\documentclass[12pt]{amsart} \textheight=574pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \topmargin=14pt \headheight=7pt % \renewcommand{\baselinestretch}{1.7} \usepackage{graphics} \begin{document} \title[Absolutely continuous spectrum: optimal results] {Absolutely continuous spectrum for one-dimensional Schr\"odinger operators with slowly decaying potentials: some optimal results} \author{Michael Christ} \address{\hskip-\parindent Michael Christ\\ Department of Mathematics \\ University of California \\ Berkeley CA 94720} \email{mchrist@math.berkeley.edu} \author{Alexander Kiselev} \address{\hskip-\parindent Alexander Kiselev\\ Mathematical Sciences Research Institute \\ 1000 Centennial Drive \\ Berkeley CA 94720 } \email{kiselev@msri.org } \subjclass{ Primary 34L40, 81Q05, 42B20 Secondary 81Q15, 42B25} \begin{abstract} The absolutely continuous spectrum of one-dimensional Schr\"odinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic Schr\"odinger operators is preserved under all perturbations $V(x)$ satisfying $|V(x)|\leq C(1+x)^{-\alpha},$ $\alpha >\frac{1}{2}.$ This result is optimal in the power scale. More general classes of perturbing potentials which are not necessarily power decaying are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schr\"odinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on new maximal function and norm estimates and almost everywhere convergence results for certain multilinear integral operators. \end{abstract} \maketitle \begin{center} \large \bf 1. Introduction and main results. \end{center} In this paper, we study the stability of the absolutely continuous spectrum of one-dimensional Schr\"odinger operators under perturbations by slowly decaying potentials. Suppose that $H_{U}$ is a Schr\"odinger operator defined on $L^{2}(0,\infty)$ by the differential expression \[ H_{U}= -\frac{d^{2}}{dx^{2}} + U(x) \] and some self-adjoint boundary condition at the origin. We assume that $U$ is some bounded function for which $H_{U}$ has absolutely continuous spectrum. The presence of the absolutely continuous spectrum has direct consequences for the physical properties of the quantum particle described by the operator $H_{U}$ (see, e.g.\ \cite{ReSi}, \cite{AvS}). If we perturb this operator by some decaying potential $V(x),$ the Weyl criterion implies that the essential spectra of the operators $H_{U}$ and $H_{U+V}$ coincide. We seek conditions on the rate of decay of $V(x)$ which ensure that the absolutely continuous spectrum of the unperturbed operator $H_{U}$ is also preserved. This problem has a long history as one of the most natural questions in quantum mechanics, and we briefly recall the main results. It has long been known that if the perturbation $V(x)$ is absolutely integrable, then the absolutely continuous spectrum of the original operator is preserved. Until recently, little more was known concerning the preservation of the absolutely continuous spectrum of Schr\"odinger operators under decaying perturbations in the general situation. Substantially more information is available in the case when $U(x)=0.$ There has been much work on proving the absolute continuity of the spectrum for the Schr\"odinger operators with potentials of slower decay, but satisfying some additional special assumptions. For example, by a result going back to Weidmann \cite{Weid}, if a potential $V$ may be represented as a sum of a function of bounded variation and an absolutely integrable function, then the spectrum of the operator $H_{V}$ on $R^{+}=(0, \infty)$ is purely absolutely continuous. Many authors developed a scattering theory for long-range potentials whose derivatives satisfy certain bounds; see for example \cite{As}, \cite{Bus}, \cite{Hor}. These results hold in any dimension and the proofs involve approximating the scattering trajectories by solutions of the classical Hamilton-Jacobi equation. The weakest conditions on the long-range part of the potential under which the wave operators are known to exist, are given in \cite{Hor}. For potentials satisfying $|V(x)| \leq C(1+|x|)^{-\frac{1}{2}-\epsilon},$ for instance, one can infer the existence of the wave operators if also $|D^{\alpha}V(x)| \leq C_{1} (1+|x|)^{-\frac{3}{2}-\epsilon}$ for every multiindex $\alpha$ with $|\alpha|= 1.$ Another class of results describes spectral behavior of specific spherically symmetric (i.e.\ essentially one-dimensional) oscillating potentials, the typical example being $V(x) = x^{-\beta} {\sin x^{\alpha}}$ with $\alpha,$ $\beta$ positive. Such potentials in general do not satisfy the derivative bounds needed for the method of the works cited above to be applicable. We mention the papers \cite{Ben}, \cite{BA}, \cite{HiS}, \cite{Mat} and \cite{White} in which further references may be found. The spectrum of the operator $H_{V}$ for such potentials turns out to be absolutely continuous with perhaps some isolated embedded eigenvalues when $\alpha =1.$ These potentials generalize the celebrated Wigner-von Neumann example \cite{Wig}. Wigner and von Neumann were the first to discover an example with Coulomb type decay at infinity, i.e.\ $V(x)=O(\frac{1}{1+|x|}),$ whose spectrum is not purely absolutely continuous and has positive eigenvalues embedded in the absolutely continuous spectrum. Moreover, Naboko \cite{Nab} and later Simon \cite{Sim1} found different constructions which show that for potentials decaying more slowly but arbitrarily close to a Coulomb rate, very striking spectral phenomena arise. Namely, for every function $C(x)$ tending monotonically to infinity as $x$ goes to infinity, no matter how slowly, there exists a potential $V(x)$ satisfying \[ |V(x)| \leq \frac{C(x)}{1+|x|}, \] for which the associated Schr\"odinger operator $H_{V}$ has a dense set of eigenvalues in $R^{+}.$ A new general class of potentials preserving the absolutely continuous spectrum of the free Schr\"odinger operator was recently found by one of us in \cite{Kis}. Namely, if the potential $V$ satisfies $|V(x)| \leq C(1+|x|)^{-\frac{3}{4}-\epsilon}$ with some $\epsilon > 0,$ with no additional assumptions, then the whole positive semi-axis $(0,\infty)$ is an essential support of the absolutely continuous part of the spectral measure.\footnote{ $E$ is an essential support of the measure $\mu$ if $\mu(X)>0$ for any $X \subset E$ of positive Lebesgue measure.} Of course, as the examples of Naboko and Simon show, rich embedded singular spectrum may occur; however it is indeed embedded in the sense that there is an underlying absolutely continuous spectrum. One can describe the set where the singular part of the spectral measure might be supported in $R^{+}$ rather explicitly in terms of the properties of the Fourier transform of $x^{\frac{1}{4}}V(x)$ \cite{Kis}. The result of \cite{Kis} was further improved in \cite{Kis1}, where a general criterion was established which implies the stability of the absolutely continuous spectrum of the operator $H_{U}$ under all perturbations $V(x)$ satisfying $|V(x)| \leq C(1+x)^{-\frac{2}{3}-\epsilon}$, under the auxiliary hypothesis that a certain operator, constructed from the generalized eigenfunctions of $H_{U},$ is bounded on $L^2(R)$. In particular, it was shown that the absolutely continuous spectra of free and periodic one-dimensional Schr\"odinger operators are stable under all perturbations by potentials satisfying $|V(x)|\leq C(1+x)^{-\frac{2}{3}-\epsilon}.$ Later, this result for the case $U=0$ was also proved by S.~Molchanov by a different method \cite{Mol1}. On the other hand, there exists work on random potentials by Kotani and Ushiroya \cite{Kot} which provides a bound for the best possible result that one can hope to prove. The results of \cite{Kot} imply that (in the case $U=0$) there exist potentials $V(x)$ satisfying $V(x) \leq C(1+x)^{-\frac{1}{2}},$ for which $H_V$ has purely singular spectrum on $R^{+}.$ We remark that by combining methods used in recent works \cite{KLS} and \cite{KRS}, one can show that the $(1+x)^{-\frac{1}{2}}$ rate of decay is also critical for perturbations of the periodic Schr\"odinger operators. In this paper we establish results on the preservation of absolutely continuous spectrum for power decaying potentials in one dimension which are of an optimal nature. In particular, we prove \\ \noindent \bf Theorem 1.1. \it Suppose that there exist $\epsilon>0$ and $p\leq 2$ so that $V(x)(1+x)^{\epsilon} \in L^{p}(R^{+}).$ Then the whole positive semi-axis $R^{+}=(0,\infty)$ is an essential support of the absolutely continuous part of the spectral measure of the operator $H_{V}.$ Moreover, for almost every $\lambda \in R^{+},$ there exist solutions $\phi_{\lambda}(x)$ and $\overline{\phi}_{\lambda}(x)$ of the generalized eigenfunction equation \[ -y'' + V(x)y=\lambda y \] with asymptotic behavior of pure WKB form in the main term: \[ \phi_{\lambda}(x)= \exp \left(i\sqrt{\lambda}x - \frac{i}{2\sqrt{\lambda}}\int\limits_{0}^{x}V(t)\,dt\right)(1+o(1)). \] \noindent \bf Corollary. \it If $V(x) = O(1+x)^{-r}$ for some $r>1/2$ then an essential support of the absolutely continuous part of the spectral measure of $H_V$ on $L^2(R^+)$ equals $[0,\infty)$. \\ \rm We have a similar result for perturbations of periodic Schr\"odinger operators. Let $U$ be continuous and periodic, let $S=\cup_{n=0} ^{\infty}(a_{n},b_{n})$ be the band spectrum of the unperturbed operator with potential $U$, and let $\theta(x, \lambda)$ be the Bloch functions for that operator. \\ \noindent \bf Theorem 1.2. \it If the potential $V(x)$ is as in Theorem 1.1, then the set $S$ is an essential support of the absolutely continuous part of the spectral measure of the operator $H_{U+V}.$ For a.e.\ $\lambda \in S,$ there exist solutions $\psi_{\lambda}(x),$ $\overline{\psi}_{\lambda}(x)$ of the equation \begin{equation} -y''+(U(x)+V(x))y = \lambda y \end{equation} with the asymptotic behavior \begin{equation} \psi_{\lambda}(x)= \theta(x,\lambda)\exp \left( \frac{i}{2\Im(\theta\overline{\theta}')} \int\limits_{0}^{x}V(t)|\theta^{2}(t,\lambda)|\,dt\right)(1+o(1)) \end{equation} as $y\to\infty$. \rm Both theorems will follow from a certain general criterion. Suppose that $H_{U}$ is an operator for which all solutions of the equation \begin{equation} -y'' + U(x)y = \lambda y \end{equation} are bounded for almost every $\lambda \in S,$ where $S$ is a certain set of positive Lebesgue measure. It is known (see, e.g.\ \cite{Sim1}, \cite{Sto}) that in this case the set $S$ belongs to an essential support of the absolutely continuous part of the spectral measure. Let us pick a family of solutions $\theta(x,\lambda),$ $\lambda \in S,$ of the equation (3), such that $\theta (x,\lambda)$ are uniformly bounded over $S$ and $\theta(x,\lambda),$ $\overline{\theta }(x,\lambda)$ are linearly independent for every $\lambda \in S.$ It is easy to see that we can always find such family. We have \\ \noindent \bf Theorem 1.3. \it Suppose that the potential $V(x)$ is such that there exist $\epsilon>0$ and $p\leq 2$ so that $(1+x)^{\epsilon}V(x)\in L^{p}.$ Assume that there exist measurable functions $\theta(x,\lambda)$ satisfying the above conditions, such that the operator \begin{equation} (Kf)(\lambda) = \chi(S)\int\limits_{0}^{\infty}\theta(x,\lambda)^{2} \exp \left(\frac{i}{\Im(\theta\overline{\theta}')}\int\limits_{0}^{x}V(t) |\theta(t,\lambda)|^{2}\,dt\right)f(x)\,dx \end{equation} satisfies an $L_{2}(R^{+},dx)-L^{2}(S,d\lambda)$ bound on functions $f$ of compact support. Then the absolutely continuous spectrum of $H_{U}$ supported on the set $S$ is preserved under perturbation by $V$, that is, the set $S$ belongs to an essential support of the absolutely continuous part of the spectral measure of operator $H_{U+V}.$ Moreover, for almost every $\lambda \in S,$ there exist solutions $\psi_{\lambda}(x),$ $\overline{\psi}_{\lambda}(x)$ of the equation (1) with the asymptotic behavior (2). \\ \noindent \it Remarks. \rm 1. The assumption of the boundedness of all solutions at almost all energies corresponding to the essential support of absolutely continuous spectrum is rather natural. Almost all known examples of one-dimensional Schr\"odinger operators with the absolutely continuous spectrum satisfy this assumption. Only recently there appeared rigorous counterexamples to the conjecture that this is true in general (see \cite{Mol}), but the corresponding potentials are of rather special form and in particular are not bounded from below. \rm 2. All three main theorems that we prove have natural analogues for the whole axis problems. We will not focus on this aspect; all proofs may be generalized to the whole axis case in a straightforward manner following \cite{Kis1}. \rm 3. We have not been able to treat potentials that are assumed merely to belong to $L^{p}$ for some $1

x$ and the transformation (7) is non-singular. By the assumption of the theorem, the off-diagonal terms are absolutely integrable. The diagonal terms are purely imaginary and hence Levinson's theorem is applicable. Asymptotic behavior of the solutions (10) follows directly from the explicit solution of the equation (8) with diagonal terms omitted and application of transformations we applied to the original system of equations. \end{proof} \rm To complete the proof of Theorem 1.3, we need to construct the function $q(x,\lambda)$ verifying (9) and conditions given in Theorem 2.2 for almost every $\lambda \in S$. The main problem is that if we try to solve the equation \begin{eqnarray} q'(x,\lambda) & + & \frac{i}{2\Im(\theta \overline{\theta})}V(x) \overline{\theta}^{2}(x,\lambda)\exp (2ip(x,\lambda)) \nonumber \\ && \qquad +\ \frac{i}{2\Im(\theta \overline{\theta})}V(x) \theta^{2}(x,\lambda) \exp(-2ip(x,\lambda))q^{2}(x,\lambda) =0 \end{eqnarray} by iteration, we obtain expressions involving multilinear integral operators of certain type. We need to show that these expressions converge for a.e.\ $\lambda \in S$ in order to ensure $q(x,\lambda) \stackrel{x \rightarrow \infty}{\longrightarrow} 0$ for a.e.\ $\lambda,$ and to make sure that (9) is satisfied after some number of iterations. The first approximation to the solution would be \[ q^{(0)}(x,\lambda) = \frac{i}{2\Im(\theta \overline{\theta})} \int\limits_{x}^{\infty}V(t)\overline{\theta}^{2}(t,\lambda) \exp (2ip(t,\lambda))\,dt. \] Again, we have to justify this formula by proving that the conditional integral is well-defined for almost every $\lambda.$ This is relatively simple and has been already done in \cite{Kis1}. In the next section, we formulate the main result from \cite{Kis1} that we will use and make a few comments. % \newpage \vspace{0.2cm} \begin{center} \large \bf 3. Almost everywhere convergence for integral operators. \end{center} \vspace{0.2cm} Let an operator $K$ be defined on all measurable bounded functions $f$ of compact support by \begin{equation} (Kf)(\lambda) = \int\limits_{0}^{\infty} k(\lambda,x) f(x)\, dx, \end{equation} where $k(\lambda,x)$ is a measurable and bounded function on $I \times R^{+}.$ To study the a.e.\ convergence of the integral defining $Kf(\lambda)$ on functions from $L^{p},$ we study the corresponding maximal function. Denote by $M_{K}f(\lambda)$ the maximal function \begin{equation} M_{K}f(\lambda) = {\rm sup}_{N} \left| \int\limits_{0}^{N}k(\lambda,x)f(x)\,dx \right|. \end{equation} The following is a mild generalization of a result proved in \cite{Kis1}. \\ \noindent \bf Theorem 3.1. \it Let $p,q$ be exponents satisfying $1\le pp \geq 1$ and $q>p.$ Let $n \geq 2$. Let $A$ be any set of ordered pairs $\alpha = (i_{\alpha}, i'_{\alpha}),$ with $1 \leq i_{\alpha}, i'_{\alpha} \leq n.$ Let $|A|$ denote the cardinality of $A.$ By $\chi_{E}(x)$ we denote a characteristic function which is equal to one when $x \in E$ and is zero otherwise. Consider the multilinear operator $T_{n}$ given by \begin{equation} T_{n}(f_{1}, \dots f_{n})(\lambda) = \int\limits_{R^{n}} \prod\limits_{j=1}^{n} f_{j}(x_{j})k_{j}(x_{j},\lambda) \prod\limits_{\alpha \in A} \chi_{R^{+}}(x_{i_{\alpha}}-x_{i'_{\alpha}})\,dx, \end{equation} $x=(x_{1}, \dots x_{n}).$ Notice that if there were no ``diagonal" characteristic functions, the expression (16) would decompose into a product of one-dimensional integrals, and the analysis would become trivial. \\ \noindent \it Remark. \rm We do not rule out the possibility that some of the characteristic functions in (16) are contradictory and the whole expression is zero. \\ Our goal in this section is to prove the following property: \\ \noindent \bf Theorem 4.1. \it Suppose that the multilinear operator $T_{n}$ is given by (16) with kernels $k_{j}(\lambda,x_{j})$ satisfying (15). Then for any functions $f_{i} \in L^{p}(R^{+},dx),$ $i=1, \dots n,$ such that the integral (16) converges absolutely for a.e.\ $\lambda,$ we have \[ \|T_{n}(f_{1},...f_{n})\|_{s_{n}} \leq C_{n}\prod_{i=1}^{n} \|f_{i}\|_{p}, \] where $s_{n}^{-1} = nq^{-1}.$ The constant $C_{n}$ depends only on $n$ and constants in the norm bounds (15) for operators $K_{i}.$ \\ \noindent \it Remarks. \rm The conclusion of Theorem 4.1 holds in particular when $s_{n}<1.$ Our proof will yield a more general inequality, in which $f_j\in L^{p_j}$, the exponents $p_j$ vary freely in a certain range, and $s_n^{-1} = \sum_j q_j^{-1}$ where $q_j^{-1} = 1 - p_j^{-1}$; the case where all exponents $p_j$ are equal suffices for our applications, and we restrict attention to it in order to simplify computations slightly. \\ \noindent \rm By assumption, the value of $T_{n}(f_{1}, \dots f_{n})(\lambda)= g(\lambda)$ is well-defined for a.e.\ $\lambda$ by the absolutely convergent integral. Our strategy will be to divide the domain of integration into disjoint pieces and represent the function $g$ as a sum of terms coming from integration over these disjoint pieces, formally: \[ g(\lambda)=\sum_{i=1}^{\infty}g_{i}(\lambda).\] Because of the absolute convergence, we have that the sum $\sum_{i=1}^{n}g_{i} (\lambda)$ converges to $g(\lambda)$ for a.e.\ $\lambda$ as $n \rightarrow \infty.$ We show, choosing the functions $g_{i}$ in a convenient way, that the sum also converges absolutely in the appropriate space $L^{s_{n}},$ thus proving Theorem 4.1. In the proof of Theorem 4.1, we will need a certain representation of the function \[ f(x_{1})f(x_{2})\chi_{R^{+}}(x_{2} - x_{1}) \] as a sum of products of two functions depending only on $x_{1}$ and $x_{2}$ respectively. Let us first introduce a decomposition of $R^{+}$ associated with the function $f.$ Normalize the function $f$ so that $\|f\|_{p}^{p}=1.$ By $\chi_{E}$ we will denote the characteristic function of the set $E.$ Let $E(1,1)$ and $E(1,2)$ be disjoint intervals such that \[ \|f(x)\chi_{E(1,1)} \|^{p}_{p} = \|f(x)\chi_{E(1,2)} \|^{p}_{p} = 2^{-1}, \] $E(1,1) \cup E(1,2)=R^{+}$ and $E(1,1)$ lies entirely to the right of $E(1,2)$ (i.e. for any $x \in E(1,1),$ $y \in E(1,2)$ we have $x \leq y$). We note that $E(1,2)$ is half-infinite and assume $E(1,1)$ contains its right end for the above decomposition to hold. We also remark that the decomposition is not necessarily unique ($f$ might vanish on some set so that this decomposition will be non-unique), and we just take some decomposition. In future we will omit such inessential details. We continue to decompose each of the intervals $E(1,l)$ in a similar manner, obtaining on the $m^{{\rm th}}$ step $2^{m}$ intervals $\{ E(m,l)\}_{l=1}^{2^{m}},$ such that $\cup_{l=1}^{2^{m}}E(m,l)=R^{+},$ $\|f(x)\chi (E(m,l)) \|_{p}^{p}=2^{-m}$ for $j=1,...2^{m},$ the intervals are disjoint and $E(m,l)$ lies entirely to the left from $E(m,i)$ if $l < i.$ In notation $E(m,l),$ we refer to $m$ as ``generation" of this interval and to $l$ as ``index". Of importance, in particular, will be the following evident property of intervals $\{E(m,l): 1 \le m \le \infty, 1 \le l \le 2^{m} \}:$ any two intervals are either disjoint or one is contained in another. We proceed to decompose the ``diagonal" characteristic functions in a convenient way. \begin{figure} \includegraphics{il.eps} \caption{Decomposition of $\chi_{R^{+}}(x_{2}-x_{1})$}\label{Fi:A} \end{figure} \noindent \bf Lemma 4.2. \it The following identity holds: \begin{equation} \chi_{R^{+}}(x_{2} - x_{1}) f(x_{1})f(x_{2})= \left( \sum\limits_{m=1}^{\infty} \sum\limits_{ \stackrel{\scriptstyle l=1} {\scriptstyle l \,{\rm odd}}} ^{2^{m}} \chi_{E(m,l)}(x_{1}) \chi_{E(m,l+1)}(x_{2}) \right)f(x_{1})f(x_{2}). \end{equation} \begin{proof} \rm Figure 1 illustrates the decomposition that we perform. Let us denote by $H_{12}$ the set \[ H_{12} = \{ x \in R^{2}, x=(x_{1},x_{2}) | x_{1} < x_{2} \}, \] and by ${\rm supp} f$ the closure of the set of the points $x$ such that for every interval $I,$ such that $x \in I,$ $|f(x)|$ is positive on the set of positive Lebesgue measure in $I.$ The claim will follow if we show that \[ H_{12} \cap ({\rm supp}_{x_{1}}f \times {\rm supp}_{x_{2}}f)= \bigcup_{m=1}^{\infty} \bigcup^{2^m}_{ \stackrel{\scriptstyle l=1} {\scriptstyle l \,{\rm odd}}} (E(m,l) \times E(m,l+1) ) \] and the sets under the union on the right hand side are disjoint. The latter fact is easy to see: if $E(m,l) \subset E(s,i),$ $l$ odd, $s\neq m,$ then necessarily $m>s$ and $E(m,l+1)$ also belongs to $E(s,i),$ not $E(s,i+1).$ On the other hand, we show that for every $y_{2},y_{1} \in {\rm supp} f,$ $y_{1} < y_{2},$ there exist two sets $E(m,l),$ $E(m,l+1),$ with $l$ odd, such that $y_{1} \in E(m,l)$ and $y_{2} \in E(m,l+1).$ Let $\|f \chi_{(y_{1}, y_{2})}\|_{p}^{p}=a>0.$ Here we assume that $f$ is normalized and use the condition that $y_{1},$ $y_{2}$ lie in ${\rm supp}f$ to infer that $a>0.$ Choose $s$ so that $2^{-s} \geq a \geq 2^{-s-1}.$ If $y_{1},$ $y_{2}$ lie in one set of generation $s,$ $E(s,l),$ then necessarily $y_{1} \in E(s+1,2l-1)$ and $y_{2} \in E(s+ 1,2l).$ If $y_{1}$ and $y_{2}$ lie in different sets of generation $m$, $E(s,l)$ and $E(s,l+1),$ then either $l$ is odd or $y_{1} \in E(s-1,l/2^{r})$ and $y_{2} \in E(s-1,l/2^{r}+1),$ where $r$ is such that $l/2^{r}$ is odd. \end{proof} \rm \noindent \it Remark. \rm In particular, if ${ \rm supp} f = R^{+},$ we get a representation of diagonal characteristic function $\chi_ {R^{+}}(x_{2} - x_{1})$ as a sum of products of characteristic functions of some intervals in $x_{1}$ and $x_{2}$ variables. We now begin proof of Theorem 4.1. \begin{proof} \rm Since $T_n$ is multilinear, we may assume without loss of generality throughout the proof that $\|f_{i}\|_{p}^p=\frac{1}{n}$ for all $i=1,...n.$ Let \[ f(x) = \left( \sum\limits_{i=1}^{n}|f_{j}(x)|^{p} \right)^{\frac{1}{p}}. \] Consider the family of the intervals $\{E(m,l)\}$ associated with the function $f.$ An important property of this family is that \begin{equation} \|f_{i}(x)\chi_{E(m,l)}\|^{p}_{p} \leq 2^{-m} \end{equation} for all $i,l,m$. Write \[ A=\{ \alpha_{1}, \dots \alpha_{|A|} \}. \] We begin by substituting the result of Lemma 4.2 into formula (16): \begin{equation} T_{n}(\overline{f})= \sum\limits_{m_{1}}^{\infty} \cdots \sum\limits_{m_{|A|}=1}^{\infty} \sum\limits_{l_{1}}' \cdots \sum\limits_{l_{|A|}}'\,\int\limits_{R^{n}} dx \prod\limits_{j=1}^{n} k_{j}(\lambda, x_{j})f_{j}(x_{j}) \prod\limits_{t=1}^{|A|} \chi_{E(m_{t},l_{t})}(x_{i_{t}}) \chi_{E(m_{t},l_{t}+1)}(x_{i'_{t}}), \end{equation} where $\sum\limits_{l_{t}}'$ means the sum over all odd integers $l_{t} \in [1,2^{m_{t}}],$ and $i_{t}=i_{\alpha_{t}},$ $i_{t}'=i_{\alpha_{t}}'.$ Thus \begin{equation} | T_n(\overline{f})(\lambda) | \leq \sum\limits_{m_{1}}^{\infty} \cdots \sum\limits_{m_{|A|}=1}^{\infty} F^{\overline{m}} (\overline{f})(\lambda), \end{equation} where $\overline{m}= (m_{1},m \dots _{|A|})$ and \begin{equation} F^{\overline{m}}(\overline{f})(\lambda)= \sum\limits_{l_{1}}' \cdots \sum\limits_{l_{|A|}}' \, \int\limits_{R^{n}} dx \prod\limits_{j=1}^{n} |K_{j}(f_{j}\chi_{G(j,\overline{l})})(\lambda)|, \end{equation} where $\overline{l}=(l_{1},...l_{|A|})$ (all variables $l_{t}$ take only odd values), the set $G(j,\overline{l})$ depends on $\overline{m}$ (and on $A$), and \[ G(j,\overline{l}) = \left[ \bigcap_{t: j=i_{t}}E(m_{t},l_{t}) \right] \bigcap \left[ \bigcap_{t:j=i'_{t}}E(m_{t},l_{t}+1) \right]. \] We aim to prove that \begin{equation} \|F^{\overline{m}}(\overline{g})\|_{s_{n}} \leq C_{n}2^{-\gamma_{n} |\overline{m}|}\prod\limits_{j=1}^{n}\|g_{j}\|_{p}^{(1-\beta_{n})} \end{equation} for any $g_{1},...g_{n}$ which satisfy (18) (that is, $\| g_i \chi_{E(m,l)}\|_p^p \le 2^{-m}$ for all $i,m,l$). Here $\gamma_{n}$ is some positive constant which depends only on $n,$ $C_{n}$ depends on $n$ and constants in norm bounds (15) for the operators $K_{j},$ and $\beta_{n}$ satisfies \[ 1 \geq 1-\beta_{n} > {\rm max}(\frac{p}{q}, \frac{p}{2}) . \] Invoking the decomposition (20) and summing over $\overline{m}$, Theorem 4.1 follows directly from (22). We may proceed to prove (22) by induction on $|A|.$ The case $|A|=0$ is immediate from the hypothesis on $K_{j}$ by H\"older's inequality (in this case $\overline{m}$ is a $0$-tuple). It will be convenient to consider the graph $\Upsilon$ with vertices $\{1,...n\}$ and edges $(i_{t},i'_{t})$ joining $i_{t}$ to $i_{t}'$ for any $t$ (and no other edges). To each edge we associate the generation $m_{t}$ which corresponds to a generation in the decomposition (17) of $\chi_{R^{+}}(x_{i_{t}}-x_{i_{t}'})$ that we fixed in the sum (21). It suffices to treat the case where $\Upsilon$ is connected; the general case then follows by H\"older's inequality. Fix $\overline{m}.$ Relabel the indices so that $m_{1} \leq ... \leq m_{|A|}.$ For simplicity of notation, we also relabel pairs $(i_{t},i_{t}')$ so that $m_{t}$ still denotes the generation in the decomposition of $\chi_{R^{+}}(x_{i_{t}}-x_{i_{t}'}).$ Let $N$ be the largest index for which $m_{N}=m_{1}.$ For many values of $\overline{l},$ the set $G(j,\overline{l})$ is empty for some $j$. Such terms contribute zero to the sum (21); this observation underlies the estimate (22) for $F^{\overline{m}}(\overline{f})$. To take this into account, drop from the sum (21) all terms for which there exists $j$ such that $G(j,\overline{l})=\emptyset;$ such terms contribute $0$ to $F^{\overline{m}}(\overline{g})$. We say that an index $l$ \it remains \rm if the corresponding term has not been dropped. We have the following \\ \noindent \bf Lemma 4.3. \it For any $1 \leq j \leq n,$ either \begin{equation} G(j,\overline{l}) \subset E(m_{1},l_{1}) \,\,\,\forall \overline{l} \,\,\,{\rm remaining} \end{equation} or \begin{equation} G(j,\overline{l}) \subset E(m_{1},l_{1}+1) \,\,\,\forall \overline{l} \,\,\,{\rm remaining}. \end{equation} Let $B_{1}=\{j:$ (23) holds $\}$, $B_{2} = \{ j:$ (24) holds $\}$. Then for any $2 \leq t \leq N$ (that is, if $m_{t}=m_{1}$), \begin{equation} l_{t}=l_{1}\,\,\,\forall \overline{l} \,\,\, {\rm remaining}. \end{equation} Finally for each $t>N$ (so $m_{t}>m_{1}$), either $i_{t},$ $i'_{t}$ are both in $B_{1}$ for all remaining $l$ or they are both in $B_{2}$ for all remaining $l.$ We say that $t \in A_{1},$ $t \in A_{2}$ respectively. \\ \begin{proof} \rm For any $m \geq 1,$ $l$ odd, set \[ \tilde{E}(m,l) = E(m,l) \cup E(m,l+1). \] First we prove that if $\overline{l}$ remains, then \begin{equation} \tilde{E}(m_{t},l_{t}) \subset \tilde{E}(m_{1},l_{1}) \end{equation} for all $t.$ Notice that both sets in (26) also belong to the family $E(m,l)$ (they are $E(m_{t}-1, \frac{l_{t}-1}{2}),$ $E(m_{1}-1, \frac{l_{1}-1}{2})$ respectively; we may assume $E(0,0)=R^{+}$). Therefore, to prove (26) it is sufficient to show that the two sets in (26) intersect, since in this case one is contained in another by the martingale-type property. Recall that $m_{1}$ is the generation which is fixed in decomposition of the characteristic function $\chi_{R^{+}}(x_{i_{1}}-x_{i_{1}'})$ in the sum (21) for $F^{\overline{m}}.$ Pick any other $m_{t}$ which is fixed in the decomposition of the characteristic function $\chi_{R^{+}}(x_{i_{t}}-x_{i_{t}'}).$ Since the graph $\Upsilon$ is connected, we can find a path in $\Upsilon$ which connects either $i_{t}$ or $i_{t}'$ with either $i_{1}$ or $i_{1}',$ and does not contain the edges $(i_{1},i_{1}'),$ $(i_{t},i_{t}').$ Suppose that this path goes from $i_{1}$ to $i_{t}$ and passes successively through the edges with the corresponding generations $m_{t_{1}},...m_{t_{r}}.$ This path does not depend on $\overline{l}$. For $G(j,\overline{l})$ to be non-zero for all $j,$ we must have \[ \tilde{E}(m_{1},l_{1}) \cap \tilde{E}(m_{t_{1}},l_{t_{1}}) \neq \emptyset, \,\,\,\, \tilde{E}(m_{t},l_{t}) \cap \tilde{E}(m_{t_{r}}, l_{t_{r}}) \neq \emptyset, \,\,\,\,\,{\rm and} \] \begin{equation} \tilde{E}(m_{t_{i}},l_{t_{i}}) \cap \tilde{E}(m_{t_{i+1}},l_{t_{i+1}}) \neq \emptyset \,\,\,\,{\rm for\,\,\, all\,\,\,} i=1,...r-1. \end{equation} Hence by our assumption that $m_{1} \leq m_{t}$ for all $t$ we see that $\tilde{E}(m_{t_{1}},l_{t_{1}}) \subset \tilde{E}(m_{1},l_{1}).$ But then by (27) also $\tilde{E}(m_{1},l_{1}) \cap \tilde{E}(m_{t_{2}},l_{t_{2}}) \neq \emptyset,$ hence $\tilde{E}(m_{t_{2}},l_{t_{2}}) \subset \tilde{E}(m_{1},l_{1}).$ We continue in the same way concluding that $\tilde{E}(m_{t},l_{t}) \subset \tilde{E}(m_{1},l_{1})$ and hence (26) holds. The statements (23), (24) and (25) of the lemma now follow immediately from the martingale-type property of the sets $E(m,l)$ and the definition of the set $G(j,\overline{l}).$ To obtain (25), note that because $l_1$ and $l_t$ are odd, when $l_t = l_1$ the inclusion $E(m_t,l_t)\subset \tilde E(m_1,l_1)$ forces $E(m_t,l_t) = E(m_1,l_1)$. To prove the final statement, suppose that we know in addition that $m_{t}>m_{1}.$ We can find a path in $\Upsilon$ which goes from $i_{t}$ or $i_{t}'$ to a vertex adjacent to an edge $(i_{s},i_{s}')$ with the corresponding generation $m_{s}$ equal to $m_{1}$ (i.e. with $s \leq N$), and contains only edges with the corresponding generations strictly less than $m_{1}.$ An argument analogous to the above shows that in this case $\tilde{E}(m_{t},l_{t})$ is contained either in $E(m_{1},l_{s})$ or in $E(m_{1},l_{s}+1)$ for all remaining $\overline{l},$ depending on whether the vertex to which the path leads coincides with $i_{s}$ or $i_{s}'$ respectively. By (25) the lemma is proven. \end{proof} \rm \noindent \it Remark. \rm It may happen that there exist two (or more) different paths from $i_{t}$ (or $i_{t}'$), one of which leads to a vertex $j_{1}$ where $f_{j_{1}}(x_{j_{1}})$ is multiplied by $\chi_{E(m_{1},l_{1})}(x_{j_{1}})$, while the other leads to a vertex $j_{2}$ where $f_{j_{2}}(x_{j_{2}})$ is multiplied by $\chi_{E(m_{1},l_{1}+1)}(x_{j_{2}})$. (The simplest way for this to happen is for $A$ to include two pairs $(i,j)$ and $(j,i)$.) In this case, $G(j,\overline{l})$ is zero and hence $\overline{l}$ does not remain. \\ By Lemma 4.3 \begin{eqnarray*} F^{\overline{m}}(\overline{f})(\lambda) & = & \left( \sum\limits_{l_{1}} ' \sum\limits_{l_{N+1}} '\cdots \sum\limits_{l_{|A|}} ' \prod\limits_{j \in B_{1}}|K_{j}(f_{j}\chi_{E(m_{1},l_{1})}\chi_{G(j, \overline{l})})(\lambda)| \right. \\ & & \cdot \ \left. \prod\limits_{j \in B_{2}}|K_{j}(f_{j}\chi_{E(m_{1},l_{1}+1)}\chi_{G(j, \overline{l})})(\lambda)| \right) \\ & \leq & \sum\limits_{l_{1}} 'F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}} (\overline{f})(\lambda)F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}} (\overline{f})(\lambda), \end{eqnarray*} where \[ F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{f})(\lambda)= \sum\limits_{l_{t}:t \in A_{1}} ' \prod\limits_{j \in B_{1}} |K_{j}(f_{j}\chi_{E(m_{1},l_{1})}\chi_{G(j,l)})(\lambda)| \] and $\sum\limits_{l_{t}:t \in A_{1}} '$ denotes the sum over all $l_{t}$ such that $1 \leq l_{t} \leq 2^{m_{t}},$ $l_{t}$ is odd, $t \in A_{1},$ and we write $\overline{m}^{(1)}= (m_{t})_{t \in A_{1}}.$ Note that $F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{f})(\lambda)$ depends only on those $f_{j}$ for which $j \in B_{1}.$ The factor $F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{f})(\lambda)$ is defined similarly, but $\chi_{E(m_{1},l_{1})}$ is replaced by $\chi_{E(m_{1},l_{1}+1)}$. We may rewrite for each $j \in B_{1}$ \[ G(j,l) = E(m_{1},l_{1}) \bigcap G^{1}(j, (l_{t})_{t \in A_{1}}), \] where \[ G^{1}(j, (l_{t})_{t \in A_{1}}) = \left[ \bigcap\limits_{t \in A_{1}, j=i_{t}} E(m_{t},l_{t}) \right] \bigcap \left[ \bigcap\limits_{t \in A_{1}, j=i_{t}'}E(m_{t},l_{t}+1) \right]. \] Indeed, by Lemma 4.3 all other sets that enter in the definition of $G(j,l)$ belong to $E(m_{1},l_{1}+1)$ and hence are absent for $\overline{l}$ which remain. Thus $F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{f})(\lambda)$ and $F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{f})(\lambda)$ are expressions of the same form as the original $F^{\overline{m}}.$ Since $0<|A_{1}|<|A|,$ both $F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{f})(\lambda)$ and $F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{f})(\lambda)$ may be estimated by induction on $|A|.$ Therefore \begin{equation} \|F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{f})(\lambda)\|_ {s_{|B_{1}|}} \leq C2^{-\gamma_{|B_{1}|} |m^{(1)}|} \prod\limits_{j \in B_{1}} \|f_{j}\chi_{E(m_{1},l_{1})}\|_{p}^{1-\beta_{|B_{1}|}}; \end{equation} a similar bound also holds for $F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{f})(\lambda)$. Using (28), we are ready to estimate $\|F^{\overline{m}}(\overline{f})\|_{s_{n}}.$ We distinguish between two cases: $s_{n} \le 1$ and $s_{n} \geq 1.$ Suppose first that $s_{n} \le 1$. Then \begin{equation} \|F^{\overline{m}}(\overline{g})\|_{s_{n}}^{s_{n}} \leq \sum\limits_{l_{1}}' \|F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{g})(\lambda)\| _{s_{|B_{1}|}}^{s_{n}} \|F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{g})(\lambda)\|_ {s_{|B_{2}|}}^{s_{n}}. \end{equation} We used the fact that $\| \sum h_{i}(x) \|^{s}_{s} \leq \sum \|h_{i}(x)\|_{s}^{s}$ when $s<1$ and H\"older's inequality. Plugging the estimate (28) and a similar bound for $F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{g})(\lambda)$ into (29), we find \begin{eqnarray} \|F^{\overline{m}}(\overline{g})\|_{s_{n}}^{s_{n}} & \leq & C_{n} 2^{-(\gamma_{|B_{1}|}|m^{(1)}|+\gamma_{|B_{2}|}|m^{(2)}|)s_{n}} \nonumber \\ & & \cdot \ \sum\limits_{l_{1}} '\left( \prod\limits_{j \in B_{1}} \|g_{j}\chi_{E(m_{1},l_{1})} \|_{p}^{(1-\beta_{|B_{1}|})s_{n}} \prod\limits_{j \in B_{2}}\|g_{j} \chi_{E(m_{1},l_{1}+1)}\|_{p}^{(1-\beta_{|B_{2}|})s_{n}}\right). \end{eqnarray} Pick $0 < a_{1}, a_{2} <1$ such that \[ a_{1}(1-\beta_{|B_{1}|}) = a_{2} (1-\beta_{|B_{2}|}) = \frac{p}{q}. \] We can find such $a_{1},$ $a_{2}$ by the induction assumption. The sum in (30) may be estimated by H\"older's inequality in the following way: \begin{eqnarray*} \lefteqn{ \sum\limits_{l_{1}}' \left(\prod\limits_{j \in B_{1}} \|g_{j}\chi_{E(m_{1},l_{1})} \|_{p}^{(1-\beta_{|B_{1}|})s_{n}} \prod\limits_{j \in B_{2}}\|g_{j} \chi_{E(m_{1},l_{1}+1)}\|_{p}^{(1-\beta_{|B_{2}|})s_{n}}\right) } \\ & & \qquad \le \prod\limits_{j \in B_{1}} {\rm max}_{l_{1}}\|g_{j}\chi_{E(m_{1},l_{1})}\|_{p}^ {(1-a_{1})(1-\beta_{|B_{1}|})s_{n}} \nonumber \\ & & \qquad\qquad \cdot \ \ \prod\limits_{j \in B_{2}}{\rm max}_{l_{1}}\|g_{j}\chi_{E(m_{1},l_{1}+1)}\|_{p}^ {(1-a_{2})(1-\beta_{|B_{2}|})s_{n}}\prod\limits_{j=1}^{n}\|g_{j}\|_{p}^ {\frac{p}{q}s_{n}}. \end{eqnarray*} Thus \begin{eqnarray*} \|F^{\overline{m}}(\overline{g})\|_{s_{n}} & \leq & C_{n}2^{-{\rm min}(\gamma_ {|B_{1}|}, \gamma_{|B_{2}|})(|m^{(1)}|+|m^{(2)}|)} \\ & & \cdot \ \ \prod\limits_{j=1}^{n} {\rm sup}_{l}\|g_{j}\chi_{E(m_{1},l)}\|_{p}^{{\rm min}(1-\beta_{|B_{1}|}, 1-\beta_{|B_{2}|})-\frac{p}{q}}\prod\limits_{j=1}^{n} \|g_{j}\|_{p}^ {\frac{p}{q}}. \end{eqnarray*} Obviously $\|g_{j}\|_{p} \geq {\rm sup}_{l}\|g_{j}\chi_{E(m_{1},l)}\|_{p},$ and $ {\rm sup}_{l}\|g_{j}\chi_{E(m_{1},l)}\|_{p} \leq 2^{-m_{1}}$ by (18). Pick $\beta_{n}$ so that \[ {\rm min}(1-\beta_{|B_{1}|}, 1-\beta_{|B_{2}|}) > 1-\beta_{n} > {\rm max}(\frac{p}{q}, \frac{p}{2}), \] and \[ \gamma_{n} = {\rm min}\big( [{\rm min}(1-\beta_{|B_{1}|},1-\beta_{|B_{2}|})- (1-\beta_{n})], \gamma_{|B_{1}|}, \gamma_{|B_{2}|}\big). \] Then \begin{equation} \|F^{\overline{m}}(\overline{g})\|_{s_{n}} \leq C_{n} 2^{-\gamma_{n}|m|} \prod\limits_{j=1}^{n} \|g_{j}\|_{p}^{1-\beta_{n}}. \end{equation} There are only finitely many pairs $B_{1},$ $B_{2}$ such that $|B_{1}|+ |B_{2}|=n,$ and hence the constants $\gamma_{n},$ $\beta_{n}$ may be chosen to be independent of $B_1,B_2$. The case $s_{n} \ge 1$ is similar. Using the triangle inequality and H\"older's inequality, we get \begin{eqnarray*} \|F^{\overline{m}}(\overline{g})\|_{s_{n}} & \le & \sum\limits_{l_{1}}' \|F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{g})(\lambda)\| _{s_{|B_{1}|}} \|F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{g})(\lambda)\|_ {s_{|B_{2}|}} \\ & \leq & C_{n}2^{-{\rm min}(\gamma_{|B_{1}|}, \gamma_{|B_{2}|})(|m^{(1)}| +|m^{(2)}|)} \\ && \cdot\ \sum\limits'_{l_{1}} \left(\prod\limits_{j \in B_{1}} \|g_{j}\chi_{E(m_{1},l_{1})} \|_{p}^{(1-\beta_{|B_{1}|})} \prod\limits_{j \in B_{2}}\|g_{j} \chi_{E(m_{1},l_{1}+1)}\|_{p}^{(1-\beta_{|B_{2}|})}\right). \end{eqnarray*} Provided that \begin{equation} \frac{1}{p} (|B_{1}|(1-\beta_{|B_{1}|})+|B_{2}|(1-\beta_{|B_{2}|}))>1, \end{equation} we can apply the same argument as in the case $s_{n}<1$ to prove (31). But (32) holds for all $|B_{1}|,$ $|B_{2}|\geq 1$ since by induction hypothesis $1-\beta_{r}> \frac{p}{2}$ for all $r 0$ there exists $N_{\delta}$ such that whenever $\min_i D_{i} \geq N_{\delta}$, we have \[ \left| T_{n}^{\infty,...\infty}(f_{1},...f_{n})(\lambda) - T^{D_{1},...D_{n}}_{n}(f_{1},...f_{n})(\lambda) \right| < \delta. \] \rm Our first result is the following maximal estimate: \\ \noindent \bf Theorem 5.2. \it Let \[ M_{T}(f_{1},...f_{n})(\lambda) = {\rm sup}_{D_{1},...D_{n}} |T_{n}^{D_{1},...D_{n}}(f_{1},...f_{n})(\lambda)|. \] Then for any $f_{i} \in L^{p},$ $1\leq p<2,$ we have \begin{equation} \|M_{T}(f_{1},...f_{n})\|_{s_{n}} \leq C_{n}\prod\limits _{i=1}^{n}\|f_{i}\|_{p}, \end{equation} where $s_{n}^{-1} =nq^{-1}$ and $q^{-1}+p^{-1}=1.$ \\ \begin{proof} \rm The proof uses a well-known device going back to Kolmogorov and Seliverstov. Namely, it is sufficient to show that for any measurable $D_{i}(\lambda),$ $i=1,...n,$ we have \[ \|T_{n}^{D_{1}(\lambda),...D_{n}(\lambda)}(f_{1},...f_{n})\|_{s_{n}} \leq C_{n} \prod\limits_{i=1}^{n} \|f_{i}\|_{p} \] with the constant $C_{n}$ independent of $D_{i}(\lambda).$ But the expression $T_{n}^{D_{1}(\lambda),...D_{n}(\lambda)}(f_{1},...f_{n}) (\lambda)$ is obtained from $T_{n}(f_{1},...f_{n})(\lambda)$ by replacing the kernels $k_{i}(\lambda,x)$ with the kernels \[ \tilde{k}_{i}(\lambda,x)= k_{i}(\lambda,x) \chi_{R^{+}}( D_{i}(\lambda)-x_{i}). \] By Theorem 3.1, these kernels satisfy the estimates (15) for all $p,$ $q$ such that $p<2$ and $q^{-1}=1-p^{-1},$ with constants in the norm bounds that do not depend on $D_{i}(\lambda).$ Therefore Theorem 4.1 is applicable and directly leads to (33). \end{proof} \rm As one may expect, the maximal estimate (33) implies a.e. convergence. \\ \noindent \bf Theorem 5.3. \it The operator $T_{n}(f_{1},...f_{n})$ converges for almost every $\lambda$ on any functions $f_{i} \in L^{p},$ $1\leq p <2.$ \\ \begin{proof} \rm Suppose, on the contrary, that for some $\epsilon>0$ we have a set $S_{\epsilon}$ of positive Lebesgue measure such that for any $N$ there exist $D_{ij}>N,$ $i=1,...n,$ $j=1,2$ such that \[ |T_{n}^{D_{11},...D_{n1}}(f_{1},...f_{n})(\lambda)- T_{n}^{D_{12},...D_{n2}}(f_{1},...f_{n})(\lambda)|>\epsilon \] for every $\lambda \in S_{\epsilon}.$ Let us denote $f_{i,N}(x_{i})= f_{i}(x_{i})\chi_{R^{+}}(N-x_{i}).$ For any $D_{i}>N,$ we have \[ |T_{n}^{D_{1},...D_{n}}(f_{1},...f_{n})(\lambda)- T_{n}^{D_{1},...D_{n}}(f_{1,N},...f_{n,N})(\lambda)| \leq \sum\limits_{j=1}^{n} |T_{n}^{D_{1},...D_{n}}(g_{1j},...g_{nj})(\lambda)|, \] where $g_{ij}=f_{i,N}$ if $i < j,$ $g_{ij}=f_{i}$ if $i>j,$ and $g_{ij}=f_{i}-f_{i,N}$ if $i=j.$ In other words, we expand the difference into a telescopic sum. To estimate each term in the sum, we can apply Theorem 5.2. We get \begin{eqnarray*} && {\rm sup}_{D_{1},...D_{n}}\|T_{n}^{D_{1},...D_{n}}(f_{1},...f_{n}) (\lambda)-T_{n}^{D_{1}, \dots D_{n}}(f_{1,N},...f_{n,N}) (\lambda)\|_{s_{n}}^{s_{n}} \\ && \qquad\qquad\qquad\le \sum\limits_{j=1}^{n}\| T_{n}^{D_{1}, \dots D_{n}}(g_{1j}, \dots g_{nj})(\lambda)\|_{s_{n}}^{s_{n}} \\ && \qquad\qquad\qquad\le C_{n}\sum\limits_{j=1}^{n}\left( \prod\limits_{i=1}^{n}\|g_{ij}\|_{p} \right)^{s_{n}}. \end{eqnarray*} Clearly, the right-hand side goes to zero as $N \rightarrow \infty,$ since every product contains the norm of $f_{i}-f_{i,N}$ for some $i.$ On the other hand, the left-hand side is by assumption bounded from below by $(\frac{\epsilon}{2})^{s_{n}}|S_{\epsilon}|$ for every $N.$ This gives a contradiction. \end{proof} \rm A straightforward adjustment of the above argument allows us to pass for almost every $\lambda$ to the infinite limit in any order (e.g.\ first in $D_{n},$ then in $D_{n-1}$ and so on). In order to prove a final Lemma that we will need in the iteration process, we need to consider a smaller class of multilinear operators than we did before. These are exactly the operators that appear in the process of solving equation (11) by iteration. \\ \noindent \bf Definition 5.4. \it We say that the multilinear transform $T_{n}$ belongs to the class $M_{n},$ if \begin{equation} T_{n}(f_{1},...f_{n})(\lambda)= \int\limits_{0}^{\infty} ... \int\limits_{0}^{\infty}dx_{1}...dx_{n} \prod\limits_{j=1}^{n}k_{j} (\lambda, x_{j})f_{j}(x_{j})\prod\limits_{j=2}^{n} \chi_{R^{+}} (x_{j}-x_{\sigma(j)}), \end{equation} where the kernels $k_{j}(\lambda,x)$ satisfy (15), and the function $\sigma$ takes values in $1,...n$ and satisfies $\sigma (j)0$ and $1\le p < 2$, for each index $1\le i\le n$, a function $f_i$ satisfies $(1+x)^{ \epsilon}f_{i}(x)\in L^p$. Then for every $\delta<\epsilon$, for almost every $\lambda\in R$, there exists $C(\lambda,n)<\infty$ such that for every nonnegative $N\in R$, \[ \left| T_{n}(f_1 - f_{1,N},f_{2} \dots f_{n})(\lambda)\right| \leq C(\lambda,n)(1+N)^{-n\delta}. \] \begin{proof} \rm The structure of operators of class $M_n$ is such that the integral defining $T_n$ extends only over those $x$ satisfying $x_j\ge x_1$ for every $j$, because of the requirement that $\sigma(j)1$. Since $f_1- f_{1,N}$ is supported where $x>N$, we therefore have for every $N,\lambda$ \[ T_{n}(f_1 - f_{1,N},f_{2} \dots f_{n})(\lambda) = T_{n}(f_1 - f_{1,N},f_{2} - f_{2,N} \dots f_{n} - f_{n,N})(\lambda). \] Fix $f_1,\dots f_n$. Since $\|f_j - f_{j,N}\|_p \le C\cdot(1+N)^{-\epsilon}$ by hypothesis, Theorem 5.2 now yields \[ 2^{nr\epsilon} \| \sup_{N\ge 2^r} |T_n(f_1-f_{1,N},f_2,\dots f_n)|(\lambda) \|_{L^{s_n}(d\lambda)} \le C < \infty \] for every nonnegative integer $r$, with $C$ independent of $r$. Consequently \[ \sum_{r=0}^\infty 2^{nr\delta} \sup_{N\ge 2^r} |T_n(f_1-f_{1,N},f_2,\dots f_n)|(\lambda) < \infty \] for almost every $\lambda$. To obtain the conclusion of the lemma, given any $N\ge 1$, choose $r$ so that $2^r\le N<2^{r+1}$ and apply this inequality. \end{proof} \rm \vspace{0.2cm} \begin{center} \large \bf 6. Conclusion of the proof of main results. \end{center} \vspace{0.2cm} With the general machinery built up in Sections 4 and 5 in hand, we are now in a position to complete the proofs of our main results. First we prove Thereom 1.3. \\ \begin{proof} \rm We recall that it suffices to show that under the assumptions of the theorem for a.e. $\lambda \in S$ we can find a function $q(x,\lambda)$ such that $q(x,\lambda) \in C^{1},$ $q(x,\lambda) \rightarrow 0,$ and the condition (9) holds: \begin{eqnarray*} q'(x,\lambda) & + & \frac{i}{2\Im(\theta \overline{\theta})}V(x)\overline{\theta}^{2}(x,\lambda) \exp (2ip(x,\lambda)) \\ &&\qquad + \frac{i}{2\Im(\theta \overline{\theta})}V(x) \theta^{2}(x,\lambda)\exp(-2ip(x,\lambda))q^{2}(x,\lambda) \qquad \in L^{1}. \end{eqnarray*} By assumption, the kernels \[ k_{1}(\lambda,x) =\overline{\theta}^{2}(x,\lambda)\exp (2ip(x,\lambda)) \] and $k_{2}(\lambda, x) = \overline{k_{1}(\lambda,x)}$ satisfy $L^{2}-L^{2}$ and trivial $L^{1}-L^{\infty}$ estimates. Therefore by interpolation, the corresponding operators map $L^p$ to $L^{q}$, where $q^{-1} = 1 - p^{-1}$ and hence $q>p$, for every $1\frac{1}{2}.$ Then by Lemma 6.1 we find that the expression on the right hand side is absolutely integrable for almost every $\lambda$, since it is the product of some $L^{p}$ function $V(x)$ with a factor which for almost every $\lambda$ is $O((1+x)^{-\frac{1}{2}-\eta})$ for some $\eta>0$, and hence belongs to $L^q$ where $q^{-1}=1-p^{-1}$. Hence $q^{(n)}(x,\lambda)$ satisfies the condition (9) and therefore we can take $q(x,\lambda)=q^{(n)}(x,\lambda)$ for a.e. $\lambda \in S.$ Then the first claim of Theorem 1.3 follows from Theorem 2.2 and Lemma 2.1. As a set of full measure in $S$ it suffices to take the set where all multilinear transforms $T_{j}(V,...V)$ (composed from $k_{1}(\lambda,x),$ $k_{2}(\lambda,x)$) of order not larger than $2^{n+1}-1$ converge. It remains to prove the formula (2) for the asymptotic behavior of the eigenfunctions. Notice that the asymptotic behavior stated in Theorem 2.2 differs from the asymptotic behavior we need to show to prove Theorem 1.3 only by the presence of an additional multiplier \begin{equation} \exp \left( \frac{i}{2\Im(\theta \overline{\theta})} \int\limits_{0}^{x}(1-|q|^{2})^{-1}V\Re(\theta^{2}q\exp(-2ip))\,dt \right) \end{equation} in the asymptotic formula for the solutions in Theorem 2.2. But note that the limit of the integral \[ \int\limits_{0}^{N}(1-|q(t,\lambda)|^{2})^{-1}V(t) \Re(\theta^{2}(t,\lambda)q(t,\lambda)\exp(-2ip(t,\lambda)))\,dt \] as $N \rightarrow \infty$ exists for a.e.\ $\lambda.$ Indeed, we can expand $(1-|q|^{2})^{-1}$ into absolutely convergent series in $|q^{2}|.$ Then the whole expression becomes represented as a sum of multilinear transforms $T_{j}^{N,\infty,...\infty}(V_{x},...V)(\lambda).$ Starting from some $l,$ the integrand will become absolutely integrable over the whole axis for a.e.\ $\lambda$ by Lemma 5.5; in the remaining finite sum every term is convergent by Theorem 5.3. Therefore, for a.e.\ $\lambda,$ the expression (36) can be written as $C(\lambda)(1+o(1))$ and hence can be omitted in the asymptotic expression for the solutions. This completes the proof. \end{proof} \rm We now prove Theorems 1.1 and 1.2. \\ \begin{proof} \rm It remains only to verify the $L^{2}-L^{2}$ bounds for the corresponding operator (4). It is convenient to choose $\theta(x,\lambda)=\exp (i\sqrt{\lambda}x)$ in the free case and $\theta (x,\lambda)$ Bloch functions in the periodic case. The corresponding $L^{2}-L^{2}$ bounds were already shown in \cite{Kis1}. For the sake of completeness and since the argument is not very long, we provide here a sketch of the proof for the free case. For the periodic case, the proof is analogous given standard information on the properties of the Bloch functions, see \cite{Kis1} for details. Without loss of generality, we can restrict our attention to some interval $(a,b),$ $01,$ we see that an operator with the kernel $Z(x,y)$ maps $L^{2}$ to $L^{2}$ (by Schur's test, for example). Therefore also \[ \|Kf\|^{2}_{L^{2}(a,b)} \leq C(\phi, a,b)\|f\|^{2}_{2}. \] \end{proof} \rm We conclude the paper by formulating one simple generalization of Theorem 1.3 (which implies the corresponding generalizations of Theorems 1.1 and 1.2). \\ \noindent \bf Theorem 6.2 \it Fix a potential $V(x).$ Suppose that there exists a monotone differentiable function $d(x),$ $d(x)>0,$ $d(x) \stackrel{x \rightarrow \infty}{\longrightarrow} 0,$ $d'(x) \leq 0,$ such that $V(x)d(x)^{-1} \in L^{p}$ with some $p<2$ and $V(x)d(x)^{N} \in L^{1}$ for some integer $N.$ Then under the assumptions of Theorem 1.3, all conclusions of Theorem 1.3 hold for perturbation of $H_{U}$ by $V(x).$ \\ \begin{proof} \rm Going through the proofs of Lemma 5.5 and Theorem 1.3, we substitute $x^{-\epsilon}$ with $d(x).$ Given that $d(x)^{-1}V(x) \in L^{p},$ $p<2,$ all proofs go unchanged, and in the final step instead of $V(x)(1+x)^{-\delta (2n-1) }$ being absolutely integrable we need to check that for some $N,$ \[ V(x)d(x)^{N} \in L^{1}. \] This is exactly what we assumed in the statement of Theorem 6.2. \end{proof} \rm We note one additional particular class of potentials which we may treat using Theorem 6.2. Namely, take any $V(x)$ such as in Theorem 1.3, take a sequence $\{x_{n}\}$ and insert intervals of arbitrary size $I_{n}$ at each point $x_{n}.$ Let $\tilde{V}(x)$ be a potential obtained from $V$ by adding such intervals $I_{n}$ where $\tilde{V}$ is zero. Then it is easy to construct a function $d(x)$ with the properties as in Theorem 6.2. For the details of such construction we refer to \cite{Kis} (where it was derived in a slightly different context). %\newpage %\vspace{0.2cm} \begin{center} \large \bf Appendix: Singular potentials. \end{center} \vspace{0.2cm} In Appendix, we discuss the preservation of the absolutely continuous spectrum for potentials with strong local singularities. The proof turns out to be almost entirely parallel to the non-singular case, so we mostly sketch the arguments with a few exceptions. We should note, however, that the result has some interest in it. In the explicit construction of power decaying potentials such that the corresponding Schr\"odinger operators have purely singular spectrum \cite{Rem} one can try to use the possible singularity of the potential to get singular spectrum under stronger decay conditions. The results of the appendix show that such plan does not work out, at least the fundamental exponent $\frac{1}{2}$ and virtually all results we have shown before extend to the situation where strong local singularities are allowed. We will consider the potentials from the spaces $l^{p}(L^{1}),$ $1\leq p \leq 2,$ with the norm given by \[ \|f\|_{l^{p}(L^{1})} = \left( \sum\limits_{n=0}^{\infty} \left( \int\limits_{n}^{n+1} |f(x)|\,dx \right)^{p} \right)^{\frac{1}{p}}. \] The main result we show is \\ \noindent \bf Theorem A.1. \it Suppose that $U$ is continuous periodic (in particular, the free case $U=0$ is of course included). Let the perturbation $V(x)$ be such that there exist $p\leq 2$ and $\epsilon>0$ with $V(x)x^{-\epsilon} \in l^{p}(L^{1}).$ Then all conclusions of Theorems 1.2 hold, in particular the absolutely continuous spectrum $S$ of the operator $H_{U}$ is preserved under the perturbation by $V$ and for almost every $\lambda \in S$ there exist solutions of the equation (1) with the pure WKB asymptotic behavior in the main term (i.e. with asymptotic behavior given by (2)). \\ \rm As before, Theorem A.1 is a corollary of the following general criterion (an analogue of Theorem 1.3): \\ \noindent \bf Theorem A.2. \it Suppose that the potential $V(x)$ is such that there exist $\epsilon>0$ and $p \leq 2$ so that $x^{\epsilon} V(x)\in l^{p}(L^{1}).$ Assume that an operator \[ (Kf)(\lambda) = \chi(S)\int\limits_{0}^{\infty}\theta(x,\lambda)^{2} \exp \left(\frac{i}{\Im(\theta\overline{\theta}')}\int\limits_{0}^{x}V(t)| \theta(t,\lambda)|^{2}\,dt\right)f(x)\,dx \] satisfies the bound \[ \|Kf(\lambda)\|_{L^{2}(S)} \leq C \|f\|_{l^{2}(L^{1})} \] on functions $f$ of compact support. Then the absolutely continuous spectrum of $H_{U}$ supported on the set $S$ is preserved under perturbation by $V,$ i.e. the set $S$ belongs to the essential support of the absolutely continuous part of the spectral measure of operator $H_{U+V}.$ Moreover, for almost every $\lambda \in S,$ there exist solutions $\psi_{\lambda}(x),$ $\overline{\psi}_{\lambda}(x)$ of the equation (1) with the asymptotic behavior (2). \\ \rm We will indicate the changes in the proof of Theorem 1.3 which are necessary to prove Theorem A.2. First, we need the following substitute for Theorem 3.1. \\ \noindent \bf Lemma A.3. \it Suppose that $1\le p<\infty$, that $q>p$, and that $K$ is a bounded linear operator from $l^p(L^1(R))$ to $L^q(R)$. Then the maximal function given by \[ (M_{K}f)(\lambda) = {\rm sup}_{N} | K(f \chi_{R^+}(N-x))(\lambda)| \] satisfies \[ \|M_{K}f\|_{q} \leq C_{q}\|f\|_{l^{p}(L^{1})} \] for the same pair of exponents $p,q$. Moreover the integral defining $(M_{K}f)(\lambda)$ converges for almost every $\lambda$, for any $f \in l^{p}(L^{1})$. \begin{proof} \rm Given the function $f$ with $\|f\|_{l^{p}(L^{1})}=1,$ we consider the family of intervals $E(m,l)$ similar to the $L^{p}$ case. Namely, first we consider intervals $E(1,1)$ and $E(1,2),$ such that their union is the whole half-axis, the first interval lies to the left of the second and \[ \|f\chi_{E(1,1)}\|^{p}_{l^{p}(L^{1})}= \|f\chi_{E(1,2)}\|^{p}_{l^{p} (L^{1})}. \] We note that in contrast to the $L^{p}$ case, we cannot in general say that $\|f\chi_{E(1,i)}\|^{p}_{l^{p}(L^{1})}=2^{-1}.$ However, it is easy to see that \[ \|f\chi_{E(1,i)}\|^{p}_{l^{p}(L^{1})} \leq 2^{-1}. \] If the sets $E(1,1),$ $E(1,2)$ are unions of the integer intervals $(l,l+1),$ we have equality. Otherwise each set $E(1,i)$ is a union of integer intervals and a set $A_{i},$ such that $A_{1} \cup A_{2} =(m,m+1)$ for some $m.$ In this case \[ \left( \int |f\chi_{A_{1}}|\,dx \right)^{p} + \left( \int |f\chi_{A_{2}}|\,dx \right)^{p} \leq \left( \int\limits_{m}^{m+1}|f(x)|\,dx \right)^{p} \] (since $p \ge 1$), and hence \[ \|f\chi_{E(1,2)}\|^{p}_{l^{p}(L^{1})}+ \|f\chi_{E(1,1)}\|^{p}_{l^{p} (L^{1})} \leq 1. \] We decompose each of the intervals $E(1,i)$ further as in $L^{p}$ case. On the step $m,$ we obtain a family of intervals $E(m,l),$ such that $\cup_{l}E(m,l)=R^{+}, $ $E(m,l_{1})$ lies to the left of $E(m,l_{2})$ if $l_{1} \leq l_{2}.$ We choose these intervals so that the total number of the intervals of generation $m$ is $2^{m}$ and $\|f\chi_{E(m,l)}\|^{p}_{l^{p}(L^{1})} \leq 2^{-m}$ for every $l.$ Such a family is obtained by splitting every interval of given generation into two equal pieces, with the same arguments as in the first step. Let \[ (M_{K,m}f)(\lambda)={\rm sup}_{l}| M_{K}(f\chi_{E(m,l)})(\lambda)|. \] Then \[ (M_{K}f)(\lambda) \leq \sum\limits_{m=1}^{\infty}(M_{K,m}f)(\lambda). \] This follows from the construction of the family $E(m,l).$ Indeed, modulo a set on which $f$ vanishes almost everywhere, any interval $[0,N]$ may be decomposed as a disjoint union of intervals $E(m,l)$, with at most one such interval for each generation number $m$. Summing over $m$ and invoking the triangle inequality leads to the desired majorization for $M_K f$. Consequently \[ \|M_{K}f\|_{q} \leq \sum\limits_{m=1}^{\infty}\|M_{K,m}f\|_{q}. \] On the other hand, \[ \|M_{K,m}f\|_{q}^{q} \leq \sum\limits_{l=1}^{2^{m}} \|K(f\chi_{E(m,l)})\|_{q}^{q} \leq \sum\limits_{l=1}^{2^{m}} \|f\chi_{E(m,l)}\|^{q}_{l^{p}(L^{1})} \leq 2^{m(1-\frac{q}{p})}. \] Therefore, \[ \|M_{K}f\|_{q} \leq \sum\limits_{m=1}^{\infty} 2^{m\left(\frac{1}{q}-\frac{1}{p}\right)} \leq C_{q}\|f\|_{l^{p}(L^{1})} \] (we assumed $\|f\|_{l^{p}(L^{1})}=1,$ but the bound extends to all $f$ by sublinearity of $M_{K}$). Almost everywhere convergence follows from the maximal estimate in a standard way. \end{proof} \rm Now we prove Theorem A.2. \\ \begin{proof} \rm The proofs of multilinear transform properties and almost everywhere convergence estimates go exactly the same way as before. The family $E(m,l)$ has the same properties as in $L^{p}$ case, in particular martingale-type property (two sets either disjoint or one is contained in another). The Lemma 4.3 clearly remains valid. The function \[ f(x) = \left( \sum\limits_{i=1}^{n} |f_{i}(x)|^{p} \right)^{\frac{1}{p}}, \] used to construct the family $E(m,l)$ in the proof may be replaced by \[ f(x)= \sum\limits_{i=1}^{n} |f_{i}(x)|. \] The only other change we need to make in the proof is to change throughout $\|\cdot\|_{p}$ to $\|\cdot\|_{l^{p}(L^{1})}.$ \end{proof} \rm To prove Theorem A.1, we need to show the $l^{2}(L^{1})-L^{2}(S)$ bounds. \\ \begin{proof} \rm We will show the proof of the needed norm bound only for the case $U=0.$ The general periodic case follows from the properties of the Bloch functions in a way parallel to the free case. We refer to \cite{Kis1} for necessary information and a similar argument. Clearly we can restrict our attention to some compact interval $I =(a,b),$ $b>a>0.$ It is sufficient to show that the operator \[ (Kf)(\lambda)= \phi(\lambda)^{2}\int\limits_{0}^{\infty} \exp\left( 2i\sqrt{\lambda}x - \frac{i}{\sqrt{\lambda}} \int\limits_{0}^{x}V(t)\,dt \right)f(x)\,dx \] satisifes $l^{2}(L^{1})-L^{2}(I)$ bound on functions of compact support, where $\phi \in C_{0}^{\infty}(R^{+})$ and $\phi(\lambda)=1$ for $\lambda \in I.$ Note that \begin{eqnarray} (Kf)(\lambda) & = & \sum\limits_{l=0}^{\infty}\phi(\lambda) \exp\left(2i\sqrt{\lambda}l-\frac{i}{\sqrt{\lambda}} \int\limits_{0}^{l}V(t)\,dt\right) \nonumber \\ && \cdot\ \phi(\lambda) \int\limits_{0}^{1}\exp\left(2i\sqrt{\lambda}y- \frac{i}{\sqrt{\lambda}}\int\limits_{l}^{y+l}V(t)\,dt\right)f(y+l)\,dy. \end{eqnarray} We can write $Kf(\lambda)$ as follows: \[ Kf(\lambda) = \sum\limits_{l=0}^{\infty}\phi(\lambda) \exp\left(2i\sqrt{\lambda}l-\frac{i}{2\sqrt{\lambda}} \int\limits_{0}^{l}V(t)\,dt\right) f(\lambda,l), \] where the expression $f(\lambda,l)$ has the following property: for every $m \geq 0,$ \[ |\partial_{\lambda}^{m}f(\lambda,l)| \leq C_{m}(I,\phi){\rm sup}_{n} \left(\int\limits_{n}^{n+1}|V(x)|\,dx\right)^{m}f_{l} \] where $f_{l} = \int_{l}^{l+1}|f(x)|\,dx.$ This property of $f(\lambda,l)$ is evident from (38). We compute \[ \|Kf\|^{2}_{L^{2}(I)} \leq \sum\limits_{l=0}^{\infty} \sum\limits_{m=0}^{\infty} \int d\lambda \phi^{2}(\lambda) \exp\left(2i\sqrt{\lambda}(l-m)-\frac{i}{2\sqrt{\lambda}} \int\limits_{m}^{l}V(t)\,dt\right)f(\lambda,l)\overline{f}(\lambda,m). \] Let us integrate by parts, differentiating $f(\lambda,l)\overline{f}(\lambda,m)$ and integrating the rest. By virtually the same computation as one which led us to (37) we obtain \[ \left| \int d\lambda \phi(\lambda)\exp \left(2i\sqrt{\lambda}(l-m)-\frac{1}{2\sqrt{\lambda}} \int\limits_{m}^{l}V(t)\, dt\right) \right| \leq C_{r}{\rm min}(1,|l-m|)^{-\frac{r}{p}} \] for every positive integer $r.$ Therefore, taking into account the properties of $f(\lambda,n),$ we obtain \[ \|Kf\|^{2}_{L^{2}(I)} \leq C_{r}\sum\limits_{l=0}^{\infty} \sum\limits_{m=0}^{\infty} {\rm min}(1,|m-l|)^{-\frac{r}{p}}f_{m}f_{l}. \] Taking $r$ large enough ($\frac{r}{p}>1$ will do) we see that the operator $K$ satisfies the required bound. \end{proof} \newpage \begin{center} \large \bf Acknowledgment. \end{center} A.K. thanks B.~Simon for constant support and encouragement and expresses his gratitude to S.~Molchanov and C.~Remling for stimulating discussions. A.K.'s work at the MSRI has been supported in part by NSF grant DMS 902140. 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