]R_1R_\ge S_\ge+R_\ge\C{A}[R_\ge]R_1S_\ge \\ &\quad +\ge\C{A}[R_\ge]R_\ge iS_\ge\C{A}[R_1]S_\ge +R_\ge\ge\C{A}[R_\ge]iS_\ge\C{A}[R_1]S_\ge. \end{align*} Now we use (3.8), (3.9) and the relations $R_\ge\to P$, $S_\ge\to 1$ strongly as $\ge\to 0$; we get $\slim_{\ge\to 0}\C{A}[R_\ge^2]S_\ge R_1=P^\perp\C{A}[R_1]P$. Hence we get \begin{equation*} \slim_{\ge\to 0}AJ_\ge f=PAf-P^\perp\C{A}[R(i)](H-i)f, \end{equation*} in particular $||PAJ_\ge f-PAf||\to 0$. (vi) We can now prove the validity of (3.5) for all $f,g\in D(A)\cap D(H)$. (3.5) holds if $f,g$ are replaced by $f_\ge=J_\ge f$ and $g_\ge=J_\ge g$ (because $f_\ge$, $g_\ge$ belong to $\C{D}$). In the inequality obtained in this way we make $\ge\to 0$ and take into account that by what we have proved at step (v) we have: \begin{equation*} \vv=\vv \to \vv =\vv . \qed \end{equation*} \renewcommand{\qed}{} \end{proof} If $H$ is a selfadjoint operator of class $C^1(A)$ we shall denote by $\C{A}[H]$ the unique continuous sesquilinear form on $D(H)$ such that $\vv =\vv -\vv $ for all $f,g\in D(A)\cap D(H)$. If $S,T\in B(\C{H})$ and their ranges are contained in $D(H)$ then $S^*\C{A}[H]T$ will be identified with a bounded operator in $\C{H}$ by using Riesz lemma. It follows easily from (3.4) and from the argument at step (vi) of the preceding proof that \begin{equation} \label{3.11} \C{A}[R(z)]=-R(z)\C{A}[H]R(z) \text{ if } z\in\D{C}\setminus\gs(H). \end{equation} Now let $\gf,\gy\in C_0^\infty(\D{R})$ real and such that $x\gf(x)=\gy(x)\gf(x)$ for all real $x$. Then $\gf(H)\in C^1(A)$, hence for $f\in D(A)$ we have $\gf(H)f\in D(A)\cap D(H)$ and \begin{align*} \vv<\gf(H)f,i\C{A}[H]\gf(H)f>&=2\Re\vv \\ &=2\Re \vv<\gy (H)\gf(H)f,iA\gf(H)f>\\ &=\vv<\gf(H)f,i\C{A}[\gy(H)]\gf(H)f>. \end{align*} Note that $\gy(H)\in C^1(A)$. So we have $\gf(H)\C{A}[H]\gf(H)=\gf(H)\C{A}[\gy(H)]\gf(H)$. We now define the {\em strict Mourre set $\gm^A(H)$ of $H$ with respect to\/} $A$ as the set of real numbers $\gl$ such that there are a real function $\gf\in C_0^\infty(\D{R})$ with $\gf(\gl)\neq 0$ and a strictly positive real number $a$ such that $\gf(H)i\C{A}[H]\gf(H)\geq a\gf(H)^2$. This is clearly an open subset of $\D{R}$. In non-trivial practical situations it is impossible to find explicitly the set $\gm^A(H)$. For this reason it is useful to introduce the {\em Mourre set $\tilde\gm^A(H)$ of $H$ with respect to\/} $A$, defined as the set of real numbers $\gl$ for which there are a real function $\gf\in C_0^\infty(\D{R})$ with $\gf(\gl)\neq 0$, a strictly positive real number $a$, and a compact operator $K$ in $\C{H}$, such that $\gf(H)i\C{A}[H]\gf(H)\geq a\gf(H)^2+K$. It turns out that in many interesting cases one can describe $\tilde\gm^A(H)$ rather explicitly (this is related to the following invariance property: if $H, H_0$ are self-adjoint operators of class $C_{\text{\rm u}}^1(A)$ and if $(H+i)^{-1}-(H_0+i)^{-1}$ is compact, then $\tilde\gm^A(H)=\tilde\gm^A(H_0)$; see Theorem 7.2.9 in [ABG]). For this reason the next result is important. Note that $\tilde\gm^A(H)$ is an open set and $\gm^A(H)\subset\tilde\gm^A(H)$. \begin{prop} \label{p:3.2} The set $\tilde\gm^A(H)\setminus\gm^A(H)$ does not have accumulation points inside $\tilde\gm^A(H)$ and it consists of eigenvalues of $H$ of finite multiplicity. The spectrum of $H$ in $\gm^A(H)$ is purely continuous. \end{prop} \begin{proof} The assertions of the proposition follow easily (see [M1]) once we have shown that the {\em virial theorem\/} is valid, namely that if $f\in D(H)$ is an eigenvector of $H$ then $\vv =0$. Let $\gf,\gy\in C_0^\infty(\D{R})$ be real functions such that $x\gf(x)\equiv\gy(x)\gf(x)$ and $\gf(\gl)=1$, where $\gl\in\D{R}$ is such that $Hf=\gl f$. Then \begin{align*} \vv &=\vv =\vv \\ &=\vv =\lim_{\ge\to 0} \vv =0. \qed \end{align*} \renewcommand{\qed}{} \end{proof} We shall say that the self-adjoint operator $H$ (or the resolvent family $\{R(z)\}$ associated to it) {\em has a spectral gap\/} if its spectrum is not equal to $\D{R}$. Fix such an $H$, let $\gl_0$ be a real number outside the spectrum of $H$, and set $R=-R(\gl_0)$. Then $R$ is a bounded self-adjoint operator $R:\C{H}\to\C{H}$ and for $\Im z\neq 0$: \begin{equation} \label{3.12} R(z)=(\gl_0-z)^{-1}R[R-(\gl_0-z)^{-1}]^{-1}. \end{equation} \begin{prop} $H$ is of class $C^1(A)$ if and only if $R$ is of class $C^1(A)$. A real number $\gl\neq\gl_0$ belongs to $\gm^A(H)$ (resp.\ $\tilde\gm^A(H)$) if and only if $(\gl_0-\gl)^{-1}$ belongs to $\gm^A(R)$ (resp.\ $\tilde\gm^A(R)$). \end{prop} The proof of this result is straightforward and will not be given; see Proposition 8.3.4 in [ABG] and note that in our context one can replace the class $C_{\text{\rm u}}^1$ by the class $C^1$ (cf.\ Propositions 7.2.5 and 7.2.7 of [ABG] for the case of densely defined operators). \section{The Twisted Hamiltonian} \label{s:4} \setcounter{equation}{0} The main technical estimates of this article will be derived in this section. We consider a bounded everywhere defined self-adjoint operator $H$ in $\C{H}$, we denote by $E$ its spectral measure, and we assume that $H$ is of class $C^1(A)$. Furthermore, {\em we fix a real open set $J$ and a real number $a>0$ such that the following condition is satisfied: there is an open set $J_0$ with $\dist(J,\D{R}\setminus J_0)\equiv \inf\{|x-y|\mid x\in J, y\notin J_0\}=\gd>0$ and there is a number $a_0>a$ such that\/} $E(J_0)i\C{A}[H]E(J_0)\geq a_0E(J_0)$. We shall need a version of the so-called quadratic estimate of Mourre. The proof of the next proposition can be found in \S 4.4 of [BG3]; see [M1], [ABG] for similar results. \begin{prop} \label{p:4.1} Let $\{H_\ge\}_{\ge\geq 0}$ be a family of bounded operators in $\C{H}$ such that $H_0=H$, $||H_\ge-H||\to 0$ and $||\ge^{-1}\Im H_\ge+i\C{A}[H]||\to 0$ as $\ge\to 0$. Then there are strictly positive numbers $\ge_0$, $b$ such that, for each $\ge\in[0,\ge_0]$ and each $z\in\D{C}$ with $\Re z\in J$ and $\Im z>-a\ge$, the operator $H_\ge-z:\C{H}\to\C{H}$ is bijective and its inverse $G_\ge=G_\ge(z)=(H_\ge-z)^{-1}\in B(\C{H})$ satisfies the estimates \begin{equation} \label{4.1} ||G_\ge^{(\pm)}f||^2\leq\pm\frac{1}{a\ge+\Im z} \Im\vv +\frac{b\ge}{(a\ge+\Im z)[\gd^2+(\Im z)^2]}||f||^2 \end{equation} for all $f\in\C{H}$. We have set $G_\ge^{(+)}=G_\ge$, $G_\ge^{(-)}=G_\ge^*$. In particular, one has \begin{equation} \label{4.2} ||G_\ge(z)||\leq \frac{1}{a\ge+\Im z} + \left [\frac{b\ge}{(a\ge+\Im z)[\gd^2+(\Im z)^2]}\right ]^{1/2} \end{equation} \end{prop} The following consequences of the inequalities (4.1) and (4.2) will be especially useful later on: if $\Im z\geq 0$ then for $0<\ge\leq\ge_0$ one has \begin{equation} \label{4.3} ||G_\ge^{(\pm)}f||^2\leq\pm\frac{1}{a\ge} \Im\vv +\frac{b}{a\gd^2}||f||^2, \end{equation} \begin{equation} \label{4.4} ||G_\ge||\leq\frac{1}{a\ge}+\left (\frac{b}{a\gd^2}\right )^{1/2}. \end{equation} Now let us assume that the family $\{H_\ge\}$ from Proposition 4.1 has two more properties: (1) $H_\ge$ is of class $C^1(A)$ if $0<\ge<\ge_0$; (2) the map $\ge\mapsto H_\ge\in B(\C{H})$ is strongly $C^1$ on $(0,\ge_0)$. \\ Let $z$ be a complex number with $\Re z\in J$ and $\Im z\geq 0$ and let $0<\ge<\ge_0$. Then $G_\ge\in C^1(A)$ and $\C{A}[G_\ge]=-G_\ge\C{A}[H_\ge]G_\ge$. Indeed, if for $\gt\neq 0$ we set $A_\gt=(i\gt)^{-1}(e^{i\gt A} -1)$ then we clearly have $[A_\gt,G_\ge]=G_\ge[H_\ge,A_\gt]G_\ge$ and the result follows by taking the limit as $\gt\to 0$ and by using, for example, the fact that $[H_\ge,A_\gt]\to\C{A}[H_\ge]$ strongly as $\gt\to 0$. Furthermore, the map $\ge\mapsto G_\ge\in B(\C{H})$ is strongly $C^1$ on $(0,\ge_0)$ and its derivative is given by $G_\ge'\equiv\frac{d}{d\ge}G_\ge=-G_\ge H_\ge'G_\ge$ (this is an easy consequence of (4.4)). In particular we get \begin{equation} \label{4.5} G_\ge' =\C{A}[G_\ge]+G_\ge (\C{A}[H_\ge]-H_\ge')G_\ge. \end{equation} This equation plays a fundamental role in the theory. In this paper we shall choose $H_\ge$ (for $\ge\in\D{R}$) of the form $H_\ge=\gx(\ge\C{A})H$, where $\gx$ is a function of the form $\gx(x)=e^x\gq(x)$ with $\gq\in C_0^\infty(\D{R})$ real even and such that $\gq(x)=1$ on a neighbourhood of zero (a rather detailed motivation of this choice can be found in [BGSh2]). Note that the operator $H_\ge$ is not self-adjoint in general and that we have $H_\ge^*=H_{-\ge}$. We shall also need the function $\gh$ given by $\gh(x)=x(\gx(x)-\gx'(x))=-e^xx\gq'(x)$, so that $\gh\in C_0^\infty(\D{R}\setminus\{0\})$. Formally, (4.5) becomes: \begin{equation} \label{4.6} G_{\ge}' =\C{A}[G_\ge]+\ge^{-1}G_\ge\gh(\ge \C{A})[H]G_\ge. \end{equation} It is not yet clear whether the so-defined family $\{H_\ge\}$ satisfies or not the hypotheses of Proposition 4.1. In fact it does not if $H$ is only of class $C^1(A)$, as we explain in the Proposition 4.2. We first state a lemma which can be proven without difficulty and which will be needed below. \noindent {\bf Lemma.} {\em Let $\gf\in\C{M}$ be a function of class $C^1$ and such that its derivative $\gf'$ and the function $\tilde\gf(x)=x\gf'(x)$ belong to $\C{M}$. Then for each $S\in B(\C{H})$ the map $\ge\mapsto \gf(\ge\C{A})S\in B(\C{H})$ is strongly $C^1$ on $\D{R}\setminus\{0\}$, for $\ge\neq 0$ the operator $\gf'(\ge\C{A})S$ is of class $C^1(A)$, and we have $(d/d\ge)\gf(\ge\C{A})S =\ge^{-1}\tilde\gf(\ge\C{A})S=\C{A}\gf'(\ge\C{A})S$. In particular, if $\gf\in C_0^\infty(\D{R})$ then $\gf^{(k)}(\ge\C{A})S\in C^\infty(A)$ if $\ge\neq 0$ and $k\in\D{N}$, the map $\ge\mapsto\gf(\ge\C{A})S\in B(\C{H})$ is of class $C^\infty$ on $\D{R}\setminus\{0\}$ and we have $(d/d\ge)^k\gf(\ge\C{A})S=\C{A}^k\gf^{(k)}(\ge\C{A})S$.} \begin{prop} \label{p:4.2} The family $\{H_\ge\}_{\ge\in\D{R}}$ defined above satisfies the hypotheses of Proposition $4.1$ if and only if the operator $H$ is of class $C_{\text{\rm u}}^1(A)$. Assume that $H\in C_{\text{\rm u}}^1(A)$ and let $z\in\D{C}$ with $\Re z\in J$ and $\Im z>0$. {\em(a)} For $0\leq\ge\leq\ge_0$ one has $G_\ge\in C_{\text{\rm u}}^1(A)$ and $\C{A}[G_\ge]=-G_\ge\C{A}[H_\ge]G_\ge$; if $0<\ge<\ge_0$ then $G_\ge \in C^\infty(A)$. {\em(b)} The map $\ge\mapsto H_\ge$ is of class $C^1$ in norm on $\D{R}$ and is of class $C^\infty$ on $\D{R}\setminus\{0\}$. The map $\ge\mapsto G_\ge$ is of class $C^1$ in norm on the closed interval $[0,\ge_0]$, where its derivative is given by $G_{\ge}'=-G_\ge H_{\ge}'G_\ge$, and is of class $C^\infty$ on $(0,\ge_0]$. {\em(c)} Set $K_\ge=\ge^{-1}\gh(\ge\C{A})H$ for $\ge\neq 0$, where $\gh\in C_0^\infty(\D{R}\setminus\{0\})$ is given by $\gh(x)=-e^xx\gq'(x)$. Then $K_\ge\in C^\infty(A)$, $\ge\mapsto K_\ge$ is of class $C^\infty$ on $\D{R}\setminus\{0\}$, and for $0<\ge\leq\ge_0$ one has \begin{equation} \label{4.7} G_{\ge}'=\C{A}[G_\ge]+G_\ge K_\ge G_\ge. \end{equation} {\em(d)} Set $K_\ge^{(j)}=(d/d\ge)^jK_\ge$ and let $\ga >-1$ real and $p\in [1,\infty]$. Then $H$ is of class $\C{C}^{1+\ga,p}(A)$ if and only if the condition \begin{equation} \label{4.8} \Bigl[\int_0^1||\ge^{-\ga+j}K_\ge^{(j)}||^p\ge^{-1}d\ge \Bigr]^{1/p}<\infty \end{equation} holds for $j=0$. If this is the case then $(4.8)$ holds for each integer $j\geq 0$. \end{prop} \begin{proof} We define a real even function $\gr\in C_0^\infty(\D{R})$ by $\gr(0)=1$ and $\gr(x)=x^{-1}\sh x\cdot\gq(x)$ if $x\neq 0$. Then for an arbitrary bounded self-adjoint operator $H$ we have $\ge^{-1}\Im H_\ge^* =i\C{A}\gr (\ge\C{A})H\equiv S_\ge$. Assume first that $\lim_{\ge\to 0}S_\ge$ exists in norm in $B(\C{H})$ and denote by $S$ the limit. Since $C^\infty(A)$ is a subspace of the norm-closed space $C_{\text{\rm u}}^0(A)$ and $S_\ge\in C^\infty(A)$ if $\ge\neq 0$, we get $S\in C_{\text{\rm u}}^0(A)$. For $f\in D(A)$ we have \begin{equation*} \vv =\vv =2\Re\vv<(\gr(\ge\C{A})H)f,iAf> \end{equation*} which converges to $2\Re\vv $ as $\ge\to 0$. So we have $2\Re\vv =\vv $ for $f\in D(A)$, i.e.\ $i\C{A}H=S\in C_{\text{\rm u}}^0(A)$. This clearly means $H\in C_{\text{\rm u}}^1(A)$. Reciprocally, if $H\in C_{\text{\rm u}}^1(A)$ then $H$ is of class $C_{\text{\rm u}}^0(A)$ hence $||H_\ge-H||\to 0$ as $\ge\to 0$. Moreover, we shall also have $S_\ge =i\gr(\ge\C{A})\C{A}H$ and $\C{A}H\in C_{\text{\rm u}}^0(A)$, so $||S_\ge-i\C{A}H||\to 0$ as $\ge\to 0$. Hence the family $\{H_\ge\}_{\ge\geq 0}$ satisfies the hypotheses of Proposition 4.1. The proof of the assertions (a), (b) and (c) is easy, see the Lemma stated before Proposition 4.2. For part (d) we use the Theorem 2.3. Observe that $\gh\in C_0^\infty(\D{R}\setminus\{0\})$ is not identically zero on $(-\infty,0)$ and on $(0,\infty)$, so if (4.8) holds with $j=0$ then $H\in\C{C}^{1+\ga,p}(A)$. Reciprocally, if $H$ has this property then we have (4.8) for all $j$ because $\ge^jK_\ge^{(j)}=\gh_j(\ge\C{A})H$ for some $\gh_j\in C_0^\infty(\D{R}\setminus\{0\})$. \end{proof} We denote by $|||\cdot|||$ either the norm in the Banach space $\C{K}=\C{H}_{1/2,1}$ or the norm associated to it in $B(\C{K};\C{K}^*)$, and we recall that we have continuous embeddings $\C{K}\subset\C{H}\subset\C{K}^*$ and $B(\C{H})\subset B(\C{K};\C{K}^*)$. {\em From now on we assume that $H$ is (at least) of class\/} $\C{C}^{1,1}(A)$. We write $z=\gl+i\gm$ and the numbers $\ge,\gl,\gm$ are supposed to verify $0<\ge<\ge_0$, $\gl\in J$, $\gm>0$. One should think of $\gm$ rather as a parameter, but it is important that the various constants that appear below are independent of $\gm$. If $F$ is a function of $(\gl,\ge)\in J\times(0,\ge_0)$ we denote by $F^{(k,m)}\equiv\partial_\gl^k\partial_\ge^m F$ its derivative of order $k$ with respect to $\gl$ and of order $m$ with respect to $\ge$. We also set $F^{(m)}=F^{(0,m)}$. The operator $G_\ge=G_\ge(z)=G_\ge(\gl+i\gm)$ will be considered as a function of $(\gl,\ge)\in J\times(0,\ge_0)$; we clearly have for $k\in\D{N}$: \begin{equation} \label{4.9} G_\ge^{(k,0)} =\partial_\gl^kG_\ge(\gl+i\gm) =k!G_\ge^{k+1}. \end{equation} \begin{prop} \label{p:4.3} If $H$ is of class $\C{C}^{1,1}(A)$ then for each $k,m\in\D{N}$ there is a number $C<\infty$, independent of $\ge\in(0,\ge_0)$, $\gl\in J$ and $\gm>0$, such that \begin{equation} \label{4.10} |||G_\ge^{(k,m)}|||\leq C\ge^{-k-m}, \end{equation} \begin{equation} \label{4.11} ||G_\ge^{(k,m)}||_{\C{K}\to\C{H}} +||G_\ge^{(k,m)}||_{\C{H}\to\C{K}^*}\leq C\ge^{-k-m-1/2}. \end{equation} \end{prop} \begin{proof} (i) We first prove (4.10), (4.11) in the case $k=m=0$. Fix a number $\ge_1\in [0,\ge_0)$ and a family $\{f_\ge\}_{\ge_1<\ge\leq \ge_0}$ of vectors in $D(A)$ such that the function $\ge\mapsto f_\ge\in\C{H}$ is of class $C^1$. We set $F_\ge=\vv $ for $\ge_1<\ge\leq\ge_0$ and we get by using (4.7): \begin{equation*} F_\ge'=\vv +\vv +\vv . \end{equation*} Denote $\ell_\ge=||f_\ge'||+||Af_\ge||$. Then (4.3) implies \begin{align*} |F_\ge'|&\leq\ell_\ge(||G_\ge f_\ge||+ ||G_\ge^*f_\ge||)+||K_\ge||\cdot ||G_\ge f_\ge||\cdot||G_\ge^*f_\ge||\\ &\leq 2\ell_\ge a^{-1/2}(\ge^{-1/2}|F_\ge|^{1/2}+b^{1/2}\gd^{-1} ||f_\ge||)+||K_\ge||a^{-1}(\ge^{-1}|F_\ge|+b\gd^{-2}||f_\ge||^2). \end{align*} So there is a constant $c>0$, depending only on $a,b$ and $d$, such that for $\ge_1<\ge\leq\ge_0$: \begin{equation*} c^{-1}|F_\ge'|\leq\ell_\ge||f_\ge||+||K_\ge||\cdot||f_\ge||^2 +\ell_\ge\ge^{-1/2}|F_\ge|^{1/2}+||K_\ge||\ge^{-1}|F_\ge|. \end{equation*} According to Lemma 7.A.1 from [ABG] the preceding estimate implies \begin{align} \label{4.12} |F_{\ge_1}|&\leq 2\Bigl\{|F_{\ge_0}|+c\int_{\ge_1}^{\ge_0} [\ell_\gt||f_\gt ||+||K_\gt||\cdot||f_\gt||^2]d\gt \\ &\qquad+c^2\Bigl[\int_{\ge_1}^{\ge_0}\ell_\gt\gt^{-1/2}d\gt\Bigr]^2\Bigr\} \exp\int_{\ge_1}^{\ge_0}c||K_\gt||\gt^{-1}d\gt.\notag \end{align} By Proposition 4.2 (d) we have $\int_0^{\ge_0}||K_\gt||\gt^{-1}d\gt$ if and only if $H\in\C{C}^{1,1}(A)$. Now let $f\in\C{H}_{1/2,1}$ and $f_\ge=\gq((\ge-\ge_1)A)f$, with the same function $\gq$ as in the definition of $H_\ge$. If we set $\tilde\gq(x)=x\gq'(x)$ and $\gq_{(1)}(x)=x\gq(x)$, then \begin{align*} \int_{\ge_1}^{\ge_0}\ell_\gt\gt^{-1/2}d\gt &=\int_0^{\ge_0-\ge_1}(||\tilde\gq(\gs A)f||+||\gq_{(1)}(\gs A)f||) \frac{d\gs}{\gs(\gs +\ge_1)^{1/2}}\\ &\leq c'||f||_{\C{H}_{1/2,1}}=c'|||f||| \end{align*} where $c'$ is a finite constant depending only on $\ge_0$ and $\gq$. Now by using (4.12) we easily see that there is a constant $c''<\infty$ such that $|\vv |\leq c''|||f|||^2$ for $0<\ge\leq\ge_0$, $\gl\in J$, $\gm>0$ and $f\in\C{K}$. The polarization identity will then give $|||G_\ge|||\leq\text{const}$. Finally, the estimate (4.11) with $k=m=0$ is an immediate consequence of the preceding one and of (4.3). (ii) Now we treat the case where one of the numbers $k,m$ is not zero. If $m=0$ then the estimates follow easily from those with $k=m=0$ by taking into account (4.4) and (4.9), so we can assume $m\geq 1$. Then by Proposition 4.2 (b) the operator $G_\ge^{(m)}$ is a linear combination of terms of the form $G_\ge H_\ge^{(m_1)}G_\ge H_\ge^{(m_2)}\dots G_\ge H_\ge^{(m_n)}$ with $m_1,\dots,m_n\geq 1$ integers and $m_1+\dots+m_n=m$. So from (4.9) it follows that $G_\ge^{(k,m)}$ is a linear combination of terms of the form \begin{equation*} G_\ge^{k_0+1}H_\ge^{(m_1)}G_\ge^{k_1+1}H_\ge^{(m_2)} G_\ge^{k_2+1}\dots H_\ge^{(m_n)}G_\ge^{k_n+1} \end{equation*} with $m_1,\dots,m_n$ as above and $k_0,k_1,\dots,k_n\in\D{N}$ such that $k_0+k_1+\dots+k_n=k$. The norm in $B(\C{K};\C{K}^*)$ of such a term is bounded by \begin{align*} &||G_\ge||_{\C{H}\to\C{K}^*}||G_\ge||^{k_0}||H_\ge^{(m_1)})||\cdot ||G_\ge||^{k_1+1}\dots ||H_\ge^{(m_n)}||\cdot||G_\ge||^{k_n}||G_\ge||_{\C{K}\to\C{H}}\\ &\leq\text{const.}\,\ge^{-1/2}\cdot\ge^{-k_0}||H_\ge^{(m_1)}|| \cdot\ge^{-k_1-1} \dots||H_\ge^{(m_n)}||\ge^{-k_n}\cdot\ge^{-1/2} \end{align*} where we have used (4.11) with $k=m=0$ and (4.4). Similarly, the norm in $B(\C{K};\C{H})$ is bounded by \begin{align*} &||G_\ge||^{k_0+1}||H_\ge^{(m_1)}||\cdot||G_\ge||^{k_1+1}\dots ||H_\ge^{(m_n)}||\cdot||G_\ge||^{k_n}||G_\ge||_{\C{K}\to\C{H}}\leq\\ &\leq\text{const.}\ge^{-k_0-1}||H_\ge^{(m_1)}||\cdot\ge^{-k_1-1} \dots||H_\ge^{(m_n)}||\cdot\ge^{-k_n}\cdot\ge^{-1/2}. \end{align*} We see that the assertions of the proposition are a consequence of the estimate $||H_\ge^{(m)}||\leq c_m\ge^{1-m}$ for $m\geq 1$ integer and $\ge>0$. But we have \begin{align*} H_\ge^{(m)}&=\partial_\ge^m \gx(\ge\C{A})H=\C{A}^m\gx^{(m)} (\ge\C{A})H\\ &=\ge^{1-m}(\ge\C{A})^{m-1}\gx^{(m)}(\ge\C{A})\C{A}H =\ge^{1-m}\gf(\ge\C{A})\C{A}H. \end{align*} where $\gf(x)=x^{m-1}\gx^{(m)}(x)$ is a function of class $C_0^\infty(\D{R})$. Hence \begin{equation*} ||H_\ge^{(m)}||\leq \ge^{1-m}||\gf||_{\C{M}}||\C{A}H||.\qed \end{equation*} \renewcommand{\qed}{} \end{proof} \begin{lem} \label{l:4.4} Set $\tilde G_\ge=G_\ge K_\ge G_\ge$, where $K_\ge$ is as in Proposition {\em 4.2 (c)\/}. Then for each $k,m\in\D{N}$ there is a finite constant $C$, independent of $\ge,\gl,\gm$, such that \begin{equation} \label{4.13} |||\tilde G_\ge^{(k,m)}|||\leq C\ge^{-k-m-1} \sum_{j=0}^m||\ge^jK_\ge^{(j)}||. \end{equation} In particular, if $H\in\C{C}^{1+\ga}(A)$ for some $\ga>0$, then we have $|||\tilde G_\ge^{(k,m)}|||\leq c\ge^{\ga -k-m-1}$. \end{lem} \begin{proof} By Leibnitz formula, and since $K_\ge$ does not depend on $\gl$, $\tilde G_\ge^{(k,m)}$ is a linear combination of terms of the form $G_\ge^{(a,u)}K_\ge^{(w)}G_\ge^{(b,v)}$ with $a,b,u,v,w\in\D{N}$ and $a+b=k$, $u+v+w=n$. Then Proposition 4.3 implies \begin{align*} |||G_\ge^{(a,u)}K_\ge^{(w)}G_\ge^{(b,v)}||| &\leq ||G_\ge^{(a,u)}||_{\C{H}\to\C{K}^*}||K_\ge^{(w)}|| \cdot||G_\ge^{(b,v)}||_{\C{K}\to\C{H}}\\ &\leq\text{const.}\ge^{-a-u-1/2}||K_\ge^{(w)}||\cdot\ge^{-b-v-1/2}\\ &=\text{const.}\ge^{-k-m-1}||\ge^wK_\ge^{(w)}||.\qed \end{align*} \renewcommand{\qed}{} \end{proof} For the proof of the next estimates we need a generalization of the identity (4.7). Assume that we are under the hypotheses of Proposition 4.2 and let $\tilde G_\ge=G_\ge K_\ge G_\ge$. Then for all $\ell,k\in\D{N}$ with $k\geq 1$ and all $\ge\in(0,\ge_0)$, $z=\gl+i\gm$, $\gl\in J$, $\gm>0$ we have \begin{equation} \label{4.14} G_\ge^{(\ell,k)}=\ell!\C{A}^k[G_\ge^{\ell+1}] +\sum_{r=0}^{k-1}\C{A}^{k-r-1}[\tilde G_\ge^{(\ell,r)}]. \end{equation} If $\ell=0$, $k=1$ this is just (4.7). (4.14) follows from this special case by taking successively derivatives with respect to $\ge$ and $\gl$ and by using the following simple result: {\em Let $[a,b]$ be a real interval and $\{S_x\}_{a\leq x\leq b}$ a family of bounded operators on $\C{H}$ having the following properties: {\rm (i)} $x\mapsto S_x\in B(\C{H})$ is strongly of class $C^1$, with derivative $S_x'=\partial_xS_x$; {\rm (ii)} $S_x$ and $S_x'$ are of class $C^1(A)$ for all $x\in[a,b]$; {\rm (iii)} $x\mapsto\C{A}S_x'\in B(\C{H})$ is strongly continuous. \\ Then the map $x\mapsto\C{A}S_x\in B(\C{H})$ is strongly $C^1$ and its derivative is given by\/} $\partial_x\C{A}S_x=\C{A}S_x'$. Now let us fix two functions $\gf,\gy\in\C{S}(\D{R})$ and let us define the operator $L_\ge\equiv L_\ge(z):\C{H}_{-\infty}\to\C{H}_{+\infty}$ by \begin{equation} \label{4.15} L_\ge(z)=\gf(\ge A)G_\ge(z)\gy(\ge A) \end{equation} for $0<\ge<\ge_0$ and $z=\gl+i\gm$ with $\gl\in J$ and $\gm>0$. Let $\ell,m\in\D{N}$. By using Leibnitz formula and by taking into account the relation $\partial_\ge^i\gf(\ge A)=A^i\gf^{(i)}(\ge A)=\ge^{-i}\gf_i(\ge A)$ with $\gf_i(x)=x^i\gf^{(i)}(x)$ we obtain \begin{equation*} L_\ge^{(\ell,m)}=\sum_{i+j+k=m}\frac{m!}{i!j!k!} \ge^{k-m}\gf_i(\ge A)G_\ge^{(\ell,k)}\gy_j(\ge A), \end{equation*} where the indices $i,j,k$ run over $\D{N}$. If we use (4.14) the expression in the r.h.s.\ above becomes \begin{align*} L_\ge^{(\ell,m)}&=\sum_{i+j+k=m} \frac{\ell!m!}{i!j!k!} \ge^{k-m}\gf_i(\ge A)\C{A}^k[G_\ge^{\ell+1}]\gy_j(\ge A)\\ &\quad +\sum_{ \substack{ i+j+k=m\\ k\geq 1\\ n+r=k-1}} \frac{m!}{i!j!k!} \ge^{k-m}\gf_i(\ge A)\C{A}^n[\tilde G_\ge^{(\ell,r)}]\gy_j(\ge A). \end{align*} Then by taking into account the identity (2.5) we get \begin{align} \label{4.16} &\ge^mL_\ge^{(\ell,m)}=\sum_{i+j+p+q=m} \frac{\ell!m!}{i!j!p!q!} (-\ge A)^p(\ge A)^i\gf^{(i)}(\ge A)G_\ge^{\ell+1} (\ge A)^{j+q}\gy^{(j)}(\ge A)\\ &\quad +\sum_{i+j+p+q+r=m-1} \frac{m!(p+q)!(-1)^p\ge^{r+1}}{i!j!p!q!(m-i-j)!} (\ge A)^{i+p}\gf^{(i)}(\ge A)\tilde G_\ge^{(\ell,r)}(\ge A)^{j+q} \gy^{(j)}(\ge A).\notag \end{align} \begin{prop} \label{p:4.5} Let $\gf,\gy\in\C{S}(\D{R})$ and let $L_\ge=L_\ge(z)$ be defined by $L_\ge=\gf(\ge A)G_\ge\gy(\ge A)$. Then for each $\ell,m\in\D{N}$ there is a constant $C$, independent of $\ge,\gl,\gm$, such that for all $f,g\in\C{H}_{-\infty}$: \begin{align} \label{4.17} |\vv | &\leq C \sum_{ \substack{a+b=m\\ 0\leq i\leq a\\ 0\leq j\leq b} } |||\gf_{i,a}(\ge A)g|||\cdot|||\gy_{j,b}(\ge A)f||| \notag\\ &\quad +C\sum_{ \substack{a+b+c\leq m-1\\ 0\leq i\leq a\\ 0\leq j\leq b} } |||\gf_{i,a}(\ge A)g|||\cdot|||\gy_{j,b}(\ge A)f||| \cdot||\ge^cK_\ge^{(c)})||. \end{align} Here the functions $\gf_{i,a}$ and $\gy_{j,b}$ are defined by $\gf_{i,a}(x)=x^a\gf^{(i)}(x)$ and $\gy_{j,b}(x)=x^b\gy^{(j)}(x)$. \end{prop} \begin{proof} We use (4.16) and the estimates \begin{equation*} |||\ge^\ell G_\ge^{\ell+1}|||\leq C(\ell) \text{ and } |||\ge^{\ell+r+1}\tilde G_\ge^{(\ell,r)}|||\leq C(\ell,r)\sum_{0\leq c\leq r}||\ge^cK_\ge^{(c)}|| \end{equation*} which have been obtained in Proposition 4.3 and Lemma 4.4. \end{proof} It is clear that the first sum from (4.16) becomes much simpler if $\gf$ is a function such that $\gf^{(i)}(x)=\gf(x)$ for all $x$. But the only function which has this property is $\gf(x)=e^x$ and it does not belong to $\C{S}(\D{R})$. However, one can circumvent this difficulty if in place of $L_\ge$ one considers the operator $\gP_-L_\ge$, where $\gP_-=E_A((-\infty,0])$ is the spectral projection of $A$ associated with the interval $(-\infty,0]$. Then we take a function $\gf\in\C{S}(\D{R})$ such that $\gf(x)=e^x$ if $x\leq 0$. Observe that for $j,q$ fixed with $n=m-j-q>0$ one has $\sum_{i+p=n}(i!p!)^{-1}(-x)^px^i=0$. Hence, after left multiplication by $\gP_-$ of (4.16), in the first sum on the r.h.s.\ will remain only terms with $j+q=m$, so $i=p=0$. On the other hand: \begin{equation} \label{4.18} \sum_{j+q=m}\frac{m!}{j!q!} x^{j+q}\gy^{(j)}(x)=x^m\Bigl(1+\frac{d}{dx}\Bigr)^m\gy(x)\equiv\gz(x). \end{equation} Hence we obtain: \begin{align*} &\ge^m\gP_-L_\ge^{(\ell,m)} =\ell !\gP_-e^{\ge A}G_\ge^{\ell+1}\gz(\ge A)\\ &\quad +\sum_{i+j+p+q+r=m-1} \frac{m!(p+q)!(-1)^p\ge^{r+1}}{i!j!p!q!(m-i-j)!} \gP_-(\ge A)^{i+p}e^{\ge A} \tilde G_\ge^{(\ell,r)}(\ge A)^{j+q}\gy^{(j)}(\ge A). \end{align*} By the same argument as in the proof of Proposition 4.5 we get, with a slight change of notation: \begin{prop} \label{p:4.6} Let $\gy\in\C{S}(\D{R})$, define $\gz$ by $(4.18)$, and set $L_\ge=\gP_-e^{\ge A}G_\ge\gy (\ge A)$. Then for each $\ell,m\in\D{N}$ there is a constant $C$, independent of $\ge,\gl,\gm$, such that for all $f,g\in\C{H}_{-\infty}$: \begin{align} \label{4.19} &|\vv |\leq C||| \gP_-e^{\ge A}g|||\cdot|||\gz(\ge A)f||| \\ &\quad +C\sum_{ \substack{ a+b+c\leq m-1\\ 0\leq j\leq b} } |||\gP_-(\ge A)^ae^{\ge A}g|||\cdot |||(\ge A)^b\gy^{(j)}(\ge A)f|||\cdot||\ge^cK_\ge^{(c)}||.\notag \end{align} \end{prop} This estimate can be further simplified by a special choice of $\gy$. Note that if $\gy(x)=e^{-x}$ then $\gz=0$. Of course this choice is not allowed by the condition $\gy\in\C{S}(\D{R})$. However, if we take $\gy$ of class $\C{S}(\D{R})$ and such that $\gy(x)=e^{-x}$ if $x\geq 0$, then $\gP_+\gz(\ge A)f=0$ for each $f\in\C{H}_{-\infty}$. Hence Proposition 4.6 immediately implies the next one. Here $\gP_+=E_A([0,\infty))$. \begin{prop} \label{p:4.7} Let $L_\ge =\gP_-e^{\ge A}G_\ge e^{-\ge A}\gP_+$. Then for each $\ell,m\in\D{N}$ with $m\geq 1$ there is $C<\infty$, independent of $\ge,\gl,\gm$, such that for all $f,g\in\C{H}_{-\infty}$: \begin{align} \label{4.20} &|\vv |\leq \\ &\leq C\sum_{a+b+c\leq m-1} |||\gP_-(\ge A)^ae^{\ge A}g|||\cdot |||\gP_+(\ge A)^be^{-\ge A}f|||\cdot||\ge^cK_\ge^{(c)}||.\notag \end{align} \end{prop} \section{Boundary Values of Resolvent Families} \label{s:5} \setcounter{equation}{0} \subsection{} \label{s:5.1} Throughout this section $\{R(z)\}$ is a resolvent family on the Hilbert space $\C{H}$, we denote by $H$ the self-adjoint operator associated to it, and {\em we assume that $H$ has a spectral gap (its spectrum $\gs(H)$ is not the whole real line)}. We shall make several hypotheses concerning the regularity class of $H$ with respect to $A$, but these hypotheses will always imply that $H$ is $A$-regular (i.e.\ of class $\C{C}^{1,1}(A)$). In particular the open real set $\gm^A(H)$ is well defined and contains $\D{R}\setminus\gs(H)$. If $f\in\C{H}$ then $z\mapsto\vv $ is a well defined holomorphic map on the open complex set $\D{C}\setminus\gs(H)$ and this set contains the upper ($\D{C}_+$) and lower ($\D{C}_-$) half-planes (we set $\D{C}_\pm =\{z\in\D{C}\mid\pm\Im z>0\})$. Our first purpose is to prove the existence of the limits $\lim_{\gm\to\pm 0}\vv \equiv\vv $ for $\gl\in\gm^A(H)$ and to discuss the continuity and differentiability properties of the maps $\gl\mapsto\vv $ in terms of the regularity properties of $H$ and $f$ with respect to $A$. Due to the relation $\vv ^*=\vv $ we may restrict ourselves to the case $\gm\to+0$. Note also that, due to the polarization identity, it is not necessary to consider the case of $\vv $ with $g\neq f$. \begin{thm} \label{t:5.1} Assume that $H$ is of class $\C{C}^{1+\ell,1}(A)$ for some integer $\ell\geq 0$ and set $s=\ell+1/2$. Then for each $f\in\C{H}_{s,1}$ the holomorphic map $\D{C}_+\ni z\mapsto\vv $ extends to a function of class $C^\ell$ on $\D{C}_+\cup\gm^A(H)$, i.e.\ for each integer $0\leq k\leq\ell$ the holomorphic function on $\D{C}_+$ given by $(d/dz)^k\vv =\vv $ has a continuous extension to $\D{C}_+\cup\gm^A(H)$. The limit $\lim _{\gm\to+0}\vv \equiv\vv $ exists locally uniformly in $\gl\in\gm^A(H)$, the boundary value function $\gl\mapsto\vv $ is of class $C^\ell$ on $\gm^A(H)$, and for $0\leq k\leq\ell$ integer one has \begin{equation} \label{5.1} \frac{d^k}{d\gl^k}\vv = \lim_{\gm\to+0}\vv \end{equation} locally uniformly in $\gl\in\gm^A(H)$. \end{thm} \begin{proof} (i) We first show that it suffices to prove the theorem under the assumption that $H$ is a bounded everywhere defined operator. For this we use the identity (3.12) which can be written $\vv =\gz \vv $, where $\gz=(\gl_0-z)^{-1}$. The map $z\mapsto \gz$ is a holomorphic diffeomorphism of $\D{C}\setminus\{\gl_0\}$ onto $\D{C}\setminus\{0\}$ which leaves $\D{C}_+$ (and $\D{C}_-$) invariant and, by Proposition 3.3, restricts to a $C^\infty$ diffeomorphism of $\gm^A(H)\setminus\{\gl_0\}$ onto $\gm^A(R)\setminus\{0\}$. The operator $R$ belongs to $\C{C}^{1+\ell,1}(A)$, hence $Rf\in\C{H}_{s,1}$ (see the discussion before Theorem 2.2). So, by taking into account the polarization identity, it suffices to prove the theorem with $H$ replaced by $R$, which is bounded. (ii) From now on we assume that $H$ is a bounded (everywhere defined) operator. By considering a small enough neighbourhood $J$ of a point from $\gm^A(H)$, we may assume that the hypotheses made at the beginning of Section 4 are satisfied. For the rest of the proof we use the notations and the results of Section 4. Let $L_\ge =L_\ge(z)=\gf(\ge A)G_\ge(z)\gf(\ge A)$ where $\gf$ is a function in $\C{S}(\D{R})$ with $\gf(0)=1$ and $0\leq\ge\leq\ge_0$, $z=\gl+i\gm$ with $\gl\in J$, $\gm>0$. Clearly \begin{equation} \label{5.2} L_\ge^{(\ell,0)}=\partial_\gl^\ell L_\ge =(\frac{d}{dz})^\ell \gf(\ge A)G_\ge(z)\gf(\ge A)=\gf(\ge A)\ell! G_\ge(z)^{\ell+1}\gf(\ge A). \end{equation} Note that by Proposition 4.2 (b) the map $\ge\mapsto L_\ge^{(\ell,0)}\in B(\C{H})$ is strongly $C^1$ on the closed interval $[0,\ge_0]$ and $L_0^{(\ell,0)} =\partial_z^{\ell}R(z)=\ell!R(z)^{\ell+1}$. Now let us fix $f\in\C{H}_{s,1}$ and define $h(\ge)=\vv $ for $0\leq\ge\leq \ge_0$. Then for $\ge>0$ and $m\geq 0$ integer we have $h^{(m)}(\ge)=\vv $ which can be estimated as in (4.17). So there is $C<\infty$, independent of $\ge,\gl,\gm$ and $f$, such that \begin{align} \label{5.3} |\ge^mh^{(m)}(\ge)| &\leq C\sum_{ \substack{ a+b=m\\ i\leq a\\ j\leq b} } \ge^{-\ell} |||\gf_{i,a}(\ge A)f||| \cdot|||\gf_{j,b}(\ge A)f||| \\ &\quad+C|||f|||^2 \sum_{0\leq j\leq m-1}\ge^{-\ell}||\ge^jK_\ge^{(j)}||.\notag \end{align} By Proposition 4.2 (d) the condition $H\in\C{C}^{1+\ell,1}(A)$ is equivalent to the integrability with respect to the measure $\ge^{-1}d\ge$ on $(0,\ge_0)$ of the second term in the r.h.s.\ of (5.3). We claim that if $m>2\ell$ then each term of the first sum from (5.3) is also integrable (with respect to the same measure). Indeed, if $a+b=m$ then either $a>\ell$ or $b>\ell$. In the first case we have \begin{align*} \int_0^1\ge^{-\ell}|||\gf_{i,a}(\ge A)f||| \cdot|||\gf_{j,b}(\ge A)f|||\ge^{-1}d\ge &\leq C'|||f|||\int_0^1||\ge^{-\ell} \gf_{i,a}(\ge A)f||_{1/2,1}\ge^{-1}d\ge\\ &\leq C''|||f|||\cdot||f||_{s,1} \end{align*} due to the Theorem 2.1 (observe that $\gf_{i,a}$ has a zero of order $\geq a>\ell$ at the origin). Let us fix an integer $m>2\ell$. We have seen that there is a function $\gc:(0,\ge_0)\to\D{R}$, independent of $\gl$ and $\gm$, such that $|\ge^mh^{(m)}(\ge)|\leq\gc(\ge)$ and $\int_0^{\ge_0}\gc(\ge)\ge^{-1}d\ge<\infty$. So we can apply Lemma 5.2 (see below) and thus obtain \begin{equation} \label{5.4} \vv = \sum_{k=0}^{m-1} \frac{(-\ge_0)^k}{k!}\vv +\frac{(-1)^m}{(m-1)!}\int_0^{\ge_0}\vv \ge^{m-1}d\ge. \end{equation} According to Proposition 4.1, for each $\ge\in[0,\ge_0]$ the function $z\mapsto G_\ge(z)=(H_\ge-z)^{-1}$ is holomorphic in the region $\gl\in J$, $\gm>-a\ge$, where $a>0$. So each term in the sum from (5.4) extends to a holomorphic function of $z$ below the real axis if $\Re z\in J$ (see (4.16) for example). For the integral in (5.4) we can use the dominated convergence theorem in order to deduce that its limit as $\gm\to+0$ exists uniformly in $\gl\in J$. We have shown that $\lim_{\gm\to 0}\vv $ exists uniformly in $\gl\in J$. Clearly the arguments still work if $\ell$ is replaced by a smaller integer. \end{proof} In the preceding proof we used the following elementary fact: \begin{lem} \label{l:5.2} Let $h:(0,\ge_0]\to\D{C}$ be a function of class $C^m$ for some integer $m\geq 1$ and some real $\ge_0>0$. Assume that $\int_0^{\ge_0}|\ge^{m-1}h^{(m)}(\ge)|d\ge<\infty$. Then $\lim_{\ge\to 0}h(\ge)\equiv h(0)$ exists and \begin{equation} \label{5.5} h(0)=\sum_{k=0}^{m-1} \frac{(-\ge_0)^k}{k!}h^{(k)}(\ge_0)+ \frac{(-1)^m}{(m-1)!}\int_0^{\ge_0}h^{(m)}(\ge)\ge^{m-1}d\ge. \end{equation} \end{lem} It is convenient to reformulate Theorem 5.1 in slightly different terms. For an arbitrary self-adjoint operator $H$ the map $z\mapsto R(z)\in B(\C{H})$ is holomorphic on $\D{C}_+$. Recall that we have continuous embeddings \begin{equation} \label{5.6} B(\C{H})\subset B(\C{K};\C{K}^*)\subset B(\C{H}_{s,1};\C{H}_{-s,\infty}) \end{equation} if $s\geq 1/2$. So, for example, $z\mapsto R(z)\in B(\C{K};\C{K}^*)$ is a holomorphic map on $\D{C}_+$. Now assume that $H\in\C{C}^{1,1}(A)$, i.e.\ the hypothesis of Theorem 5.1 holds with $\ell=0$. Then the theorem says that the preceding function extends to a weak* continuous function on $\D{C}_+\cup \gm^A(H)$, in fact $\lim_{\gm\to+0}R(\gl+i\gm)\equiv R(\gl+i0)\in B(\C{K};\C{K}^*)$ exists in the weak* topology of $B(\C{K};\C{K}^*)$, locally uniformly in $\gl\in\gm^A(H)$. So the boundary value function $\gl\mapsto R(\gl+i0)\in B(\C{K};\C{K}^*)$ is well defined and weak* continuous on $\gm^A(H)$. According to (5.6), we may consider the map $\gl\mapsto R(\gl+i0)\in B(\C{H}_{s,1};\C{H}_{-s,\infty})$ for each $s\geq 1/2$; clearly it is a weak* continuous function (recall that $\C{H}_{-s,\infty}=\C{H}_{s,1}^*$, which defines the weak* topology of the preceding space). Now assume that $H\in\C{C}^{1+\ell,1}(A)$ for some integer $\ell\geq 1$. Then the Theorem 5.1 says that the map $\gl\mapsto R(\gl+i0)\in B(\C{H}_{s,1};\C{H}_{-s,\infty})$ is of class $C^\ell$ on $\gm^A(H)$ in the weak* topology if $s=\ell+1/2$. Moreover its weak* derivatives are given by \begin{equation} \label{5.7} \frac{d^k}{d\gl^k}R(\gl+i0)=\lim_{\gm\to+0} k!R(\gl+i\gm)^{k+1}\equiv k!R^{k+1}(\gl+i0) \end{equation} where the limit exists in the weak* topology of $B(\C{H}_{s,1};\C{H}_{-s,\infty})$, locally uniformly in $\gl\in\gm^A(H)$. Our next purpose is to describe the regularity properties of the function $\gl\mapsto R(\gl+i0)$ in terms of the classes $\gL^\ga$. For the proof of the next result we need the following lemma (proved in [BG3]): \begin{lem} \label{l:5.3} Let $J\subset\D{R}$ be an open set, $\ge_0>0$ a real number and $\tilde J=\{(\gl,\ge)\in\D{R}^2\mid\gl\in J, 0<\ge<\ge_0\}$. Let $F:\tilde J\to\D{C}$ be a function of class $C^m$ for some integer $m\geq 1$ and assume that there are real numbers $\gs,M$, with $0<\gs 0$, such that $\sum_{\ell+k=m}|\partial_\gl^\ell\partial_\ge^k F(\gl,\ge)|\leq M\ge^{\gs-m}$ on $\tilde J$. Then the limit $\lim_{\ge\to 0}F(\gl,\ge)\equiv F_0(\gl)$ exists uniformly in $\gl\in J$ and the function $F_0:J\to\D{C}$ is locally of class $\gL^\gs$. Moreover, there is a constant $C_m$ (depending only on $m$) such that \begin{equation} \label{5.8} |[(T_\gn-1)^mF_0](\gl)|\leq C_mM\gs^{-1}|\gn|^\gs \end{equation} if $\gl\in J$ and $\gn\in\D{R}$ have the properties $|\gn |<\ge_0$ and $\gl+t\gn\in J$ for all $t\in[0,m]$. In $(5.8)$ the translation operator $T_\gn$ acts according to $(T_\gn g)(\gl)=g(\gl+\gn)$. \end{lem} Moreover, we shall need the following particular case of the Theorem 2.1: {\em if $\gc:\D{R}\to\D{C}$ is a bounded Borel function and $s=\ga+1/2$ is a real number $>1/2$, and if $\gc$ has a zero of order $>\ga$ at the origin (i.e.\ $|\gc(x)|\leq c|x|^\gb$ for some $\gb>\ga$), then there is a constant $C<\infty$ such that for all\/} $\ge>0$: \begin{equation} \label{5.9} ||\gc(\ge A)||_{\C{H}_{s,\infty}\to\C{H}_{1/2,1}} +||\gc(\ge A)||_{\C{H}_{-1/2,\infty}\to\C{H}_{-s,1}} \leq C\ge^\ga. \end{equation} \begin{thm} \label{t:5.4} Let $H$ be of class $\C{C}^{1+\ga}(A)$ for some real $\ga>0$ and let us set $s=\ga+1/2$. Then the function \begin{equation} \label{5.10} \gm^A(H)\ni\gl\mapsto R(\gl+i0)\in B(\C{H}_{s,\infty};\C{H}_{-s,1}) \end{equation} is locally of class $\gL^\ga$. \end{thm} \begin{proof} (i) As explained in the first part of the proof of Theorem 5.1 it is sufficient to consider the case when $H$ is a bounded (everywhere defined) operator. From now on we keep the notations and assumptions of the part (ii) of the proof of Theorem 5.1. We first prove that for each $\ell,m\in\D{N}$ with $m>2\ga$ we have \begin{equation} \label{5.11} ||L_\ge^{(\ell,m)}||_{\C{H}_{s,\infty}\to\C{H}_{-s,1}} \leq C(\ell,m)\ge^{\ga-\ell-m} \end{equation} for a number $C(\ell,m)<\infty$ independent of $\ge\in(0,\ge_0)$, $\gl\in J$ and $\gm>0$. For this purpose we use the Proposition 4.5. Note that for each term of the first sum on the r.h.s.\ of (4.17) we have either $a>\ga$ or $b>\ga$. If, for example $a>\ga$, we use the estimate (5.9) with $c=\gf_{i,a}$ and get that the corresponding term is bounded by a constant times $\ge^\ga||g||_{s,\infty}|||f|||$, and this is better than needed (because $s>1/2$). A typical term of the second sum on the r.h.s.\ of (4.17) is dominated by $\text{const}\cdot|||g|||\cdot|||f|||\cdot||\ge^cK_\ge^{(c)}||$ and now we may use Proposition 4.2 (d). (ii) Now let $f\in\C{H}_{s,\infty}$ and $F(\gl,\ge)=\vv $. Then (5.11) gives \begin{equation} \label{5.12} |\partial_\gl^\ell\partial_\ge^mF(\gl,\ge)| \leq C(\ell,m)||f||_{s,\infty}^2\ge^{\ga-\ell-m}. \end{equation} This implies the hypothesis of Lemma 5.3, namely $|\partial_\gl^\ell\partial_\ge^kF(\gl,\ge)| \leq M\ge^{\ga-m}$ if $\ell+k=m$, with $M=\text{const.}||f||_{s,\infty}^2$. Indeed, if $\ell=0$ this is a particular case of (5.12). If $\ell\geq 1$ we integrate (5.12) $\ell$ times with respect to $\ge$ over an interval of the form $(\gt,\ge_0)$ with $0<\gt<\ge_0$; since $\ga-m<0$ we shall get $|\partial_\gl^\ell\partial_\gt^{m-\ell}F(\gl,\gt)|\leq M\gt^{\ga-m}$, which is the estimate we were looking for. Now we use Lemma 5.3. Since $F_0=\vv $ and $\C{H}_{s,\infty}=(\C{H}_{-s,1})^*$, the estimate (5.8) implies the assertion of the theorem. \end{proof} \subsection{} \label{s:5.2} $\C{K}^*=\C{H}_{-1/2,\infty}$ is the smallest space in the Besov scale associated to $A$ which contains the set $R(\gl+i0)\C{H}_\infty$ (if $\gl\in J$ is a spectral value of $H$). We show now that the operator $\gP_-R(\gl+i0$) behaves much better (similar assertions hold for $\gP_+R(\gl-i0)$). Here $\gP_-=E_A((-\infty,0])$ extends to a continuous operator in $\C{H}_{-\infty}$ which leaves invariant each $\C{H}_{s,p}$; hence the product $\gP_{-}R(\gl+i0)$ is well defined and belongs to $B(\C{K};\C{K}^*)$. Note that under the conditions of Theorem 5.5 we have $R(z)\C{H}_{s,p}\subset \C{H}_{s,p}$, hence the r.h.s.\ of (5.13) makes sense. \begin{thm} \label{t:5.1} Assume that $H$ is of class $\C{C}^{s+1/2,p}(A)$ for some real number $s>1/2$ and some $p\in[1,\infty]$. Then for all $\gl\in\gm^A(H)$ one has $\gP_-R(\gl+i0)\C{H}_{s,p}\subset\C{H}_{s-1,p}$. Let $\ell\geq 0$ be an integer such that $\ell $ is of class $C^\ell$ on $\gm^A(H)$ and one has \begin{equation} \label{5.13} \frac{d^\ell}{d\gl^\ell}\vv =\lim_{\gm\to+0}\vv<\gP_-g,\ell!R(\gl+i\gm)^{\ell+1}f> \end{equation} where the limit exists locally uniformly in $\gl\in\gm^A(H)$. \end{thm} \begin{proof} Exactly as in the proof of Theorem 5.1 it suffices to consider the case where $H$ is a bounded operator. Then, $J$ being chosen as in (ii) of the proof of Theorem 5.1, we may assume that the assumptions of Section 4 are satisfied. Let $L_\ge$ be the operator introduced in Proposition 4.6, where $\gy$ is assumed to have the property $\gy(0)=1$. We set $h(\ge)=\vv $ for $0<\ge\leq\ge_0$ and some given vectors $f\in\C{H}_{s,p}$ and $g\in\C{H}_{1+\ell-s,p'}$. Here $p'$ is defined by $1/p+1/p'=1$. Then (4.19) gives \begin{align} \label{5.14} |\ge^mh^{(m)}(\ge)|&\leq C\ge^{-\ell}|||\gP_-e^{\ge A}g||| \cdot|||\gz(\ge A)f|||\\ &\quad +C|||f|||\sum_{i+j\leq m-1}\ge^{-\ell} |||\gP_-(\ge A)^ie^{\ge A}g|||\cdot||\ge^jK_\ge^{(j)}||\notag \end{align} where $C$ is a constant independent of $\ge$, $\gl$, $\gm$, $f$ and $g$. We choose $m>\ga\equiv s-1/2$. Then the integral over the interval $(0,1)$ with respect to the measure $\ge^{-1}d\ge$ of the first term on the r.h.s.\ of (5.14) is bounded by \begin{align*} &C\left [\int_0^1|||\ge^{\ga -\ell}\gP_-e^{\ge A}g|||^{p'} \ge^{-1}d\ge\right ]^{1/p'}\cdot \left [\int_0^1|||\ge^{-\ga}\gz(\ge A)f|||^p\ge^{-1}d\ge\right ]^{1/p} \leq\\ &\leq C'||g||_{1/2-\ga+\ell,p'}||f||_{1/2+\ga,p}. \end{align*} We have used the Theorem 2.1 which is allowed by the fact that $\ga-\ell>0$, $0<\ga $, which is holomorphic on $\D{C}_+$, extends to a function of class $C^\ell$ on $\D{C}_+\cup J$ (in a sense explained in the statement of Theorem 5.1). In particular, if we take $\ell=0$ we see that $\lim_{\gm\to+0}\vv<\gP_-g,R(\gl+i\gm)f>$ exists (uniformly in $\gl\in J$) for each $f\in\C{H}_{s,p}$ and $g\in\C{H}_{1-s,p'}$. Now recall that $\C{H}_{s-1,p}=(\C{H}_{1-s,p'})^*$ if $1 |$ with $f\in\C{H}_{s,p}$ and $g\in\C{H}_{-t,q'}$. Then the second part of Theorem 5.5 can be expressed as follows: the map $\gl\mapsto\gP_-R(\gl+i0)\in B(\C{H}_{s,p};\C{H}_{s-\ell-1,p})$ is of class $C^\ell$ in the $w$-topology. \begin{thm} \label{t:5.6} Let $s,\ga$ be real numbers such that $0<\ga

s-1/2\equiv\gb$ there is a number $C(\ell,m)$, independent of $\ge,\gl,\gm$, such that \begin{equation} \label{5.16} ||L_\ge^{(\ell,m)}||_{\C{H}_{s,\infty}\to\C{H}_{s-1-\ga,1}} \leq C(\ell,m)\ge^{\ga-\ell-m}. \end{equation} We use Proposition 4.6. Then (5.9) with $\gc=\gz$ (which vanishes of order $m>\gb$ at the origin, see (4.18)) implies $|||\gz(\ge A)f|||\leq C'\ge^\gb||f||_{s,\infty}$. On the other hand the Theorem 2.1 implies for $\gb-\ga>0$: \begin{equation} \label{5.17} \ge^{\gb-\ga}|||\gP_-e^{\ge A}g||| \leq C''||g||_{1/2-\gb +\ga,\infty} =C''||g||_{1-s+\ga,\infty}. \end{equation} Hence the first term on the r.h.s.\ of (4.19) is bounded by a constant times $\ge^\ga||g||_{1-s+\ga,\infty}||f||_{s,\infty}$. Now we bound the terms of the sum from (4.19) by using $|||(\ge A)^b\gy^{(j)}(\ge A)f||| \leq C'|||f|||\leq C''||f||_{s,\infty}$ and Proposition 4.2 (d). We shall get terms of the form $C'''\ge^\gb|||\gP_-(\ge A)^ae^{\ge A}g|||\cdot||f||_{s,\infty}$. By an estimate similar to (5.17) (use the Theorem 2.1 again) we finally obtain \begin{equation*} |\vv | \leq C\ge^\ga||g||_{1-s+\ga,\infty}||f||_{s,\infty}. \end{equation*} This implies (5.16) because $\C{H}_{1-s+\ga,\infty}=(\C{H}_{s-1-\ga,1})^*$. (ii) Let $F(\gl,\ge)=\vv $ with $f\in\C{H}_{s,\infty}$ and $g\in\C{H}_{1+\ga -s,\infty}$. If $\ell,m\geq 0$ are integers and $m>\gb$ then (5.16) gives \begin{equation*} |\partial_\gl^\ell\partial_\ge^mF(\gl,\ge)| \leq C(\ell,m)||f||_{s,\infty} ||g||_{1+\ga -s,\infty}\ge^{\ga-\ell-m}. \end{equation*} Now the proof can be finished as in the case of Theorem 5.4. \end{proof} \subsection{} \label{s:5.3} If $f\notin\C{H}_{1/2,1}$ then $\gP_-R(\gl+i0)f$ has no meaning in general. However, one can give a sense to this expression if $E_A((-\infty,a))f=0$ for some $a\in\D{R}$. \begin{thm} \label{t:5.1} Assume that $H$ is of class $\C{C}^{1+\ga,r}(A)$ with $\ga>0$ real and $r\in[1,\infty]$. Let $\ell\in\D{N}$ with $\ell<\ga$, let $s$ be a real number such that $1/2-(\ga-\ell)\leq s\leq 1/2$, and let us denote $t=s-1+(\ga-\ell)$, so that $-1/2\leq t\leq -1/2+(\ga-\ell)$. Finally, let $f\in\C{H}_{s,p}$ and $g\in\C{H}_{-t,q'}$ where $p,q\in[1,\infty]$ are such that {\em(i)} if $s=1/2-(\ga-\ell)$ then $p=r'$ and $q=\infty$; {\em(ii)} if $s=1/2$ then $p=1$ and $q=r$; {\em(iii)} if $1/2-(\ga-\ell) ~~$ extends to a function of class $C^\ell$ on $\D{C}_+\cup\gm^A(H)$ and one has \begin{equation} \label{5.18} \frac{d^\ell}{d\gl^\ell}\vv<\gP_-g,R(\gl+i0)\gP_+f> =\lim_{\gm\to+0}\vv<\gP_-g,\ell!R(\gl+i\gm)^{\ell+1}\gP_+f> \end{equation} where the limit exists locally uniformly in $\gl\in\gm^A(H)$. \end{thm} \begin{proof} (i) As usual we reduce ourselves to the case when $H$ is a bounded operator, but this time the argument is slightly more involved. With the notations of part (i) of the proof of Theorem 5.1, we write \begin{align*} \vv<\gP_-g,R(z)\gP_+f>&=\gz \vv<\gP_-g,(R-\gz)^{-1}R\gP_+f>\\ &=\gz\vv<\gP_-g,(R-\gz )^{-1}\gP_+R\gP_+f>+\gz \vv<\gP_-g,(R-\gz)^{-1}\gP_-R\gP_+f>. \end{align*} The first term in the last member here is easy to treat (because $R\gP_+f\in\C{H}_{s,\infty}$ if $f\in\C{H}_{s,\infty})$. For the last term we first use Theorem 2.2, which gives $\gP_-R\gP_+f\in\C{H}_{t+3/2}$ if $f\in\C{H}_{s,\infty}$. In conclusion, for the rest of the proof we can assume that $H$ is bounded and that the hypotheses of Section 4 are fulfilled. (ii) Let $L_\ge$ be as in Proposition 4.7 and let us set $h(\ge)=\vv~~$ with $f\in\C{H}_{s,p}$ and $g\in\C{H}_{-t,q'}$. Then, according to (4.20), for each integer $m\geq 1$ (in fact this time one can take $m=1$, which simplifies the proof, but not significantly) we have \begin{align} \label{5.19} &|\ge^mh^{(m)}(\ge)|\leq\\ &\leq C\sum_{a+b+c\leq m-1} |||\ge^{\ga-\ell-\gs}\gP_-(\ge A)^ae^{\ge A}g||| \cdot|||\ge^\gs\gP_+(\ge A)^be^{-\ge A}f||| \cdot||\ge^{-\ga+c}K_\ge^{(c)}||,\notag \end{align} where $\gs$ is the real number defined by $s=1/2-\gs$, so that $0\leq\gs\leq\ga-\ell$. If $\gs=0$ (case (ii) of the theorem) then we bound a term in the sum from (5.19) by \begin{equation*} C'|||\ge^{\ga-\ell}\gP_-(\ge A)^ae^{\ge A}g||| \cdot|||f|||\cdot||\ge^{-\ga+c}K_\ge^{(c)}||. \end{equation*} Then by using Theorem 2.1 we obtain $|\ge^mh^{(m)}(\ge)|\leq\gc(\ge)$, where $\gc(\ge)$ is independent of $\gl$ and $\gm$, and \begin{equation*} \int_0^{\ge_0}|\gc(\ge)|\ge^{-1}d\ge \leq C''|||f|||\cdot||g||_{1/2-\ga +\ell,r'}. \end{equation*} If $\gs =\ga-\ell$ (case (i) of the theorem) we estimate a typical term in the r.h.s.\ of (5.19) by \begin{equation*} C'|||g|||\cdot| ||\ge^{\ga-\ell}\gP_+(\ge A)^be^{-\ge A}f|||\cdot ||\ge^{-\ga+c}K_\ge^{(c)}||. \end{equation*} Then as above we get \begin{equation*} \int_0^{\ge_0}|\gc(\ge)|\ge^{-1}d\ge \leq C''|||g|||\cdot||g||_{1/2-\ga+\ell,r'}. \end{equation*} Finally, if $0<\gs<\ga-\ell$ we can use Theorem 2.1 for each of the factors in (5.19) which contains $f$ or $g$. So the relation $1/q'+1/p+1/r=1$ and the H\"older inequality with three factors will give \begin{equation*} \int_0^{\ge_0}|\gc(\ge)|\ge^{-1}d\ge \leq C''||g||_{1/2-\ga+\ell+\gs,q'}||f||_{1/2-\gs,p}. \end{equation*} Since $1/2-\ga+\ell+\gs=1-s-(\ga-\ell)=-t$, we can finish the proof as usual (see the proof of Theorem 5.1). \end{proof} We shall reformulate Theorem 5.7 as follows. We know that for each real $s$ with $|s|<1+\ga$ and each $p\in[1,\infty]$ the operator $R(z)$ has a canonical extension to a bounded operator in $\C{H}_{s,p}$ if $z\in\D{C}_+$, and the map $z\mapsto R(z)\in B(\C{H}_{s,p})$ is holomorphic. If $s,p,t,q$ are as in Theorem 5.7 then $t ~~0$. Let $\gb ,s,t$ be real numbers such that $0<\gb<\ga$, $1/2-(\ga -\gb)\leq s\leq 1/2$ and $t=s-1+(\ga -\gb )$, so that $-1/2\leq t\leq -1/2+(\ga -\gb)$. Finally, let $p,q\in[1,\infty]$ be such that {\em(i)} if $s=1/2-(\ga-\gb)$ then $p=q=\infty$; {\em(ii)} if $s=1/2$ then $p=q=1$; {\em(iii)} if $1/2-(\ga-\gb)~~~~$ where $L_\ge$ is as in Proposition 4.7 and $f\in\C{H}_{s,p}$, $g\in\C{H}_{-t,q'}$. As in the proof of Theorems 5.4 and 5.6 we shall need the following estimate: for $\ell,m\in\D{N}$ there is a constant $C(\ell,m)$, independent of $\ge\in(0,\ge_0)$, $\gl\in J$ and $\gm>0$, such that \begin{equation} \label{5.22} ||L_\ge^{(\ell,m)}||_{\C{H}_{s,p}\to\C{H}_{t,q}}\leq C(\ell,m)\ge^{\gb-\ell-m}. \end{equation} In order to prove this we use the inequality established in Proposition 4.7. Each term in the r.h.s.\ of (4.20) is of the form $|||\gf(\ge A)g|||\cdot|||\gy(\ge A)f|||\cdot||\ge^cK_\ge^{(c)}||$ where $\gf,\gy\in\C{S}(\D{R})$ but do not vanish at zero in general. By Proposition 4.2 (d) such a term is bounded by a constant times \begin{equation} \label{5.23} \ge^\ga|||\gf(\ge A)g|||\cdot|||\gy(\ge A)f||| =\ge^\gb|||\ge^{\ga-\gb-\gs} \gf(\ge A)g|||\cdot|||\ge^\gs\gy(\ge A)f||| \end{equation} where $\gs$ could be an arbitrary real number. If $0<\gs<\ga-\gb$ then the r.h.s.\ of (5.23) can be estimated with the help of Theorem 2.1. We clearly get a bound of the form $c\ge^\gb||g||_{1/2-\ga+\gb+\gs,\infty}||f||_{1/2-\gs,\infty}$. We set $s=1/2-\gs$ and we obtain (5.22) by a simple argument. 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