\input amstex \loadbold \documentstyle{amsppt} \magnification=1200 \baselineskip=15 pt \NoBlackBoxes \TagsOnRight \topmatter \title $\boldkey L^{\boldkey p}$ Norms of the Borel Transform and the Decomposition of Measures \endtitle \rightheadtext{$L^{p}$ Norms of the Borel Transform and the Decomposition of Measures} \author B.~Simon$^{*}$ \endauthor \leftheadtext{B.~Simon} \affil Division of Physics, Mathematics, and Astronomy \\ California Institute of Technology, 253-37 \\ Pasadena, CA 91125 \endaffil \thanks $^{*}$ This material is based upon work supported by the National Science Foundation under Grant No.~DMS-9101715. The Government has certain rights in this material. \endthanks \thanks To appear in {\it Proc.~Amer.~Math.~Soc.} \endthanks \endtopmatter \document \bigpagebreak \block {\smc{Abstract.}} We relate the decomposition over $[a,b]$ of a measure $d\mu$ (on $\Bbb R$) into absolutely continuous, pure point, and singular continuous pieces to the behavior of integrals $\int\limits ^{b}_{a}(\text{Im}\,F(x+i\epsilon))^{p}\,dx$ as $\epsilon\downarrow 0$. Here $F$ is the Borel transform of $d\mu$, that is, $F(z)=\int (x- z)^{-1}\,d\mu(x)$. \endblock \vskip 0.4in \flushpar {\bf \S 1. Introduction} Given any positive measure $\mu$ on $\Bbb R$ with $$\int\frac{d\mu(x)}{1+|x|} < \infty, \tag 1.1$$ one can define its Borel transform by $$F(z)=\int\frac{d\mu(x)}{x-z}. \tag 1.2$$ We have two goals in this note. One is to discuss the relation of the decomposition of $\mu$ into components ($d\mu = d\mu_{\text{\rom ac}} +d\mu_{\text{\rom pp}}+d\mu_{\text{\rom sc}}$ with $d\mu_{\text{\rom ac}} (x)=g(x)\,dx$, $d\mu_{\text{\rom pp}}$ a pure point measure, and $d\mu_{\text{\rom sc}}$ a singular continuous measure) to integrals of powers of $\text{Im}\,F(x+i\epsilon)$. This is straightforward, and global results (e.g., involving $\int\limits^{\infty}_{-\infty}|\text{Im}\, F(x+i\epsilon)|^{2}\,dx$) are well-known to harmonic analysts (see, e.g., Koosis [5, pg.~157])---but there seems to be a point in writing down elementary proofs of the local results (e.g., involving $\int\limits^{b} _{a}|\text{Im}\,F(x+i\epsilon)|^{2}\,dx$). Secondly, by proper use of these theorems, we can simplify the proofs in [7] that certain sets of operators are $G_\delta$'s in certain metric spaces. In \S 2, we will see that $\int\limits^{b}_{a}|\text{Im}\,F(x+ i\epsilon)|^{p}\,dx$ with $p>1$ is sensitive to singular parts of $d\mu$ and can be used to prove they are absent. In \S 3, we see the opposite results when $p<1$ and the singular parts are irrelevant, so that integrals can be used for a test of whether $\mu_{\text{\rom ac}} =0$. Finally, in \S 4, we turn to the aforementioned results on $G_\delta$ sets of operators. Since we only discuss $\text{Im}\,F(z)$ and $$\text{Im}\,F(x+i\epsilon)=\epsilon\int\frac{d\mu(y)}{(x-y)^{2}+ \epsilon^{2}}, \tag 1.3$$ our results actually hold if (1.1) is replaced by $$\int\frac{d\mu(x)}{(1+|x|)^{2}}<\infty. \tag 1.4$$ It is a pleasure to thank S.~Jitomirskaya, A.~Klein, and T.~Wolff for valuable discussions. \bigpagebreak \flushpar {\bf \S 2. $\boldkey p$-norms for $\boldkey p \boldkey> \boldkey 1$} \medpagebreak \proclaim{Theorem 2.1} Fix $p>1$. Suppose that $$\sup\limits_{0<\epsilon<1} \int\limits^{b}_{a}|\text{\rom{Im}}\,F(x+i \epsilon)|^{p}\,dx <\infty. \tag 2.1$$ Then $d\mu$ is purely absolutely continuous on $(a, b)$, $\frac{d\mu_ {\text{\rom{ac}}}}{dx}\in L^{p}(a, b)$; and for any $[c, d]\subset (a,b)$, $\frac{1}{\pi}\,\text{\rom{Im}}\,F(x+i\epsilon)$ converge to $\frac{d\mu_{\text{\rom{ac}}}}{dx}$ in $L^p$. Conversely, if $[a,b] \subset (e,f)$ with $d\mu$ purely absolutely continuous on $(e, f)$, and $\frac{d\mu_{\text{\rom{ac}}}}{dx}\in L^{p}(e, f)$, then \rom(2.1\rom) holds. \endproclaim \remark{Remarks} 1. This criterion with $p=2$ is used by Klein [4], who has a different proof. 2. The $p=2$ results can be viewed as following from Kato's theory of smooth perturbations [2,6]. 3. It is easy to construct measures supported on $\Bbb R\backslash (a, b)$ so that (2.1) fails or so that the $L^p$ norm oscillates, for example, suitable point measures $\sum\alpha_{n}\delta_{x_n}$ with $x_{n}\uparrow a$. For this reason, we are forced to shrink/expand $(a,b)$ to $(c,d)/(e,f)$. \endremark \demo{Proof} Let $d\mu_{\epsilon}(x)=\pi^{-1}\text{Im}\,F(x+i \epsilon)\,dx$. Then [8] $d\mu_{\epsilon}\to d\mu$ weakly, as $\epsilon\downarrow 0$, that is, $\lim\limits_{\epsilon\downarrow 0} \int f(x)\,d\mu_{\epsilon}(x)=\int f(x)\,d\mu(x)$ for $f$ a continuous function of compact support. Let $q$ be the dual index to $p$ and $f$ a continuous function supported in $(a, b)$. Then \align \biggl|\int f\,d\mu\biggr| &=\lim\limits_{\epsilon\downarrow 0}\, \biggl|\int f\,d\mu_{\epsilon}\biggr| \\ &\leq \varlimsup\limits_{\epsilon\downarrow 0}\, \biggl[\int\limits ^{b}_{a} |f(x)|^{q}\,dx\biggr]^{1/q}\, \biggl[\int\limits^{b}_{a} \biggr(\frac{1}{\pi}\,\text{Im}\,F(x+i\epsilon)\biggr)^{p}\,dx \biggr]^{1/p} \\ &\leq C \|f\|_{q}. \endalign Thus, $f\mapsto\int f\,d\mu$ is a bounded functional on $L^q$, and thus $\chi_{(a, b)}\,d\mu = g\,dx$ for some $g\in L^{p}(a, b)$. We claim that when $\chi_{(a, b)}\,d\mu=g\,dx$ with $g\in L^{p}(a, b)$, then for any $[c, d]\subset (a, b)$, $\frac{1}{\pi}\,\text{Im}\,F(x+ i\epsilon)\to g$ in $L^{p}(c, d)$---this implies the remaining parts of the theorem. To prove the claim, write $F=F_{1}+F_{2}$ where $F_1$ comes from $d\mu_{1}\equiv\chi_{(a, b)}\,d\mu$ and $d\mu_{2}=(1-\chi_{(a,b)})\, d\mu$. $\frac{1}{\pi}\,\text{Im}\,F_{1}$ is a convolution of $g\,dx$ with an approximate delta function. So, by a standard argument, $\frac {1}{\pi}\,\text{Im}\,F_{1}\to g$ in $L^p$. On the other hand, since $\text{dist}([c,d], \Bbb R\backslash (a,b))>0$, one easily obtains a bound: $$|\text{Im}\,F_{2}(x+i\epsilon)|\leq C\epsilon \qquad \text{for } x\in [c,d].$$ So $\frac{1}{\pi}\,\text{Im}\,F_{2}\to 0$ in $L^p$. \qed \enddemo The following is a local version of Wiener's theorem: \proclaim{Theorem 2.2} $$\lim\limits_{\epsilon\downarrow 0}\,\epsilon\int\limits^{b}_{a}|\text {\rom{Im}}\,F(x+i\epsilon)|^{2}\,dx=\frac{\pi}{2}\biggl(\frac{1}{2}\, \mu(\{a\})^{2}+\frac{1}{2}\,\mu(\{b\})^{2}+\sum_{x\in (a, b)}\mu (\{x\})^{2}\biggr). \tag 2.1$$ \endproclaim \demo{Proof} Using (1.3), we see that $$\epsilon\int\limits^{b}_{a}(\text{Im}\,F(x+i\epsilon))^{2}\,dx = \int\int g_{\epsilon}(x, y)\,d\mu(x)\,d\mu(y),$$ where $$g_{\epsilon}(x, y)=\int\limits^{b}_{a}\frac{\epsilon^{3}\,dw} {((w-x)^{2}+\epsilon^{2})((w-y)^{2}+\epsilon^{2})}.$$ It is easy to see that for $0< \epsilon <1$: \roster \item"{(i)}" $g_{\epsilon}(x, y)\leq \pi\,\frac{1}{\text {dist}(x, [a,b])^{2}+1}$ \item"{(ii)}" $\lim\limits_{\epsilon\downarrow 0}\,g_{\epsilon}(x, y) =0$ if $x\neq y$ or $x\notin [a,b]$ \item"{(iii)}" $\lim\limits_{\epsilon\downarrow 0}\, g_{\epsilon}(x, y) =\frac{\pi}{2}$ if $x=y\in (a, b)$ \item"{(iv)}" $\lim\limits_{\epsilon\downarrow 0}\, g_{\epsilon}(x, y) =\frac{\pi}{4}$ if $x=y$ is $a$ or $b$. \endroster Thus, the desired result follows from dominated convergence. \qed \enddemo \remark{Remarks} 1. It is not hard to extend this to $\epsilon^{p-1} \int\limits^{b}_{a}|\text{Im}\,F(x+i\epsilon)|^{p}\,dx$ for any $p>1$. The limit has $\int\limits^{\infty}_{-\infty}(1+x^{2})^{-p/2}\,dx$ in place of $\pi$ (which can be evaluated exactly in terms of gamma functions) and $\mu (\{x\})^p$ in place of $\mu (\{x\})^2$. For the above proof extends to $p$ an even integer. Interpolation then shows that the continuous part of $\mu$ makes no contribution to the limit, and a simple argument restricts the result to a finite sum of point measure where it is easy. (Note: For 10}\,(2t)^{-1}\mu(x-t, x+t). $$By the standard Hardy-Littlewood argument (see, e.g., Katznelson [3]):$$ |\{x\mid M_{\mu}(x)>t\}|\leq C\mu(\Bbb R)/t, $$which in particular implies$$ \int\limits^{b}_{a}M_{\mu}(x)^{p}\,dx <\infty $$for all p<1. Since \frac{1}{\pi}\,\text{Im}\,F(x+i\epsilon)\leq M_{\mu}(x) for all \epsilon and \frac{1}{\pi}\,\text{Im}\,F(\cdot +i\epsilon)\to(\frac{d\mu_ {\text{\rom{ac}}}}{dx})(x) a.e.~in x, the desired result follows by the dominated convergence theorem. \qed \enddemo \remark{Remark} The reader will note that the first proof is similar to the proof in [7] that the measures with no a.c.~part are a G_\delta. In a sense, this part of our discussion in \S4 is a transform for the proof of [7] to this proof instead! \endremark \proclaim{Corollary 3.2} A measure \mu has no absolutely continuous part on (a, b) if and only if$$ \varliminf\limits_{k\to\infty} \int\limits^{b}_{a} \text{\rom{Im}}\, F(x+ik^{-1})^{1/2}\,dx = 0. $$\endproclaim \bigpagebreak \flushpar {\bf \S4. \boldkey G_{\boldsymbol\delta} properties of sets of measures and operators} \medpagebreak \proclaim{Lemma 4.1} Let X be a topological space and f_{n}:X\to \Bbb R a sequence of non-negative continuous functions. Then \bigl\{x\mid\varliminf\limits_{n\to\infty}\,F_{n}(x)=0\bigr\} is a G_\delta. \endproclaim \demo{Proof}$$\align \biggl\{x\mid\varliminf\limits_{n\to\infty}\, F_{n}(x)=0\biggr\} &=\biggl\{x\mid \forall k\,\forall N\,\exists n\geq N \ F_{n}(x)<\frac{1}{k}\biggr\} \\ &=\bigcap\limits^{\infty}_{k=1}\,\bigcap\limits^{\infty}_{N=1}\, \bigcup\limits^{\infty}_{n=N} \biggl\{x\mid F_{n}(x)<\frac{1}{k} \biggr\} \endalign $$is a G_\delta. \qed \enddemo As a corollary of this and Corollaries 2.3 and 3.2, we obtain a proof of the result of [9]. \proclaim{Theorem 4.1} Let M be the set of probability measures on [a,b] in the topology of weak convergence \rom(this is a complete metric space\rom). Then \{\mu\mid\mu \text{\rom{ is purely singular continuous}}\} is a dense G_\delta. \endproclaim \demo{Proof} By Corollary 3.2:$$ \{\mu\mid\mu_{\text{\rom{ac}}}=0\}=\biggl\{\mu\mid\varliminf\limits_{k\to \infty}\int\limits^{b}_{a}(\text{Im}\,F_{\mu}(x+ik^{-1})^{1/2}\,dx=0 \biggr\}, $$and by Corollary 2.3:$$ \{\mu\mid\mu_{\text{\rom{pp}}}=0\}=\biggl\{\mu\mid\varliminf\limits_{k\to \infty}k^{-1}\int\limits^{b}_{a}\text{Im}\,F_{\mu}(x+ik^{-1})^{2}\, dx=0\biggr\}, $$so by Lemma 4.1, each is a G_\delta. Here we use the fact that \mu\mapsto F_{\mu}(x+i\epsilon) is weakly continuous for each x, \epsilon >0 and dominated above for each \epsilon >0 so the integrals are weakly continuous. By the convergence of the Riemann-Stieltjes integrals, the point measures are dense in M, so \{\mu\mid\mu_{\text{\rom{ac}}}=0\} is dense. On the other hand, the fact that \frac{1}{\pi}\,\text{Im}\,F_{\mu}(x+i\epsilon)\,dx converge in M to d\mu shows that the a.c.~measures are dense in M, so \{\mu\mid\mu_{\text{\rom{pp}}}=0\} is dense. Thus, by the Baire category theorem, \{\mu\mid\mu_{\text{\rom{pp}}}=0\}\cap\{\mu\mid \mu_{\text{\rom{ac}}}=0\} is a dense G_\delta! \qed \enddemo Finally, we recover our results in [7]. We call a metric space X of self-adjoint operators on a Hilbert space \Cal H regular if and only if A_{n}\to A in the metric topology implies that A_{n}\to A in strong resolvent sense. (Strong resolvent convergence of self-adjoint operators means (A_{n}-z)^{-1}\varphi\overset \|\,\| \to\longrightarrow (A-z)^{-1}\varphi for all \varphi and all z with \text{Im}\,z\neq 0. Notice this implies that for any a,b,p and \epsilon >0 and any \varphi\in\Cal H, A\mapsto \int\limits^{b}_{a}\text{Im}(\varphi, (A-x-i\epsilon)^{-1}\varphi)^{p}\,dx\equiv F_{a,b,p,\epsilon,\varphi}(A) is a continuous function in the metric topology. \proclaim{Theorem 4.3} For any open set \Cal O\subset\Bbb R and any regular metric space of operators, \{A\mid A\text{\rom{ has no a.c.~spectrum in }}\Cal O\} is a G_\delta. \endproclaim \demo{Proof} Any \Cal O is a countable union of intervals so it suffices to consider the case \Cal O=(a,b). Let \varphi_{n} be an orthonormal basis for \Cal H. Then,$$ \{A\mid A\text{ has no a.c.~spectrum in }(a,b)\}=\bigcap\limits_{n} \biggl\{A\mid\varliminf\limits_{k\to\infty}\, F_{a, b, 1/2, 1/k, \varphi_{n}} (A)\biggr\} $$is a G_\delta by Lemma 4.1 and Corollary 3.2. \qed \enddemo Similarly, using Corollary 2.3, we obtain \proclaim{Theorem 4.4} For any interval [a,b] and any regular metric space of operators, \{A\mid A\text{\rom{ has no point spectrum in }}[a,b]\} is a G_\delta. \endproclaim \remark{Note} This is slightly weaker than the result in [7] but suffices for most applications. One can recover the full result of [7], namely Theorem 4.4 with [a,b] replaced by an arbitrary closed set K, by first noting that any closed set is a union of compacts, so it suffices to consider compact K. For each K, let K_{\epsilon}=\{x\mid\text{dist}(x, K)<\epsilon\}. Then one can show that if d\mu has no pure points in K, then$$ \lim\limits_{\epsilon\downarrow 0}\,\epsilon\int\limits_{K_\epsilon} (\text{Im}\,F_{\epsilon}(x+i\epsilon))^{2}\, dx = 0; $$and if it does have pure points in K, then$$ \varliminf\limits_{k\to\infty}\,k^{-1}\int\limits_{K_\epsilon} |\text{Im}\, F(x+ik^{-1})|^{2}\,dx>0$and Theorem 4.4 extends. \endremark \vskip 0.4in \Refs \endRefs \item{[1]} R.~del Rio, S.~Jitomirskaya, Y.~Last, and B.~Simon, {\it Operators with singular continuous spectrum, IV.~Hausdorff dimensions, rank one perturbations, and localization}, preprint. \item{[2]} T.~Kato, {\it Wave operators and similarity for some non-self-adjoint operators}, Math. Ann. {\bf 162} (1966), 258--279. \item{[3]} Y.~Katznelson, {\it An Introduction to Harmonic Analysis}, Dover, New York, 1976. \item{[4]} A.~Klein, {\it Extended states in the Anderson model on Bethe lattice}, preprint. \item{[5]} P.~Koosis, {\it Introduction to$H_p\$ Spaces}, London Mathematical Society Lecture Note Series {\bf 40}, Cambridge University Press, New York, 1980. \item{[6]} M.~Reed and B.~Simon, {\it Methods of Modern Mathematical Physics, IV.~Analysis of Operators}, Academic Press, New York, 1978. \item{[7]} B.~Simon, {\it Operators with singular continuous spectrum, I.~General operators}, Ann.~of Math. {\bf 141} (1995), 131--145 \item{[8]} B.~Simon, {\it Spectral analysis of rank one perturbations and applications}, to appear in Proc.~1993 Vancouver Summer School in Mathematical Physics. \item{[9]} T.~Zamfirescu, {\it Most monotone functions are singular}, Amer.~Math.~Monthly {\bf 88} (1981), 47--49. \enddocument