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\begin{document}

\begin{center}
\large \bf  Absolutely continuous spectrum for one-dimensional 
Schr\"odinger operators 
with slowly decaying potentials: some optimal results.
\end{center}

\begin{center}
Michael Christ \\
Department of Mathematics \\
University of California \\
Berkeley CA 94720 
\end{center}
\begin{center}
and
\end{center}
\begin{center}
Alexander Kiselev \\
Mathematical Sciences Research Institute \\
Berkeley CA 94720 \\
e-mail: kiselev@msri.org \\
\end{center}

\begin{abstract} 
We prove new results  on stability of the absolutely continuous spectrum 
of one-dimensional Schr\"odinger operators. In particular, our results 
imply that 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 known to be optimal on the power 
scale.
We derive a new general criterion for stability of the absolutely 
continuous spectrum 
of one-dimensional Schr\"odinger operators under slowly decaying 
perturbations. We also consider more general classes of perturbing
 potentials which are not necessarily power decaying. In all cases 
we show that the main term of the asymptotic behavior of 
the generalized eigenfunctions has WKB form for almost all energies.
 The proof uses new maximal function and norm estimates and a.e. 
convergence results for certain multilinear integral operators. 
\end{abstract}

\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 zero. We assume that $U$ is
 some bounded function such that the operator $H_{U}$ 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 criteria implies that the essential spectrum of the operators 
$H_{U}$ and $H_{U+V}$ coincide.  
We would like to find 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 been 
known for a long time that if the perturbation $V(x)$ is absolutely
integrable, then the absolutely
 continuous spectrum of the original operator is preserved. Up to a recent
 time, this result was basically all
what was known on the preservation of the absolutely continuous spectrum of
Schr\"odinger operators
under decaying perturbations in 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.
We mention a result going back to Weidmann \cite{Weid} that 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 satisfying certain conditions 
on its
derivatives. See, for example, \cite{As}, \cite{Bus}, \cite{Hor}.  These 
results hold in any
dimension and the proofs use a method of approximating the scattering
 trajectories
by the 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 describe spectral behavior of  spherically 
symmetric (i.e. essentially one-dimensional) specific oscillating 
potentials,
 the
typical example being potentials of the type $V(x) = \frac{\sin x^{\alpha}}
{x^{\beta}}$ with $\alpha,$ $\beta$ positive. Such potentials
 in general do not satisfy  conditions on the decay of the 
derivatives
 which are needed for the method of 
the above works to be applicable. We mention 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 first 
to discover an example which shows that already for potentials decaying 
as a Coulomb potential at infinity, i.e. $V(x)=O(\frac{1}{1+|x|}),$
the spectrum need not to be purely absolutely continuous and positive 
embedded eigenvalues may occur. Moreover, Naboko \cite{Nab} and  later 
Simon
\cite{Sim1} found different constructions which show that for potentials 
decaying arbitrarily slower than
at a Coulomb rate, one can already have very striking spectral phenomena.
Namely, for every function $C(x)$ going monotonously 
to
infinity as $x$ goes to infinity at an arbitrarily slow rate one can find
a potential $V(x)$ satisfying
\[ |V(x)| \leq \frac{C(x)}{1+|x|}, \]
such that the corresponding 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.
We recall that the set $E$ 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 proven, which implies the stability of the absolutely 
continuous spectrum of the operator $H_{U}$ under perturbations $V(x)$
 satisfying $|V(x)| \leq C(1+x)^{-\frac{2}{3}-\epsilon}$ given that 
certain 
operator, composed from the generalized eigenfunctions of $H_{U},$ 
satisfies
 $L^{2}-L^{2}$ estimate. In particular, it was shown that the absolutely 
continuous spectrum of free and periodic one-dimensional 
Schr\"odinger operators was 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  $V(x) \leq C(1+x)^{-\frac{1}{2}},$ 
which
lead to the purely singular spectrum on $R^{+}.$ 
We remark that 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 some results on the preservation of absolutely
 continuous spectrum of optimal nature for power decaying potentials. 
In particular, we prove \\

\noindent \bf Theorem 1.1. \it Suppose that the potential $V(x)$ is such 
that there exist $\epsilon>0$ and $p\leq 2$ so that $V(x)x^{\epsilon} \in 
L^{p}(R^{+}).$
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 a.e. $\lambda \in R^{+},$ there exist 
solutions 
$\phi_{\lambda}(x)$ and $\overline{\phi}_{\lambda}(x)$ of generalized 
eigenfunction equation
\[ -y'' + V(x)y=\lambda y \]
with the asymptotic behavior which is 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 \rm Of course this result also implies the stability of the 
absolutely continuous spectrum for all $V(x)$ satisfying $|V(x)| \leq 
C(1+x)^{-\frac{1}{2}-\epsilon}$ with some $\epsilon >0.$ 

 We have a similar result for perturbations of periodic 
Schr\"odinger operators. Namely if $U$ is continuous periodic, $S=\cup_{n=0}
^{\infty}(a_{n},b_{n})$ is 
the band spectrum of the unperturbed operator, and $\theta(x, \lambda)$
 are Bloch functions, we have \\

\noindent \bf Theorem 1.2. \it If the potential $V(x)$ is as in Theorem
 1.1, 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}

\rm Both theorems will follow from a certain general criterion. 
Suppose that $H_{U}$ is an operator such that all solutions of the equation 
\begin{equation}
 -y'' + U(x)y = \lambda y
\end{equation}
are bounded for a.e. $\lambda \in S,$ 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 $x^{\epsilon}V(x)\in L^{p}.$
 Assume that we can choose $\theta(x,\lambda)$ so that an 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 $L_{2}(R^{+},dx)-L^{2}(S,d\lambda)$ 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 $V,$ i.e. the set $S$ belongs to an 
essential support of the absolutely continuous part of the spectral 
measure of operator $H_{U+V}.$
Moreover, for a.e. $\lambda \in S,$ there exist solutions 
$\psi_{\lambda}(x),$ $\overline{\psi}_{\lambda}(x)$ of the equation (1)
with the asymptotic behavior (2).  \\

\noindent \it Remarks. \rm 1. We notice that the assumption 
of the boundedness of all solutions at a.e. 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 analogs 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. While we seem to come quite close, we cannot yet treat 
potentials under $L^{p},$ $1<p \leq 2$ assumption only. \\

\rm The main new technique we develop in this paper involves norm 
bounds and a.e. convergence results for multilinear integral 
operators of certain type, which may be of interest on its own. 
 The plan of the paper is as follows. In Section 2 we 
discuss the basic scheme of asymptotic integration. In Section 3 
we formulate first key results on the estimates for maximal functions 
of certain integral operators. In
 Section 4 we establish norm estimates for multilinear integral
 operators, and in Section 5 prove corresponding a.e. convergence results.
 Section 6 contains the conclusion of the proof of all main results.
 In the Appendix we discuss generalization of our results to the
case of potentials which may have strong local singularities.

Independently,  results related to ours were obtained by Christian Remling
 \cite{Remac} by a very different method, based in part on the 
ideas from \cite{Kis1} and \cite{Mol1}. Some of the results we prove 
here (along with some of the results of \cite{Remac}) 
were announced in \cite{CKR}. The announcement \cite{CKR} also 
contains a list of open problems  which we find most 
interesting.

\begin{center}
 \large \bf 2. Asymptotic integration and bounded eigenfunctions.
\end{center}


To prove the stability of the absolutely continuous spectrum of the operator
 $H_{U},$ we will use the following Lemma: \\

\noindent \bf Lemma 2.1. \it Let the potential $W(x)$ be locally integrable 
and satisfy
\[ {\rm sup}_{x} \int\limits_{|x-t|\leq 1} |W_{-}(t)|\,dt < \infty \]
($W_{-}$ denotes the negative part of the potential $W$).
Suppose that for every energy $\lambda$ from the set $E$ of positive 
Lebesgue
 measure, all solutions of the equation
\[ -\frac{d^{2}}{dx^{2}}y(x) + W(x)y(x)=\lambda y(x) \]
are bounded. Then the set $E$ belongs to an essential support of the 
absolutely
 continuous part of the spectral measure of the operator $H_{W}.$ \\

\rm The proof of this Lemma may be found in \cite{Sto} (see also 
\cite{Sim3}). We notice that in different formulations, the fact 
that bounded solutions imply absolutely continuous spectrum was known
 for a long time (see, e.g., \cite{Car}). Lemma 2.1 
is the most convenient statement for our purpose. 

The general plan of our proof is similar to \cite{Kis1} and
 may be described as follows. By assumption, 
we know that for every $\lambda \in S,$ all solutions of the generalized 
eigenfunction equation (3) for the unperturbed operator are bounded.
Examples with imbedded eigenvalues \cite{Nab}, \cite{Sim1}, \cite{KRS}
show that if $V(x)$ is not short-range, we cannot hope in general that for
 every $\lambda \in S$ we still have only bounded solutions for a perturbed
 equation; there may exist rather rich, dense in $S$ set for which we will 
have decaying ($L^{2}$) and therefore 
also growing solutions. Our goal will be to show that nevertheless 
for a.e. $\lambda \in S,$ we still have only bounded solutions. This will 
ensure that the absolutely continuous spectrum is preserved, although 
embedded singular spectrum may occur.

Thus, our goal is to study the solutions of the equation 
\[ -y''+(U(x)+V(x))y=\lambda y. \]
We rewrite this
equation as a system
\[ y_{1}' = \left( \begin{array}{cc} 0 & 1 \\ U+V-\lambda &
 0 \end{array} \right)y_{1}, \]
where $y_{1}$ is now a vector $\left(
 \begin{array}{c} y \\ y' \end{array} \right).$
Let us apply a variation of the parameters transformation with solutions 
of the unperturbed equation
\[ y_{1} = \left( \begin{array}{cc} \theta (x, \lambda) & \overline{\theta}
 (x, \lambda)
\\ \theta '(x, \lambda) & \overline{\theta}' (x, \lambda) \end{array} 
\right) y_{2}, \]
to bring the equation to a more symmetric form
\begin{equation} y_{2}' = \frac{i}{2\Im (\theta \overline{\theta}')}\left( 
\begin{array}{cc} V(x)|\theta (x, \lambda)|^{2} & V(x) \overline{\theta}
 (x, \lambda)^{2} \\ -V(x) \theta (x, \lambda)^{2} & -V(x) |\theta (x, 
\lambda)|^{2} \end{array} \right)y_{2}. 
\end{equation}
Notice that $2i\Im (\theta, \overline{\theta'})$ is a Wronskian of two 
solutions $\theta$ and $\overline{\theta}$ and hence is independent of $x.$
Let us introduce the notation
\[ p(x,\lambda) = \frac{1}{2\Im (\theta \overline{\theta'})} 
\int\limits_{0}^{x}V(y) |\theta(y,\lambda)|^{2}\,dy. \]
It will be convenient to apply to (5) an additional transformation
\[ y_{2} = \left( \begin{array}{cc} \exp (ip(x,\lambda))  & 0
 \\ 0 & \exp (-ip(x,\lambda)) \end{array} \right)y_{3} . \]
We arrive at the following equation for $y_{3}:$
\begin{equation}
y_{3}' = \frac{i}{2\Im (\theta \overline{\theta}')}\left( 
\begin{array}{cc} 0 & V(x) \overline{\theta}
 (x, \lambda)^{2}\exp (2ip(x,\lambda)) \\ -V(x) \theta (x, \lambda)^{2}
\exp (-2ip(x,\lambda)) & 0 \end{array} \right)y_{3}. 
\end{equation}
We  follow the idea of ``$(I+Q)$" asymptotic integration 
originating in Harris-Lutz \cite{HaLu}: to find some invertible
 transformation of equation (6) which would make off-diagonal 
terms absolutely integrable and then  apply 
Levinson's theorem (see, e.g. \cite{CoLe}) to find the asymptotic 
behavior of solutions of the resulting equation. If we succeed, 
we can go back and find the asymptotic behavior of the solutions
 of the original equation.

Let
\begin{equation}
 y_{3} = (1-|q|^{2})^{-\frac{1}{2}}\left( I+ Q \right) y_{4}, 
\end{equation}
where $I$ is an identity matrix, 
\[ Q = \left( \begin{array}{cc} 0 & q \\ \overline{q} & 0 \end{array} 
\right), \]
and $q(x,\lambda)$ is some function to be defined. 
A computation gives for $y_{4}$
\[
y_{4}' = (1-|q|^{2})^{-1} \left( \frac{i}{2\Im (\theta \overline{\theta}')}
\left( \begin{array}{cc} V \Re (\theta^{2}q \exp(-2ip)) & V
\overline{\theta}^{2}\exp(2ip) + V\theta^{2}q^{2}\exp(-2ip) \\
- V\theta^{2}\exp(-2ip) - V\overline{\theta}^{2}\overline{q}^{2}\exp(2ip) 
& -V \Re (\theta^{2}q \exp(-2ip)) \end{array} \right)\right. + \] 
\begin{equation} +\left( \left. \begin{array}{cc} \frac{1}{2}
(q\overline{q}'
-q'\overline{q}) &
-q' \\ -\overline{q}' & \frac{1}{2}(q'\overline{q}-q\overline{q}') 
\end{array}
 \right) \right) y_{4}.
\end{equation}
We summarize the main result of this section in \\

\noindent \bf Theorem 2.2. \it Suppose that for some given $\lambda$ there 
exists a function $q(x,\lambda) \in C^{1},$ $q(x,\lambda) \stackrel{x 
\rightarrow \infty}{\longrightarrow} 0,$ such that 
\begin{equation}
q'(x,\lambda) + \frac{i}{2\Im(\theta \overline{\theta})}V(x)\overline
{\theta}^{2}(x,\lambda)\exp (2ip(x,\lambda)) + \frac{i}{2\Im(\theta 
\overline{\theta})}V(x) \theta(x,\lambda)^{2}\exp(-2ip(x,\lambda))
q^{2}(x,\lambda) \in L^{1}.
\end{equation}
Then all solutions of the generalized eigenfunction equation (1) are
 bounded. Moreover, there are two solutions $\psi(x, \lambda),$ 
$\overline{\psi}(x,\lambda)$ with the asymptotic behavior 
\begin{equation}
\psi(x,\lambda) = \theta(x,\lambda) \exp (i p(x,\lambda)) \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)
\left(1+o(1)\right).
\end{equation}

\noindent \it Remark. \rm We note that in all applications that we will 
 have the last 
cumbersome term in the product giving the asymptotic behavior will turn 
out to be integrable. \\

\noindent \bf Proof. \rm The proof follows immediately from Levinson's 
theorem (see, e.g. \cite{CoLe}) and equation (8). We may consider
 this system of equations only for $x$ large enough, so that 
$|q(t,\lambda)|<1$ for all $t>x$ 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. $\Box$

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 a.e. 
$\lambda \in S.$ The main problem is that if we try to solve equation 
\begin{equation}
q'(x,\lambda) + \frac{i}{2\Im(\theta \overline{\theta})}V(x)
\overline{\theta}^{2}(x,\lambda)\exp (2ip(x,\lambda)) + 
\frac{i}{2\Im(\theta \overline{\theta})}V(x) \theta^{2}(x,\lambda)
\exp(-2ip(x,\lambda))q^{2}(x,\lambda) =0
\end{equation}
by iteration, we obtain  expressions involving multilinear integral
 operators of certain type. We need to show that these expressions 
converge 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 a.e. $\lambda.$ 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

\begin{center}
 \large \bf 3. A.e. convergence for integral operators.
\end{center}

Let the operator $K$ be defined on the 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}
In \cite{Kis1}, the following was proved: \\

\noindent \bf Theorem 3.1. \it Suppose that an operator K, defined by \rm 
(12) \it with bounded kernel $k(\lambda,x),$ satisfies the norm estimate
$\| Kf\|_{2} \leq C_{2} \|f\|_{2}$ for all bounded functions of compact
support. Then for every $q>2$ and $p$ such that $q^{-1}+p^{-1}=1,$
we have the following estimate for the maximal function:
\begin{equation}
 \|M_{K}f\|_{q} \leq C_{q} \|f\|_{p} \;\: for \;\: every \;\: f \in L_{p}
\end{equation}
As a consequence, the integral 
\[ \int^{N}_{0} k(\lambda,x)f(x)\,dx \]
 converges
as $N \rightarrow \infty$ for almost every value of 
$\lambda$ if $f \in L_{p}.$  \\

\rm For the proof of Theorem 3.1 we refer to \cite{Kis1}; we also 
sketch in the appendix a proof of a very similar result 
(Lemma A.3). The proof of Theorem 3.1 may be obtained from that
 proof by a simple modification.

Finally, we note that it is a general principle for singular 
integral operators of this type that a couple of $L^{p}$-type 
norm estimates, $L^{p_{1}}-L^{q_{1}}$ and $L^{p_{2}}-L^{q_{2}}$ valid 
on functions of compact support imply the 
estimates of type (14) (but may be in general involving Lorentz spaces)
for the corresponding maximal function (and hence 
a.e. convergence) for all 
intermediate $L^{p},$ $p_{1}< p < p_{2}.$ See \cite{Kis2} for more
 details. 

\begin{center}
\large \bf 4. Norm estimates for multilinear transforms.
\end{center}

In this section, we study the questions related to the norm estimates  
for certain multilinear transforms. The results of this and next sections
 will enable us to fulfill the plan sketched in the end of Section 2 and
 find the function $q(x,\lambda)$ with the needed properties
 for a.e. $\lambda$ by iteration of (11). 

Suppose that the functions $k_{i}(\lambda, x),$ $i=1,...n$ are defined on
 $I \times R^{+},$ where $I$ is some measurable set in $R.$ We assume that
 the 
operators 
\[ (K_{i}f)(\lambda) = \int\limits_{R^{+}} k_{i}(\lambda, x)f(x)\,dx \]
satisfy the bounds 
\begin{equation}
 \|K_{i}f\|_{L^{q}(I, d\lambda)} \leq C_{i}\|f\|_{L^{p}(R^{+}, dx)} 
\end{equation}
on functions of compact support
for some $2>p \geq 1$ 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},..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}
$x=(x_{1},...x_{n}).$ 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: \\
\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 $f_{i} \in L^{p}(R^{+},dx),$ $i=1,...n,$
 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 Remark. \rm Notice that the conclusion of Theorem 4.1 holds 
in particular when $s_{n}<1.$ \\

\noindent \rm  By assumption, the value of $T_{n}(f_{1},...f_{n})(\lambda)=
g(\lambda)$ 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 $\{E(m,l) \}_{m=1,\, l=1}^{\infty, \,\,\, j=2^{m}}:$
 any two intervals are either disjoint or one is contained in another. 

We  proceed to decompose the ``diagonal" characteristic functions
 in a convenient way. \\
\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_{l=1, 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}

\noindent \bf Proof. \rm 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 
\[ H_{12} \cap ({\rm supp}_{x_{1}}f \times {\rm supp}_{x_{2}}f)=
 \cup_{m=1}^{\infty} \cup_{l=1, l {\rm odd}}(E(m,l) 
\times E(m,l+1) 
) \]
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. $\Box$ \\

\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. \\

\noindent \bf Proof of Theorem 4.1. \rm  Without loss of
 generality, we assume throughout the proof that 
$\|f_{i}\|_{p}=\frac{1}{n}$ 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 
\[ A=\{ \alpha_{1},...\alpha_{|A|} \}. \]
We begin by substituting the result of Lemma 4.2 into formula (16):
\begin{equation}
 T_{n}(\overline{f})= 
 \sum\limits_{m_{1}}^{\infty}...\sum\limits_{m_{|A|}=1}^{\infty}
 \sum\limits_{l_{1}=1} '...\sum\limits_{l_{|A|}=1} '\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(\overline{f})(\lambda) \leq 
\sum\limits_{m_{1}}^{\infty}...\sum\limits_{m_{|A|}=1}^{\infty}
F^{\overline{m}}
(\overline{f})(\lambda), 
\end{equation}
where $\overline{m}= (m_{1},...m_{|A|})$ and 
\begin{equation}
 F^{\overline{m}}(\overline{f})(\lambda)=
\sum\limits_{l_{1}=1} '...\sum\limits_{l_{|A|}=1} '\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 
\[ G(j,\overline{l}) = \left[ \bigcap_{t: j=i_{t}}E(m_{t},l_{t}) \right] 
\bigcap
\left[ \bigcap_{t:j=i'_{t}}E(m_{t},l_{t}+1) \right].  \]
For many values of $\overline{l},$ the set $G(j,\overline{l})$ is
 empty for some $j.$
Such terms contribute zero to the sum, and 
this is an observation underlying the estimate that we are going to 
derive for $F^{\overline{m}}(\overline{f}).$ 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).
 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 
\[ 1 \geq 1-\beta_{n} > {\rm max}(\frac{p}{q}, \frac{p}{2}) . \]
We may proceed to prove (22) by induction on $|A|.$ The case 
$|A|=0$ is immediate from hypothesis on $K_{j}$ by H\"older inequality
(in this case $\overline{m}=0$). 

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 $m_{N}=m_{1}.$ 
Drop from the sum (21) all terms for which there exists $j$ such that 
$G(j,\overline{l})=\emptyset;$ such terms contribute $0.$ 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 $\}$.
Moreover if $2 \leq t \leq N$ (that is, if $m_{t}=m_{1}$) then
\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 $t \in B_{1},$ $t \in B_{2}$ respectively.  \\

\noindent \bf 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
by martingale-type property one is contained in another.
 
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 $m_{t_{1}},...m_{t_{r}}.$ 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 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.
$\Box$ \\

\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 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 vertex $j_{2}$ where 
$f_{j_{2}}(x_{j_{2}})$ is multiplied by 
$\chi_{E(m_{1},l_{1}+1)}(x_{j_{2}}).$ In this case, $G(j,\overline{l})$
is zero and hence $\overline{l}$ is not remaining. \\

By Lemma 4.3
\[ F^{\overline{m}}(\overline{f})(\lambda) = \sum\limits_{l_{1}} '
\sum\limits_{l_{N+1}} '...\sum\limits_{l_{|A|}} '
\prod\limits_{j \in B_{1}}|K_{j}(f_{j}\chi_{E(m_{1},l_{1})}\chi_{G(j,
\overline{l})})(\lambda)| 
\prod\limits_{j \in B_{2}}|K_{j}(f_{j}\chi_{E(m_{1},l_{1}+1)}\chi_{G(j,
\overline{l})})(\lambda)| \leq \]
\[ \leq \sum\limits_{l_{1}} 'F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}
(\overline{f})(\lambda)F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}
(\overline{f})(\lambda), \]
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 $m^{(1)}=
(m_{t})_{t \in A_{1}}.$ 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 $E(m_{1},l_{1})$ is replaced by 
$E(m_{1},l_{1}+1).$ 

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 which 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 the 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}
the 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: $s_{n}<1$ and $s_{n}>1.$ Suppose first that $s_{n}<1.$
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{equation}
 \|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}}
\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{equation}
 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 induction assumption. 
The sum in (30) may be estimated by H\"older's inequality in the following 
way:
\[ \sum\limits_{l_{1}} 
'\left(\prod\limits_{j \in B_{1}} \|g_{j}\chi_{E(m_{1},l_{1})}
\|_{p}^{(1-\beta_{|B_{1}|})s_{n}} \prod\limits_{j \in B_{2}}\|g_{j}
\chi_{E(m_{1},l_{1}+1)}\|_{p}^{(1-\beta_{|B_{2}|})s_{n}}\right) \leq \] \[ \leq
\prod\limits_{j \in B_{1}}
{\rm max}_{l_{1}}\|g_{j}\chi_{E(m_{1},l_{1})}\|_{p}^
{(1-a_{1})(1-\beta_{|B_{1}|})s_{n}}\prod\limits_{j \in B_{2}}{\rm max}_{l_{1}}\|g_{j}\chi_{E(m_{1},l_{1}+1)}\|_{p}^
{(1-a_{2})(1-\beta_{|B_{2}|})s_{n}}\prod\limits_{j=1}^{n}\|g_{j}\|_{p}^
{\frac{p}{q}s_{n}}. \]
Thus
\[ \|F^{\overline{m}}(\overline{g})\|_{s_{n}} 
\leq C_{n}2^{-{\rm min}(\gamma_
{|B_{1}|}, \gamma_{|B_{2}|})(|m^{(1)}|+|m^{(2)}|)}\prod\limits_{j=1}^{n}
{\rm sup}_{l}\|g_{j}\chi_{E(m_{1},l)}\|_{p}^{{\rm min}(1-\beta_{|B_{1}|},
1-\beta_{|B_{2}|})-\frac{p}{q}}\prod\limits_{j=1}^{n} \|g_{j}\|_{p}^
{\frac{p}{q}}. \]
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 
\[ \gamma_{n} = {\rm min}(({\rm min}(1-\beta_{|B_{1}|},1-\beta_{|B_{2}|})-
(1-\beta_{n})), \gamma_{|B_{1}|}, \gamma_{|B_{2}|}). \]
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 is only finite number of pairs $B_{1},$ $B_{2}$ such that $|B_{1}|+
|B_{2}|=n,$ and hence the constants $\gamma_{n},$ $\beta_{n}$ may be chosen
universally. 

The case $s_{n}>1$ is similar. Using triangle inequality and H\"older's 
inequality, we get
\[ \|F^{\overline{m}}(\overline{g})\|_{s_{n}} \leq
\sum\limits_{l_{1}}' 
\|F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{g})(\lambda)\|
_{s_{|B_{1}|}}
\|F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{g})(\lambda)\|_
{s_{|B_{2}|}} \leq  \]
\[  \leq C_{n}2^{-{\rm min}(\gamma_{|B_{1}|}, \gamma_{|B_{2}|})(|m^{(1)}|
+|m^{(2)}|)}\sum\limits_{l_{1}}' \left(\prod\limits_{j \in B_{1}} 
\|g_{j}\chi_{E(m_{1},l_{1})}
\|_{p}^{(1-\beta_{|B_{1}|})} \prod\limits_{j \in B_{2}}\|g_{j}
\chi_{E(m_{1},l_{1}+1)}\|_{p}^{(1-\beta_{|B_{2}|})}\right). \]
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 $r <n.$ This completes 
the proof of (22).

Because of inequality (20) and (22) we find
\[ \|T_{n}(\overline{f})\|_{s_{n}} \leq C_{n} \]  
for all $f_{i}$ such that $\|f_{i}\|_{p} \leq \frac{1}{n}$ with 
$C_{n}$ depending only on $n.$ Hence by multilinearity Theorem 4.1 holds. 
$\Box$  

\begin{center}
\large \bf 5. A.e. convergence of multilinear transforms 
\end{center} 

In the proof of the a.e. convergence, an important role will be played by 
the following operators. Let 
$D_{1}(\lambda),...D_{n}(\lambda)$
 be measurable functions of $\lambda$ 
mapping $I$ to $R^{+}\cup\{\infty\}.$ Let us denote 
\[ T^{D_{1}(\lambda),...D_{n}(\lambda)}_{n}(f_{1},...f_{n})
(\lambda) \]
an operator obtained from $T_{n}$ by replacing the
 kernels $k_{i}(\lambda,x_{i}),$ $i=1,...n$ with 
\[
 \tilde{k}_{i}(\lambda, x_{i}) = k_{i}(\lambda, x_{i})
\chi_{R^{+}}(D_{i}(\lambda)-x_{i}), \] 
$i=2,...n.$ Throughout this section, we will assume that the kernels
$k_{i}(\lambda,x)$ are bounded and the
integral operators $K_{i}$ corresponding to these kernels satisfy
$L^{2}-L^{2}$ estimates. The results we prove here directly 
extend to more general
situations, however the above conditions are 
exactly the case that we will need
in applications and it is convenient to  restrict our attention to it.
 We give 
the following natural definition: \\

\noindent \bf Definition 5.1. \it We say that the operator 
$T_{n}$ converges on functions $f_{1},...f_{n}$ for some $\lambda$ 
if  the expressions
\[  T^{D_{1},...D_{n}}_{n}(f_{1},...f_{n})(\lambda) \]
converge to a  finite limit as all $D_{i}$ tend to infinity in any 
way. Namely, there exists a number, which we 
denote $T_{n}^{\infty,...\infty}(f_{1},...f_{n})(\lambda)$ such that
 for any $\delta >0$ there exists $N_{\delta}$ such that for 
any $D_{i} \geq N_{\delta},$ $i=1,...n$ (possibly infinite) 
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 $q^{-1}+p^{-1}=1.$  \\

\noindent \bf Proof. \rm The proof uses a well-known trick going 
back to Kolmogorov and Silvester. 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). $\Box$ \\

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 a.e. $\lambda$ on any functions $f_{i} \in L^{p},$ 
$1\leq p <2.$ \\

\noindent \bf 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
\[ {\rm sup}_{D_{1},...D_{n}}\|T_{n}^{D_{1},...D_{n}}(f_{1},...f_{n})
(\lambda)-T_{n}^{D_{1},...D_{n}}(f_{1,N},...f_{n,N})
(\lambda)\|_{s_{n}}^{s_{n}} \leq \] \[ \leq \sum\limits_{j=1}^{n}\|
T_{n}^{D_{1},...D_{n}}(g_{1j},...g_{nj})(\lambda)\|_{s_{n}}^{s_{n}} 
\leq C_{n}\sum\limits_{j=1}^{n}\left( \prod\limits_{i=1}^{n}\|g_{ij}\|_{p}
\right)^{s_{n}}. \]
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. 

A straightforward adjustment of the above argument allows (for a.e. $\lambda$)
to pass to the 
infinite limit in any order (i.e. for example first in $D_{n},$ then 
in $D_{n-1}$ and so on). $\Box$  

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 which appear in the process 
of iterating equation (11).  \\

\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 $k_{j}(\lambda,x)$ satisfy (15), and function $\sigma$ 
takes values in $1,...n$ and satisfies $\sigma (j)<j.$ \\

\noindent \rm The operator $T_{n}$ depends on the kernels $k_{j}$ and the
 choice of function $\sigma.$ It
 will be convenient to visualize the structure of the integral  (34)
defining the operator $T_{n}$ 
 as follows. To each operator $T_{n}$ defined by (34) with some $\sigma$
 we associate a graph $\Gamma$ with $n$ vertices $\{v_{j}\}_{j=1}^{n}.$
 We think of the vertex $v_{j}$ as of 
function $f_{j}(x_{j}).$ Two vertices $v_{i},$ $v_{j}$ are connected by the
 directed edge $e_{ij}$ (from $i$ to $j$) iff the product under integral in
 (34) contains the characteristic function 
$\chi_{R^{+}}(x_{j}-x_{i}).$ In this case we say that $v_{i}$ precedes 
$v_{j},$
 $v_{i} \prec v_{j}.$   It is easy to see that under all conditions
 on $T_{n}$
 the graph $\Gamma$ is a connected tree.  The directions on 
the edges (i.e.
 relation ``$\prec$") induce a partial order on the vertices 
of $\Gamma$ 
in which $v_{1}$ is the 
minimal element. We note also that a graph $\Gamma$ 
together with the mentioned above partial order and the 
correspondence between kernels $k_{i}(\lambda,x_{i})$ and vertices
 determine the operator $T_{n}$ in a unique way.  \\

\noindent \bf Lemma 5.5. \it Suppose that functions $f_{i}(x),$ 
$i=1,...n$ satisfy $(1+x)^{-\epsilon}f_{i}(x)\in L_p$ 
for some $\epsilon>0$ and
 $p \leq 2.$ 
 Then for a.e. $\lambda$
\[
\left| T_{n}(f_{1,N},f_{2}...f_{n})(\lambda)\right| 
\leq C(\lambda)(1+N)^{-n\delta}
\]
for any $\delta<\epsilon.$

\noindent \bf Proof. \rm The proof is an exercise in integration by parts.
Fix $\delta<\epsilon.$  
Let us introduce notation $f_{i}^{\delta}(x)$ for a function 
\[ f_{i}^{\delta}(x) = f_{i}(x)(1+x)^{\delta}. \]
By assumption, $f^{\delta}(x) \in L^{p}$ with $p<2.$ The proof uses 
induction in $n;$ the claim is that if $m$ arguments in the operator 
$T_{n}(g_{1},...g_{n})$ are functions $f_{i}$ (without weight)
 and the rest are the functions $f_{j}^{\delta}
(x_{j}),$ then 
\[ \left| T_{n}(g_{1,N},g_{2},...g_{n})
\right|
 \leq C(\lambda)(1+ N)^{-m\delta} \]
for a.e. $\lambda.$ For $n=1,$ the induction hypothesis is easily 
justified by
 Theorem 3.1 and integration by parts. Suppose the induction statement 
is true
 for $l \leq n-1.$ It also true for $n$ and $m=0$ by Theorem 5.3. 
We will now
 show  that if the induction 
statement  is true also for $n$ and any $m < m_{0} \leq n-1$ then it is
 true for $n,$ $m_{0}.$ That will conclude the proof.  
Since $m_{0} \geq 1,$ find some function $g_{l}$ which is equal to  
function $f_{l}$ without weight. Recall that we have a correspondence
between $T_{n}$ and a tree graph $\Gamma$ (describing effectively how
the integrals nest). Take a path on $\Gamma$ from 
vertex $v_{1}$ to $v_{l}.$ Assume that this path passes through the 
vertices
  (in this order) $v_{1},v_{l(1)},...v_{l(s)},v_{l}.$ 
We integrate by parts $(s+1)$ times, on the first 
step
 in $x_{1},$ differentiating the part of the under-integral expression 
\[ \left( \,\,\int\limits_{x_{1}}^{\infty}dx_{l(1)}...\right) \]
and integrating the rest from $x_{l(1)}$ to infinity. 
Such integration by parts is well-defined for a.e. $\lambda$ by 
Theorem 5.3; 
we may regularize all infinite integrals in the original expression by
 integrating to some number $M$ instead of infinity in all of them and then
 after integration by parts pass 
to the limit $M \rightarrow \infty,$ 
obtaining for a.e. $\lambda$ a well-defined equality by Theorem 5.3. We
 obtain the off-diagonal terms which have the right decay 
($C(\lambda)N^{-\delta m_{0}}$) by induction hypothesis in
 $n$ (they are products of the operators 
of order less than $n$). The graph corresponding
 to the new integral term is isomorphic to $\Gamma,$ but
 the minimal vertex is now $v_{l(1)}$ (new integral term includes
 $\chi_{R^{+}}(x_{1} - x_{l(1)})$ rather than $\chi_{R^{+}}
(x_{l(1)}- x_{1})).$
 Integration by parts allows to ``travel" along the graph $\Gamma.$
 We continue integration by parts for a.e. $\lambda$ so that in $s$ steps
 the vertex $v_{l}$ becomes minimal. All off-diagonal terms that we obtain
 are well-behaved (a.e. $\lambda$) by
 induction hypothesis in $n.$ The final integral
term looks as follows:
\[ \int\limits_{N}^{\infty}dx_{l} k_{l}(\lambda, x_{l})
(f_{l}(x_{l})(1+x_{l})^{\delta})(1+x_{l})^{-\delta}
\left(\,\int\limits_{x_{l}}^{\infty}...\right)...
\left(\,\int\limits_{x_{l}}^{\infty}...\right). \]
Integrating by parts one last time in $x_{l},$ 
differentiating $(1+x_{l})^{-\delta}$ and integrating the rest, we obtain 
\[ (1+N)^{-\delta} \int\limits_{N}^{\infty}dx_{l}
k_{l}(\lambda,x_{l})f_{l,\delta}(x_{l})
\left(\,\int\limits_{x_{l}}^{\infty}...\right)...
\left(\,\int\limits_{x_{l}}^{\infty}...\right)- \]
 \[-
\int\limits_{N}^{\infty} dx_{l}(1+x_{l})^{-1-\delta} 
\int\limits dx_{l}'k_{l}(\lambda,x_{l}')f_{l,\delta}
(x_{l}')\left(\, \int\limits_{x_{l}'}^{\infty}...\right)...
\left(\,\int\limits_{x_{l}'}^{\infty}...\right). \]
These expressions are estimated from above by $C(\lambda)
N^{-\delta m_{0}}$ by our induction hypothesis in $m$ 
for $n$ fixed. This completes the proof of the lemma. $\Box$

\begin{center}
 \large \bf 6. Conclusion of the proof of main results.
\end{center}

Given the general results of Sections 4 and 5, we complete the proof 
of main results. \\

\noindent \bf Proof of Theorem 1.3. \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:
\[ q'(x,\lambda) + 
\frac{i}{2\Im(\theta
 \overline{\theta})}V(x)\overline{\theta}^{2}(x,\lambda)
\exp (2ip(x,\lambda)) + \frac{i}{2\Im(\theta \overline{\theta})}V(x) 
\theta^{2}(x,\lambda)\exp(-2ip(x,\lambda))q^{2}(x,\lambda) \in L^{1}. \]
By assumption, 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 and therefore 
the theory developed in Sections 4 and 5 applies to the multilinear 
operators $T_{n}$ from classes $M_{n}$ composed from the kernels 
$k_{1}(\lambda, x),$ $k_{2}(\lambda, x).$ We will 
construct the function $q(x,\lambda)$ by iteration. 
Let 
\[ q^{(0)}(x,\lambda) = -\frac{i}{2\Im(\theta \overline{\theta})}
\int\limits_{x}^{\infty}\overline{\theta}^{2}(t,\lambda)\exp
 (2ip(t,\lambda))V(t)\,dt, \]
well-defined for a.e. $\lambda \in S$ by Theorem 3.1. Given 
$q^{(n-1)}(x,\lambda),$ we define 
\begin{equation}
q^{(n)}(x,\lambda)=  -\frac{i}{2\Im(\theta \overline{\theta})}
\int\limits_{x}^{\infty}\overline{\theta}^{2}\exp (2ip)V(t)\,dt
 - \frac{i}{2\Im(\theta \overline{\theta})}\int\limits_{x}^{\infty}
 \theta^{2}\exp(-2ip)V(t)(q^{(n-1)})^{2}\,dt.
\end{equation}

We need the following \\

\noindent \bf Lemma 6.1. \it $q^{(n)}(x,\lambda)$ is equal to a sum
 of multilinear transforms of classes $M_{j}$  (composed from the 
kernels $k_{1}(\lambda,x)$ and $k_{2}(\lambda,x)$ and with all arguments
 equal to $V(x)$) and is defined a.e. $\lambda \in
 S.$ Moreover, 
\[ q^{(n)}(x,\lambda) - q^{(n-1)}(x,\lambda) = \sum\limits_{i} 
T_{j_{i},x}^{\infty,...\infty}(V,...V), \]
where sum is finite and the orders $j_{i}$ satisfy $2n-1 \leq j_{i} 
\leq 2^{n+1}-1.$ 

\noindent \bf Proof. \rm We use induction. For $n=0,$ all statements
 are obvious (assume $q^{(-1)}(x,\lambda)=0$). Suppose they are also
 true for $m \leq n-1.$ The fact that $q^{(n)}(x,\lambda)$ is a sum 
of multilinear transforms of classes $M_{l}$ with some 
$l$ follows immediately from the induction hypothesis and 
formula (35). The fact that $q^{(n)}$ is well-defined for a.e. 
$\lambda \in S$ is then a consequence of Theorem 5.3. 
Note that 
\[ q^{(n)}(x,\lambda)- q^{(n-1)}(x,\lambda) = \]
\[ = -\frac{i}{2\Im(\theta \overline{\theta})}
\int\limits_{0}^{\infty}\theta^{2}\exp(-2ip)V(t)
(q^{(n-1)}(t,\lambda)-q^{(n-2)}(t,\lambda))
(q^{(n-1)}(t,\lambda)+q^{(n-2)}(t,\lambda))\,dt. \]
By induction assumption, every term on the right-hand side is a
multilinear transform from class $M_{j}$ of order $j$ no less then
 $(2n-3)+2=2n-1$ and no higher than $(2^{n}-1)2+1= 2^{n+1}-1.$ $\Box$ \\

\noindent \rm Now we can complete the proof of 
Theorem 1.3. If $x^{\epsilon}V(x) \in L^{p},$ 
$p\leq 2,$ Lemma 5.5 implies that for any multilinear operator of order 
$l_{i}$
\[ \left| T_{l_{i},x}^{\infty,...\infty}(V,...V)(\lambda)
 \right| \leq C(\lambda)(1+x)^{-\delta l_{i}} \]
for a.e. $\lambda \in S$ and any $\delta<\epsilon.$ 
Notice that  
\[ (q^{(n)})'(x,\lambda)+ \frac{i}{2\Im(\theta \overline{\theta})}
V(x)\overline{\theta}^{2}(x,\lambda)\exp (2ip(x,\lambda)) + 
\frac{i}{2\Im(\theta \overline{\theta})}V(x) \theta^{2}(x,\lambda)
\exp(-2ip(x,\lambda))(q^{(n)})^{2}(x,\lambda) = \]
\[ = \frac{i}{2\Im(\theta \overline{\theta})}V(x) \theta^{2}(x,\lambda)
\exp(-2ip(x,\lambda))\left((q^{(n)})^{2}(x,\lambda)-
(q^{(n-1)})^{2}(x,\lambda)\right). 
\]
Pick $n$ so that 
$(2n-1)\delta>\frac{1}{2}.$ Then by Lemma 6.1  we find that
the expression on the right hand side is a.e. $\lambda$ absolutely 
integrable,
 since it is a.e. $\lambda$ a product of some bounded function, $L^{p}$ 
function $V(x)$ and $L^{q}$ function $(1+x)^{-\frac{1}{2}-\eta},$ $\eta>0$ 
(take $q^{-1}=1-p^{-1}$).
 
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 $T_{l}(V,...V)$ 
(composed from $k_{1}(\lambda,x),$ $k_{2}(\lambda,x)$) of order not lager 
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 
$T_{l,x}^{N,\infty,...\infty}(V,...V)(\lambda).$ 
Starting from some $l,$ the under-integral expression 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. $\Box$ \\

\noindent \bf Proof of Theorems 1.1 and 1.2. \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 
$(a,b),$ $0<a<b<\infty.$ Consider $\phi(\lambda) \in C_{0}^{\infty}
(0,\infty),$ 
$1\geq \phi(\lambda) \geq 0$ such that $\phi(\lambda) =1$ when 
$\lambda 
\in (a,b).$ Clearly it suffices to show $L^{2}-L^{2}$ bound on functions 
of 
compact support for an operator
\[ Kf(\lambda) = \phi(\lambda)\int\limits_{0}^{\infty}\exp 
\left(2i\sqrt{\lambda}x-\frac{i}{\sqrt{\lambda}}\int\limits_{0}^{x}V(t)\,dt
 \right) f(x)\, dx.
\] 
We have 
\[ \|Kf\|_{L^{2}(a,b)}^{2} = \int\limits_{0}^{\infty}
\int\limits_{0}^{\infty}
dxdy f(x)f(y)\int d\lambda \phi(\lambda) \exp 
\left( 2i\sqrt{\lambda}(x-y)-\frac{i}{\sqrt{\lambda}}
\int\limits_{y}^{x}V(t)\,dt \right). \]
Let us denote by $Z(x,y)$ the kernel 
\[ Z(x,y) = \int d \lambda \phi(\lambda) \exp 
\left( 2i\sqrt{\lambda}(x-y)-
\frac{i}{\sqrt{\lambda}}\int\limits_{y}^{x}V(t)\,dt \right). \]
Let us integrate by parts in $\lambda$ in the expression 
for $Z(x,y)$ $N$ times, integrating $\exp (2i\sqrt{\lambda}(x-y))$ and 
differentiating the rest. We obtain 
\begin{equation} 
|Z(x,y)| \leq C(\phi, a,b, N) {\rm min}(1,|x-y|^{-N})\left| 
\int\limits_{y}^{x}
V(t)\,dt \right|^{N} \leq C(\phi, a,b, N){\rm min}
(1,|x-y|^{-N(1-\frac{1}{q})})\|V\|_{p}. 
\end{equation}
Taking $N$ large enough, for instance such that $N(1-\frac{1}{q})>1,$ 
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
\[ \|Kf\|^{2}_{L^{2}(a,b)} \leq C(\phi, a,b)\|f\|^{2}_{2}. 
\,\,\,\,\,\,\,\,\,
\Box \]

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 $V(x).$ \\

\noindent \bf 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. $\Box$ \\

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).  \\

\begin{center}
 \large \bf Appendix: Singular potentials. 
\end{center}

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 a.e. 
$\lambda \in S$
 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 analog 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 a.e. $\lambda \in S,$ 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 
of Theorem 3.1.  \\

\noindent \bf Lemma A.3. \it Suppose that an integral operator $K$ with 
uniformly bounded kernel $k(\lambda,x)$ satisfies $l^{2}(L^{1})-L^{2}$ 
estimate. Then the maximal function given by 
\[ (M_{K}f)(\lambda) = {\rm sup}_{N}\left| 
\int\limits_{0}^{N}k(\lambda,x)f(x)
\,dx \right| \]
satisfies 
\[ \|M_{K}f\|_{q} \leq C_{q}\|f\|_{l^{p}(L^{1})} \]
for any $p<2$ and $q^{-1}=1-p^{-1}.$ Therefore, the integral defining 
$(M_{K}f)(\lambda)$ converges a.e. $\lambda$ for any $f \in l^{p}(L^{1}).$

\noindent \bf 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 $p>1$), 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 from 
$E(m,l_{2})$ 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 family is obtained by splitting every interval of given power into two 
equal pieces, with the same arguments as on 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 $E(m,l).$ 
Hence 
\[ \|M_{K}f\|_{q} \leq \sum\limits_{n=1}^{\infty}\|M_{K,m}f\|_{q}.
 \]
On the other hand, 
\[ \|M_{K,m}f\|_{q}^{q} \leq \sum\limits_{l=1}^{2^{m}} 
\|K(f\chi_{E(m,l)})\|_{L^{q}}^{q} \leq \sum\limits_{l=1}^{2^{m}} 
\|f\chi_{E(m,l)}\|^{q}_{l^{p}(L^{1})} \leq 2^{m(1-\frac{q}{p})}. \]
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 $M_{K}$). A.e. convergence follows from the maximal estimate
 in a standard way.
$\Box$ \\

\noindent \bf Proof of Theorem A.2.
\rm The proofs of multilinear transform properties and a.e. 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
changes 
we need to make in the proof is to
 change throughout $\|\cdot\|_{p}$ to $\|\cdot\|_{l^{p}(L^{1})}.$ 
$\Box$

To prove Theorem A.1, we need to show the $l^{2}(L^{1})-L^{2}(I)$ bounds. \\

\noindent \bf Proof of Theorem A.1. \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{equation} 
(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)
\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{equation}
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. $\Box$ 


\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. 
M.C.'s work has been supported in part by NSF grant DMS96-23007.

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