ABSOLUTELY CONTINUOUS SPECTRUM FOR ONE-DIMENSIONAL SCHRÖDINGER OPERATORS WITH

SLOWLY DECAYING POTENTIALS: SOME OPTIMAL RESULTS

MICHAEL CHRIST AND ALEXANDER KISELEV

Abstract. The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic Schrödinger operators is preserved under all perturbations V (x) satisfying |V (x)| ≤ C(1+x)^{−α}, α > \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 Schrödinger 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.

1. Introduction and main results.

In this paper, we study the stability of the absolutely continuous spectrum of onedimensional Schrödinger operators under perturbations by slowly decaying potentials. Suppose that H_{U} is a Schrödinger operator defined on L^{2}(0, ∞) 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. [22], [2]). 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 Schrödinger operators under decaying perturbations in the general situation.

1991 Mathematics Subject Classification. Primary 34L40, 81Q05, 42B20 Secondary 81Q15, 42B25.

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 Schrödinger operators with potentials of slower decay, but satisfying some additional special assumptions. For example, by a result going back to Weidmann [30], 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, ∞) is purely absolutely continuous. Many authors developed a scattering theory for longrange potentials whose derivatives satisfy certain bounds; see for example [1], [5], [11]. 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 [11]. For potentials satisfying |V (x)| ≤ C(1 + |x|)^{− \frac{1}{2} −ϵ}, for instance, one can infer the existence of the wave operators if also |D^{α}V (x)| ≤ C_{1}(1 + |x|)^{− \frac{3}{2} −ϵ} for every multiindex α with |α| = 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^{−β}sin x^{α} with α, β 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 [3], [4], [10], [19] and [31] 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 α = 1. These potentials generalize the celebrated Wigner-von Neumann example [29]. 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 [21] and later Simon [25] 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)| ≤ \frac{C(x)}{1 + |x|},

for which the associated Schrödinger 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 Schrödinger operator was recently found by one of us in [12]. Namely, if the potential V satisfies |V (x)| ≤ C(1 + |x|)^{− \frac{3}{4} −ϵ} with some ϵ > 0, with no additional assumptions, then the whole positive semi-axis (0, ∞) is an essential support of the absolutely continuous part of the spectral measure.^{1} 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) [12].

The result of [12] was further improved in [13], 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)| ≤ C(1 + x)^{− \frac{2}{3} −ϵ}, 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 Schrödinger operators are stable under all perturbations by potentials satisfying |V (x)| ≤ C(1+x)^{− \frac{2}{3} −ϵ}. Later, this result for the case U = 0 was also proved by S. Molchanov by a different method [20].

On the other hand, there exists work on random potentials by Kotani and Ushiroya [17] which provides a bound for the best possible result that one can hope to prove. The results of [17] imply that (in the case U = 0) there exist potentials V (x) satisfying V (x) ≤ 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 [15] and [16], one can show that the (1 + x)^{− \frac{1}{2}} rate of decay is also critical for perturbations of the periodic Schrödinger 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

Theorem 1.1. Suppose that there exist ϵ > 0 and p ≤ 2 so that V (x)(1 + x)^{ϵ} ∈ L^{p}(R^{+}). Then the whole positive semi-axis R^{+} = (0, ∞) is an essential support of the absolutely continuous part of the spectral measure of the operator H_{V} . Moreover, for almost every λ ∈ R^{+}, there exist solutions ϕ_{λ}(x) and \overline{ϕ}_{λ}(x) of the generalized eigenfunction equation

−y′′ + V (x)y = λy

1E is an essential support of the measure μ if μ(X) > 0 for any X ⊂ E of positive Lebesgue measure.

with asymptotic behavior of pure WKB form in the main term:

ϕ_{λ}(x) = exp (i\sqrt{λ}x − \frac{i}{2\sqrt{λ}} ∫_{0}^{x} V (t) dt) (1 + o(1)).

Corollary. 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, ∞).

We have a similar result for perturbations of periodic Schrödinger operators. Let U be continuous and periodic, let S = ∪_{n=0}^{∞}(a_{n}, b_{n}) be the band spectrum of the unperturbed operator with potential U, and let θ(x, λ) be the Bloch functions for that operator.

Theorem 1.2. 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. λ ∈ S, there exist solutions ψ_{λ}(x), \overline{ψ}_{λ}(x) of the equation

−y′′ + (U(x) + V (x))y = λy (1)

with the asymptotic behavior

ψ_{λ}(x) = θ(x, λ) exp (\frac{i}{2ℑ(θ\overline{θ}′)} ∫_{0}^{x} V (t)|θ^{2}(t, λ)| dt) (1 + o(1)) (2)

as y → ∞.

Both theorems will follow from a certain general criterion. Suppose that H_{U} is an operator for which all solutions of the equation

−y′′ + U(x)y = λy (3)

are bounded for almost every λ ∈ S, where S is a certain set of positive Lebesgue measure. It is known (see, e.g. [25], [28]) 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 θ(x, λ), λ ∈ S, of the equation (3), such that θ(x, λ) are uniformly bounded over S and θ(x, λ), \overline{θ}(x, λ) are linearly independent for every λ ∈ S. It is easy to see that we can always find such family. We have

Theorem 1.3. Suppose that the potential V (x) is such that there exist ϵ > 0 and p ≤ 2 so that (1+x)^{ϵ}V (x) ∈ L^{p}. Assume that there exist measurable functions θ(x, λ) satisfying the above conditions, such that the operator

(Kf)(λ) = χ(S) ∫_{0}^{∞}θ(x, λ)^{2} exp (\frac{i}{ℑ(θ\overline{θ}′)} ∫_{0}^{x} V (t)|θ(t, λ)|^{2} dt) f(x) dx (4)

satisfies an L_{2}(R^{+}, dx) − L^{2}(S, dλ) 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 λ ∈ S, there exist solutions ψ_{λ}(x), \overline{ψ}_{λ}(x) of the equation (1) with the asymptotic behavior (2).

Remarks. 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 Schrödinger 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 [18]), but the corresponding potentials are of rather special form and in particular are not bounded from below.

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 [13].

3. We have not been able to treat potentials that are assumed merely to belong to L^{p} for some 1 < p ≤ 2.

The main new technique we develop in this paper involves norm bounds and almost everywhere convergence results for a class of multilinear integral operators, which may be of interest in its own right. 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 similar to Theorem 1.1 were obtained by Remling [24] by a very different method, based in part on the ideas from [13] and [20]. Some of the results we prove here (along with some of the results of [24]) were announced in [7]. The announcement [7] also contains a list of open problems which we find most interesting.

2. Asymptotic integration and bounded eigenfunctions.

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

Lemma 2.1. Let the potential W (x) be locally integrable and satisfy

sup_{x} ∫_{|x−t|≤1} |W_{−}(t)| dt < ∞

(W_{−} denotes the negative part of the potential W). Suppose that for every energy λ from the set E of positive Lebesgue measure, all solutions of the equation

−\frac{d^{2}}{dx^{2}} y(x) + W (x)y(x) = λy(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} .

The proof of this Lemma may be found in [28] (see also [26]). We notice that in different formulations, the fact that bounded solutions imply absolutely continuous spectrum was known for a long time (see, e.g., [6]). Lemma 2.1 is the most convenient statement for our purpose.

The general plan of our proof is similar to [13] and may be described as follows. By assumption, we know that for every λ ∈ S, all solutions of the generalized eigenfunction equation (3) for the unperturbed operator are bounded. Examples with imbedded eigenvalues [21], [25], [16] show that if V (x) is not short-range, we cannot hope in general that for every λ ∈ 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. λ ∈ 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 = λy.

We rewrite this equation as a system

y′_{1} = ( \begin{matrix}0 & 1 \\ U + V − λ & 0 \end{matrix} ) y_{1},

where y_{1} is now a vector ( \begin{matrix}y \\ y′ \end{matrix} ) . Let us apply a variation of the parameters trans-

formation with solutions of the unperturbed equation

y_{1} = ( \begin{matrix}θ(x, λ) & \overline{θ}(x, λ) \\ θ′(x, λ) & \overline{θ}′(x, λ) \end{matrix} ) y_{2},

to bring the equation to a more symmetric form

y′_{2} = \frac{i}{2ℑ(θ\overline{θ}′)} ( \begin{matrix}V (x)|θ(x, λ)|^{2} & V (x)\overline{θ}(x, λ)^{2} \\ −V (x)θ(x, λ)^{2} & −V (x)|θ(x, λ)|^{2} \end{matrix} ) y_{2}. (5)

Notice that 2iℑ(θ, \overline{θ′}) is a Wronskian of two solutions θ and \overline{θ} and hence is independent of x. Let us introduce the notation

p(x, λ) = \frac{1}{2ℑ(θ\overline{θ′})} ∫_{0}^{x} V (y)|θ(y, λ)|^{2} dy.

It will be convenient to apply to (5) an additional transformation

y_{2} = ( \begin{matrix}exp(ip(x, λ)) & 0 \\ 0 & exp(−ip(x, λ)) \end{matrix} ) y_{3}.

We arrive at the following equation for y_{3} :

y′_{3} = \frac{i}{2ℑ(θ\overline{θ}′)} ( \begin{matrix}0 & V (x)\overline{θ}(x, λ)^{2} exp(2ip(x, λ)) \\ −V (x)θ(x, λ)^{2} exp(−2ip(x, λ)) & 0 \end{matrix} ) y_{3}. (6)

We follow the idea of “(I + Q)” asymptotic integration originating in Harris-Lutz [9]: 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. [8]) 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

y_{3} = (1 − |q|^{2})^{− \frac{1}{2}} (I + Q) y_{4}, (7)

where I is an identity matrix,

Q = ( \begin{matrix}0 & q \\ \overline{q} & 0 \end{matrix} ) ,

and q(x, λ) is some function to be defined. A computation gives for y_{4}

y′_{4} = (1 − |q|^{2})^{−1} \frac{i}{2ℑ(θ\overline{θ}′)}[

( \begin{matrix}V ℜ(θ^{2}q exp(−2ip)) & V \overline{θ}^{2} exp(2ip) + V θ^{2}q^{2} exp(−2ip) \\ −V θ^{2} exp(−2ip) − V \overline{θ}^{2}\overline{q}^{2} exp(2ip) & −V ℜ(θ^{2}q exp(−2ip)) \end{matrix} ) y_{4}

+ ( \begin{matrix}\frac{1}{2}(q\overline{q}′ − q′\overline{q}) & −q′ \\ −\overline{q}′ & \frac{1}{2}(q′\overline{q} − q\overline{q}′) \end{matrix} ) y_{4}]. (8)

We summarize the main result of this section in

Theorem 2.2. Suppose that for some given λ there exists a function q(x, λ) ∈ C^{1}, such that q(x, λ) → 0 as x → +∞, such that

q′(x, λ) + \frac{i}{2ℑ(θ\overline{θ})}V (x)\overline{θ}^{2}(x, λ) exp(2ip(x, λ))

+\frac{i}{2ℑ(θ\overline{θ})}V (x)θ(x, λ)^{2} exp(−2ip(x, λ))q^{2}(x, λ) ∈ L^{1}. (9)

Then all solutions of the generalized eigenfunction equation (1) are bounded. Moreover, there are two solutions ψ(x, λ), \overline{ψ}(x, λ) with the asymptotic behavior

ψ(x, λ) = θ(x, λ) exp(ip(x, λ))

· exp (\frac{i}{2ℑ(θ\overline{θ})} ∫_{0}^{x}(1 − |q|^{2})^{−1}V ℜ(θ^{2}q exp(−2ip)) dt) (1 + o(1)) . (10)

Remark. 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.

Proof. The proof follows immediately from Levinson’s theorem (see, e.g. [8]) and equation (8). We may consider this system of equations only for x large enough, so that |q(t, λ)| < 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.

To complete the proof of Theorem 1.3, we need to construct the function q(x, λ) verifying (9) and conditions given in Theorem 2.2 for almost every λ ∈ S. The main problem is that if we try to solve the equation

q′(x, λ) + \frac{i}{2ℑ(θ\overline{θ})}V (x)\overline{θ}^{2}(x, λ) exp(2ip(x, λ))

+ \frac{i}{2ℑ(θ\overline{θ})}V (x)θ^{2}(x, λ) exp(−2ip(x, λ))q^{2}(x, λ) = 0 (11)

by iteration, we obtain expressions involving multilinear integral operators of certain type. We need to show that these expressions converge for a.e. λ ∈ S in order to ensure q(x, λ) ⟶^{x→∞} 0 for a.e. λ, and to make sure that (9) is satisfied after some number of iterations. The first approximation to the solution would be

q^{(0)}(x, λ) = \frac{i}{2ℑ(θ\overline{θ})} ∫_{x}^{∞}V (t)\overline{θ}^{2}(t, λ) exp(2ip(t, λ)) dt.

Again, we have to justify this formula by proving that the conditional integral is well-defined for almost every λ. This is relatively simple and has been already done in [13]. In the next section, we formulate the main result from [13] that we will use and make a few comments.

3. Almost everywhere convergence for integral operators.

Let an operator K be defined on all measurable bounded functions f of compact support by

(Kf)(λ) = ∫_{0}^{∞}k(λ, x)f(x) dx, (12)

where k(λ, x) is a measurable and bounded function on I × R^{+}. To study the a.e. convergence of the integral defining Kf(λ) on functions from L^{p}, we study the corresponding maximal function. Denote by M_{K}f(λ) the maximal function

M_{K}f(λ) = sup_{N} |∫_{0}^{N} k(λ, x)f(x) dx| . (13)

The following is a mild generalization of a result proved in [13].

Theorem 3.1. Let p, q be exponents satisfying 1 ≤ p < q ≤ ∞. Suppose that K is a bounded linear operator from L^{p}(R) to L^{q}(R). Then the maximal function M_{K} also maps L^{p} to L^{q} boundedly, that is,

∥M_{K}f∥_{q} ≤ C_{q}∥f∥_{p} for every f ∈ L^{p} (14)

As a consequence, the integral

∫_{0}^{N} k(λ, x)f(x) dx

converges as N → ∞ for almost every value of λ, for every f ∈ L^{p}.

For the proof of a slightly less general result we refer to [13]; we will also sketch in the appendix a proof of a very similar result (Lemma A.3). Theorem 3.1 may be obtained from that proof by a simple modification.

The following variant may also be proved by the same method. Denote by ∥K∥_{p,q} the norm of K, as an operator from L^{p} to L^{q}.

Theorem 3.2. Let (X, μ), (Y, ν) be σ-finite measure spaces. Suppose that 1 ≤ p < q ≤ ∞. Then for any bounded linear operator K from L^{p}(X) to L^{q}(Y), and for any sequence of measurable sets {E_{n} ⊂ X : n ∈ Z} such that E_{n} ⊂ E_{n+1} for every n, the maximal function

M_{K}f(y) = sup_{n} |K(f · χ_{E_{n}})(y)|

is bounded from L^{p}(X) to L^{q}(Y). Moreover

∥M_{K}∥_{p,q} ≤ A∥K∥_{p,q},

where A < ∞ depends only on p, q.

See [14] for other results along these lines.

4. Norm estimates for multilinear transforms.

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

Suppose that the functions k_{i}(λ, x), i = 1, n … are defined on I × R^{+}, where I is some measurable set in R. We assume that the operators

(K_{i}f)(λ) = ∫_{R^{+}} k_{i}(λ, x)f(x) dx

satisfy the bounds

∥K_{i}f∥_{L^{q}(I,dλ)} ≤ C_{i}∥f∥_{L^{p}(R^{+},dx)} (15)

on functions of compact support for some 2 > p ≥ 1 and q > p.

Let n ≥ 2. Let A be any set of ordered pairs α = (i_{α}, i′_{α}), with 1 ≤ i_{α}, i′_{α} ≤ n. Let |A| denote the cardinality of A. By χ_{E}(x) we denote a characteristic function which is equal to one when x ∈ E and is zero otherwise.

Consider the multilinear operator T_{n} given by

T_{n}(f_{1}, … f_{n})(λ) = ∫_{R^{n}} ∏_{j=1}^{n} f_{j}(x_{j})k_{j}(x_{j}, λ) ∏_{α∈A} χ_{R^{+}}(x_{i_{α}} − x_{i′_{α}}) dx, (16)

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.

Remark. 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:

Theorem 4.1. Suppose that the multilinear operator T_{n} is given by (16) with kernels k_{j}(λ, x_{j}) satisfying (15). Then for any functions f_{i} ∈ L^{p}(R^{+}, dx), i = 1, … n, such that the integral (16) converges absolutely for a.e. λ, we have

∥T_{n}(f_{1}, ...f_{n})∥_{s_{n}} ≤ C_{n} ∏_{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}.

Remarks. 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} ∈ L^{p_{j}}, the exponents p_{j} vary freely in a certain range, and s_{n}^{−1} = ∑_{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.

By assumption, the value of T_{n}(f_{1}, … f_{n})(λ) = g(λ) is well-defined for a.e. λ 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(λ) = ∑_{i=1}^{∞} g_{i}(λ).

Because of the absolute convergence, we have that the sum ∑_{i=1}^{n} g_{i}(λ) converges to g(λ) for a.e. λ as n → ∞. 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})χ_{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 χ_{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)χ_{E(1,1)}∥_{p}^{p} = ∥f(x)χ_{E(1,2)}∥_{p}^{p} = 2^{−1},

E(1, 1) ∪ E(1, 2) = R^{+} and E(1, 1) lies entirely to the right of E(1, 2) (i.e. for any x ∈ E(1, 1), y ∈ E(1, 2) we have x ≤ 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^{th} step 2^{m} intervals {E(m, l)}_{l=1}^{2^{m}}, such that ∪_{l=1}^{2^{m}}E(m, l) = R^{+}, ∥f(x)χ(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 ≤ m ≤ ∞, 1 ≤ l ≤ 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. Lemma 4.2. The following identity holds:

χ_{R^{+}}(x_{2} − x_{1})f(x_{1})f(x_{2}) = (∑_{m=1}^{∞} ∑_{{l=1 \atop l odd}}^{2^{m}} χ_{E(m,l)}(x_{1})χ_{E(m,l+1)}(x_{2})) f(x_{1})f(x_{2}). (17)

Proof. Figure 1 illustrates the decomposition that we perform. Let us denote by H_{12} the set

H_{12} = {x ∈ R^{2}, x = (x_{1}, x_{2})|x_{1} < x_{2}},

and by suppf the closure of the set of the points x such that for every interval I, such that x ∈ I, |f(x)| is positive on the set of positive Lebesgue measure in I. The claim will follow if we show that

H_{12} ∩ (supp_{x_{1}}f × supp_{x_{2}}f) = ⋃_{m=1}^{∞} ⋃_{{l=1 \atop l odd}}^{2^{m}}(E(m, l) × 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) ⊂ E(s, i), l odd, s ≠ 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} ∈ suppf, y_{1} < y_{2}, there exist two sets E(m, l), E(m, l + 1), with l odd, such that y_{1} ∈ E(m, l) and y_{2} ∈ E(m, l + 1). Let ∥fχ_{(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 suppf to infer that a > 0. Choose s so that 2^{−s} ≥ a ≥ 2^{−s−1}. If y_{1}, y_{2} lie in one set of generation s, E(s, l), then

Figure 1. Decomposition of χ_{R^{+}}(x_{2} − x_{1})

necessarily y_{1} ∈ E(s + 1, 2l − 1) and y_{2} ∈ 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} ∈ E(s − 1, l/2^{r}) and y_{2} ∈ E(s − 1, l/2^{r} + 1), where r is such that l/2^{r} is odd.

Remark. In particular, if suppf = R^{+}, we get a representation of diagonal characteristic function χ_{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.

Proof. 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) = (∑_{i=1}^{n} |f_{j}(x)|^{p})^{\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

∥f_{i}(x)χ_{E(m,l)}∥_{p}^{p} ≤ 2^{−m} (18)

for all i, l, m. Write

A = {α_{1}, … α_{|A|}}.

We begin by substituting the result of Lemma 4.2 into formula (16):

T_{n}(\overline{f}) = ∑_{m_{1}}^{∞} ⋯ ∑_{m_{|A|}=1}^{∞} ∑_{l_{1}}^{′} ⋯ ∑_{l_{|A|}}^{′} ∫_{R^{n}} dx ∏_{j=1}^{n} k_{j}(λ, x_{j})f_{j}(x_{j}) ∏_{t=1}^{|A|} χ_{E(m_{t},l_{t})}(x_{i_{t}})χ_{E(m_{t},l_{t}+1)}(x_{i′_{t}}), (19)

where ∑_{l_{t}}^{′} means the sum over all odd integers l_{t} ∈ [1, 2^{m_{t}}], and i_{t} = i_{α_{t}}, i′_{t} = i′_{α_{t}}. Thus

|T_{n}(\overline{f})(λ)| ≤ ∑_{m_{1}}^{∞} ⋯ ∑_{m_{|A|}=1}^{∞} F \overline{m}(\overline{f})(λ), (20)

where \overline{m} = (m_{1}, m …_{|A|}) and

F \overline{m}(\overline{f})(λ) = ∑_{l_{1}}^{′} ⋯ ∑_{l_{|A|}}^{′} ∫_{R^{n}} dx ∏_{j=1}^{n} |K_{j}(f_{j}χ_{G(j,\overline{l})})(λ)|, (21)

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}) = [ ⋂_{t:j=i_{t}} E(m_{t}, l_{t})] ⋂ [ ⋂_{t:j=i′_{t}} E(m_{t}, l_{t} + 1)] .

We aim to prove that

∥F \overline{m}(\overline{g})∥_{s_{n}} ≤ C_{n}2^{−γ_{n}|\overline{m}|} ∏_{j=1}^{n} ∥g_{j}∥_{p}^{(1−β_{n})} (22)

for any g_{1}, ...g_{n} which satisfy (18) (that is, ∥g_{i}χ_{E(m,l)}∥_{p}^{p} ≤ 2^{−m} for all i, m, l). Here γ_{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 β_{n} satisfies

1 ≥ 1 − β_{n} > 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 Hölder’s inequality (in this case \overline{m} is a 0-tuple). It will be convenient to consider the graph Υ 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 χ_{R^{+}}(x_{i_{t}} −x_{i′_{t}}) that we fixed in the sum (21). It suffices to treat the case where Υ is connected; the general case then follows by Hölder’s inequality.

Fix \overline{m}. Relabel the indices so that m_{1} ≤ ... ≤ 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 χ_{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}) = ∅; such terms contribute 0 to F \overline{m}(\overline{g}). We say that an index l remains if the corresponding term has not been dropped.

We have the following

Lemma 4.3. For any 1 ≤ j ≤ n, either

(23) G(j, \overline{l}) ⊂ E(m_{1}, l_{1}) ∀\overline{l} remaining

or

(24) G(j, \overline{l}) ⊂ E(m_{1}, l_{1} + 1) ∀\overline{l} remaining.

Let B_{1} = {j : (23) holds }, B_{2} = {j : (24) holds }. Then for any 2 ≤ t ≤ N (that is, if m_{t} = m_{1}),

(25) l_{t} = l_{1} ∀\overline{l} remaining.

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 ∈ A_{1}, t ∈ A_{2} respectively.

Proof. For any m ≥ 1, l odd, set

Ẽ(m, l) = E(m, l) ∪ E(m, l + 1).

First we prove that if \overline{l} remains, then

Ẽ(m_{t}, l_{t}) ⊂ Ẽ(m_{1}, l_{1}) (26)

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 χ_{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 χ_{R^{+}}(x_{i_{t}} − x_{i′_{t}}). Since the graph Υ is connected, we can find a path in Υ 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

Ẽ(m_{1}, l_{1}) ∩ Ẽ(m_{t_{1}}, l_{t_{1}}) ≠ ∅, Ẽ(m_{t}, l_{t}) ∩ Ẽ(m_{t_{r}}, l_{t_{r}}) ≠ ∅, and

Ẽ(m_{t_{i}}, l_{t_{i}}) ∩ Ẽ(m_{t_{i+1}}, l_{t_{i+1}}) ≠ ∅ for all i = 1, ...r − 1. (27)

Hence by our assumption that m_{1} ≤ m_{t} for all t we see that Ẽ(m_{t_{1}}, l_{t_{1}}) ⊂ Ẽ(m_{1}, l_{1}). But then by (27) also Ẽ(m_{1}, l_{1}) ∩ Ẽ(m_{t_{2}}, l_{t_{2}}) ≠ ∅, hence Ẽ(m_{t_{2}}, l_{t_{2}}) ⊂ Ẽ(m_{1}, l_{1}). We continue in the same way concluding that Ẽ(m_{t}, l_{t}) ⊂ Ẽ(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}) ⊂ Ẽ(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 Υ 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 ≤ 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 Ẽ(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.

Remark. 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 χ_{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 χ_{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

F \overline{m}(\overline{f})(λ) = (∑_{l_{1}}^{′} ∑_{l_{N+1}}^{′} ⋯ ∑_{l_{|A|}}^{′} ∏_{j∈B_{1}} |K_{j}(f_{j}χ_{E(m_{1},l_{1})}χ_{G(j,\overline{l})})(λ)|

· ∏_{j∈B_{2}} |K_{j}(f_{j}χ_{E(m_{1},l_{1}+1)}χ_{G(j,\overline{l})})(λ)|)

≤ ∑_{l_{1}}^{′} F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{f})(λ)F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{f})(λ),

where

F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{f})(λ) = ∑_{l_{t}:t∈A_{1}}^{′} ∏_{j∈B_{1}} |K_{j}(f_{j}χ_{E(m_{1},l_{1})}χ_{G(j,l)})(λ)|

and ∑_{l_{t}:t∈A_{1}}^{′} denotes the sum over all l_{t} such that 1 ≤ l_{t} ≤ 2^{m_{t}}, l_{t} is odd, t ∈ A_{1},

and we write \overline{m}^{(1)} = (m_{t})_{t∈A_{1}}. Note that F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{f})(λ) depends only on those f_{j} for which j ∈ B_{1}. The factor F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{f})(λ) is defined similarly, but χ_{E(m_{1},l_{1})} is replaced by χ_{E(m_{1},l_{1}+1)}.

We may rewrite for each j ∈ B_{1}

G(j, l) = E(m_{1}, l_{1}) ⋂ G^{1}(j, (l_{t})_{t∈A_{1}}),

where

G^{1}(j, (l_{t})_{t∈A_{1}}) = [ ⋂_{t∈A_{1},j=i_{t}} E(m_{t}, l_{t})] ⋂ [ ⋂_{t∈A_{1},j=i′_{t}} E(m_{t}, l_{t} + 1)] .

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})(λ) and F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{f})(λ) 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})(λ) and F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{f})(λ) may be estimated by induction on |A|. Therefore

∥F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{f})(λ)∥_{s_{|B_{1}|}} ≤ C2^{−γ_{|B_{1}|}|m^{(1)}|} ∏_{j∈B_{1}} ∥f_{j}χ_{E(m_{1},l_{1})}∥_{p}^{1−β_{|B_{1}|}}; (28)

a similar bound also holds for F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{f})(λ).

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

∥F \overline{m}(\overline{g})∥_{s_{n}}^{s_{n}} ≤ ∑_{l_{1}}^{′} ∥F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{g})(λ)∥_{s_{|B_{1}|}}^{s_{n}}∥F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{g})(λ)∥_{s_{|B_{2}|}}^{s_{n}}. (29)

We used the fact that ∥ ∑ h_{i}(x)∥_{s}^{s} ≤ ∑ ∥h_{i}(x)∥_{s}^{s} when s < 1 and Hölder’s inequality. Plugging the estimate (28) and a similar bound for F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{g})(λ) into (29), we find

∥F \overline{m}(\overline{g})∥_{s_{n}}^{s_{n}} ≤ C_{n}2^{−(γ_{|B_{1}|}|m^{(1)}|+γ_{|B_{2}|}|m^{(2)}|)s_{n}}

· ∑_{l_{1}}^{′} ( ∏_{j∈B_{1}} ∥g_{j}χ_{E(m_{1},l_{1})}∥_{p}^{(1−β_{|B_{1}|})s_{n}} ∏_{j∈B_{2}} ∥g_{j}χ_{E(m_{1},l_{1}+1)}∥_{p}^{(1−β_{|B_{2}|})s_{n}}) . (30)

Pick 0 < a_{1}, a_{2} < 1 such that

a_{1}(1 − β_{|B_{1}|}) = a_{2}(1 − β_{|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 Hölder’s inequality in the following way:

∑_{l_{1}}^{′} ( ∏_{j∈B_{1}} ∥g_{j}χ_{E(m_{1},l_{1})}∥_{p}^{(1−β_{|B_{1}|})s_{n}} ∏_{j∈B_{2}} ∥g_{j}χ_{E(m_{1},l_{1}+1)}∥_{p}^{(1−β_{|B_{2}|})s_{n}})

≤ ∏_{j∈B_{1}} max_{l_{1}}∥g_{j}χ_{E(m_{1},l_{1})}∥_{p}^{(1−a_{1})(1−β_{|B_{1}|})s_{n}}

· ∏_{j∈B_{2}} max_{l_{1}}∥g_{j}χ_{E(m_{1},l_{1}+1)}∥_{p}^{(1−a_{2})(1−β_{|B_{2}|})s_{n}} ∏_{j=1}^{n} ∥g_{j}∥_{p}^{\frac{p}{q} s_{n}}.

Thus

∥F \overline{m}(\overline{g})∥_{s_{n}} ≤ C_{n}2^{−min(γ_{|B_{1}|},γ_{|B_{2}|})(|m^{(1)}|+|m^{(2)}|)}

· ∏_{j=1}^{n} sup_{l}∥g_{j}χ_{E(m_{1},l)}∥_{p}^{min(1−β_{|B_{1}|},1−β_{|B_{2}|})− \frac{p}{q}} ∏_{j=1}^{n} ∥g_{j}∥p^{\frac{p}{q}} .

Obviously ∥g_{j}∥_{p} ≥ sup_{l}∥g_{j}χ_{E(m_{1},l)}∥_{p}, and sup_{l}∥g_{j}χ_{E(m_{1},l)}∥_{p} ≤ 2^{−m_{1}} by (18). Pick β_{n} so that

min(1 − β_{|B_{1}|}, 1 − β_{|B_{2}|}) > 1 − β_{n} > max(\frac{p}{q}, \frac{p}{2}),

and

γ_{n} = min([min(1 − β_{|B_{1}|}, 1 − β_{|B_{2}|}) − (1 − β_{n})], γ_{|B_{1}|}, γ_{|B_{2}|}).

Then

∥F \overline{m}(\overline{g})∥_{s_{n}} ≤ C_{n}2^{−γ_{n}|m|} ∏_{j=1}^{n} ∥g_{j}∥_{p}^{1−β_{n}}. (31)

There are only finitely many pairs B_{1}, B_{2} such that |B_{1}| + |B_{2}| = n, and hence the constants γ_{n}, β_{n} may be chosen to be independent of B_{1}, B_{2}.

The case s_{n} ≥ 1 is similar. Using the triangle inequality and Hölder’s inequality, we get

∥F \overline{m}(\overline{g})∥_{s_{n}} ≤ ∑_{l_{1}}^{′} ∥F_{A_{1},B_{1}}^{\overline{m}^{(1)},l_{1}}(\overline{g})(λ)∥_{s_{|B_{1}|}}∥F_{A_{2},B_{2}}^{\overline{m}^{(2)},l_{1}}(\overline{g})(λ)∥_{s_{|B_{2}|}}
≤ C_{n}2^{−min(γ_{|B_{1}|},γ_{|B_{2}|})(|m^{(1)}|+|m^{(2)}|)}

· ∑_{l_{1}}^{′} ( ∏_{j∈B_{1}} ∥g_{j}χ_{E(m_{1},l_{1})}∥_{p}^{(1−β_{|B_{1}|})} ∏_{j∈B_{2}} ∥g_{j}χ_{E(m_{1},l_{1}+1)}∥_{p}^{(1−β_{|B_{2}|})}) .

Provided that

\frac{1}{p}(|B_{1}|(1 − β_{|B_{1}|}) + |B_{2}|(1 − β_{|B_{2}|})) > 1, (32)

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}| ≥ 1 since by induction hypothesis 1 − β_{r} > \frac{p}{2} for all r < n. This completes the proof of (22), and hence of Theorem 4.1.

5. Almost everywhere convergence of multilinear transforms

In the proof of the a.e. convergence, an important role will be played by the following operators. Let D_{1}(λ), ...D_{n}(λ) be measurable functions of λ mapping I to R^{+} ∪ {∞}. Let us denote by

T_{n}^{D_{1}(λ),...D_{n}(λ)}(f_{1}, ...f_{n})(λ)

an operator obtained from T_{n} by replacing the kernels k_{i}(λ, x_{i}), i = 1, ...n with

k̃_{i}(λ, x_{i}) = k_{i}(λ, x_{i})χ_{R^{+}}(D_{i}(λ) − x_{i}),

i = 2, ...n. Throughout this section, we will assume that the kernels k_{i}(λ, x) are bounded and the integral operators K_{i} corresponding to these kernels satisfy L^{2} − L^{2} estimates. The results we prove here extend directly 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:

Definition 5.1. We say that the operator T_{n} converges on functions f_{1}, ...f_{n} for some λ if the expressions

T_{n}^{D_{1},...D_{n}}(f_{1}, ...f_{n})(λ)

converge to a finite limit as min_{i} D_{i} tends to infinity. Namely, there exists a number, which we denote T_{n}^{∞,...∞}(f_{1}, ...f_{n})(λ) such that for any δ > 0 there exists N_{δ} such that whenever min_{i} D_{i} ≥ N_{δ}, we have

|T_{n}^{∞,...∞}(f_{1}, ...f_{n})(λ) − T_{n}^{D_{1},...D_{n}}(f_{1}, ...f_{n})(λ)| < δ.

Our first result is the following maximal estimate:

Theorem 5.2. Let

M_{T} (f_{1}, ...f_{n})(λ) = sup_{D_{1},...D_{n}}|T_{n}^{D_{1},...D_{n}}(f_{1}, ...f_{n})(λ)|.

Then for any f_{i} ∈ L^{p}, 1 ≤ p < 2, we have

∥M_{T} (f_{1}, ...f_{n})∥_{s_{n}} ≤ C_{n} ∏_{i=1}^{n} ∥f_{i}∥_{p}, (33)

where s_{n}^{−1} = nq^{−1} and q^{−1} + p^{−1} = 1.

Proof. 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}(λ), i = 1, ...n, we have

∥T_{n}^{D_{1}(λ),...D_{n}(λ)}(f_{1}, ...f_{n})∥_{s_{n}} ≤ C_{n} ∏_{i=1}^{n} ∥f_{i}∥_{p}

with the constant C_{n} independent of D_{i}(λ). But the expression T_{n}^{D_{1}(λ),...D_{n}(λ)}(f_{1}, ...f_{n})(λ) is obtained from T_{n}(f_{1}, ...f_{n})(λ) by replacing the kernels k_{i}(λ, x) with the kernels

k̃_{i}(λ, x) = k_{i}(λ, x)χ_{R^{+}}(D_{i}(λ) − 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}(λ). Therefore Theorem 4.1 is applicable and directly leads to (33).

As one may expect, the maximal estimate (33) implies a.e. convergence.

Theorem 5.3. The operator T_{n}(f_{1}, ...f_{n}) converges for almost every λ on any functions f_{i} ∈ L^{p}, 1 ≤ p < 2.

Proof. Suppose, on the contrary, that for some ϵ > 0 we have a set S_{ϵ} 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})(λ) − T_{n}^{D_{12},...D_{n2}}(f_{1}, ...f_{n})(λ)| > ϵ

for every λ ∈ S_{ϵ}. Let us denote f_{i,N}(x_{i}) = f_{i}(x_{i})χ_{R^{+}}(N − x_{i}). For any D_{i} > N, we have

|T_{n}^{D_{1},...D_{n}}(f_{1}, ...f_{n})(λ) − T_{n}^{D_{1},...D_{n}}(f_{1,N}, ...f_{n,N})(λ)| ≤ ∑_{j=1}^{n} |T_{n}^{D_{1},...D_{n}}(g_{1j}, ...g_{nj})(λ)|,

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

sup_{D_{1},...D_{n}}∥T_{n}^{D_{1},...D_{n}}(f_{1}, ...f_{n})(λ) − T_{n}^{D_{1},...D_{n}}(f_{1,N}, ...f_{n,N})(λ)∥_{s_{n}}^{s_{n}}

≤ ∑_{j=1}^{n} ∥T_{n}^{D_{1},...D_{n}}(g_{1j}, … g_{nj})(λ)∥_{s_{n}}^{s_{n}}

≤ C_{n} ∑_{j=1}^{n} (∏_{i=1}^{n} ∥g_{ij}∥_{p})^{s_{n}} .

Clearly, the right-hand side goes to zero as N → ∞, 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{ϵ}{2})^{s_{n}}|S_{ϵ}| for every N. This gives a contradiction.

A straightforward adjustment of the above argument allows us to pass for almost every λ 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.

Definition 5.4. We say that the multilinear transform T_{n} belongs to the class M_{n}, if

T_{n}(f_{1}, ...f_{n})(λ) = ∫_{0}^{∞}... ∫_{0}^{∞}dx_{1}...dx_{n} ∏_{j=1}^{n} k_{j}(λ, x_{j})f_{j}(x_{j}) ∏_{j=2}^{n} χ_{R^{+}}(x_{j} − x_{σ(j)}), (34)

where the kernels k_{j}(λ, x) satisfy (15), and the function σ takes values in 1, ...n and satisfies σ(j) < j for every j.

Recall the notation f_{i,N}(x) = f_{i}(x)χ_{R^{+}}(N − x).

Lemma 5.5. Suppose that for some ϵ > 0 and 1 ≤ p < 2, for each index 1 ≤ i ≤ n, a function f_{i} satisfies (1 + x)^{ϵ}f_{i}(x) ∈ L^{p}. Then for every δ < ϵ, for almost every λ ∈ R, there exists C(λ, n) < ∞ such that for every nonnegative N ∈ R,

|T_{n}(f_{1} − f_{1,N}, f_{2} … f_{n})(λ)| ≤ C(λ, n)(1 + N)^{−nδ}.

Proof. The structure of operators of class M_{n} is such that the integral defining T_{n} extends only over those x satisfying x_{j} ≥ x_{1} for every j, because of the requirement that σ(j) < j for every j > 1. Since f_{1} −f_{1,N} is supported where x > N, we therefore have for every N, λ

T_{n}(f_{1} − f_{1,N}, f_{2} … f_{n})(λ) = T_{n}(f_{1} − f_{1,N}, f_{2} − f_{2,N} … f_{n} − f_{n,N})(λ).

Fix f_{1}, … f_{n}. Since ∥f_{j} − f_{j,N}∥_{p} ≤ C · (1 + N)^{−ϵ} by hypothesis, Theorem 5.2 now yields

2^{nrϵ}∥ sup_{N≥2^{r}} |T_{n}(f_{1} − f_{1,N}, f_{2}, … f_{n})|(λ)∥_{L^{sn}(dλ)} ≤ C < ∞

for every nonnegative integer r, with C independent of r. Consequently

∑_{r=0}^{∞} 2^{nrδ} sup_{N≥2^{r}} |T_{n}(f_{1} − f_{1,N}, f_{2}, … f_{n})|(λ) < ∞

for almost every λ. To obtain the conclusion of the lemma, given any N ≥ 1, choose r so that 2^{r} ≤ N < 2^{r+1} and apply this inequality.

6. Conclusion of the proof of main results.

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.

Proof. We recall that it suffices to show that under the assumptions of the theorem for a.e. λ ∈ S we can find a function q(x, λ) such that q(x, λ) ∈ C^{1}, q(x, λ) → 0, and the condition (9) holds:

q′(x, λ) + \frac{i}{2ℑ(θ\overline{θ})}V (x)\overline{θ}^{2}(x, λ) exp(2ip(x, λ))

+\frac{i}{2ℑ(θ\overline{θ})}V (x)θ^{2}(x, λ) exp(−2ip(x, λ))q^{2}(x, λ) ∈ L^{1}.

By assumption, the kernels

k_{1}(λ, x) = \overline{θ}^{2}(x, λ) exp(2ip(x, λ))

and k_{2}(λ, x) = \overline{k_{1}(λ, x)} satisfy L^{2} − L^{2} and trivial L^{1} − L^{∞} 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 < p < 2. 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}(λ, x), k_{2}(λ, x).

We will construct the function q(x, λ) by iteration. Let

q^{(0)}(x, λ) = −\frac{i}{2ℑ(θ\overline{θ})} ∫_{x}^{∞} \overline{θ}^{2}(t, λ) exp(2ip(t, λ))V (t) dt,

well-defined for almost every λ ∈ S by Theorem 3.1. Given q^{(n−1)}(x, λ), we define

q^{(n)}(x, λ) = −\frac{i}{2ℑ(θ\overline{θ})} ∫_{x}^{∞} \overline{θ}^{2} exp(2ip)V (t) dt − \frac{i}{2ℑ(θ\overline{θ})} ∫_{x}^{∞}θ^{2} exp(−2ip)V (t)(q^{(n−1)})^{2} dt. (35)

Define

V_{x}(y) = V (y)χ_{R^{+}}(y − x).

We need the following

Lemma 6.1. q^{(n)}(x, λ) is equal to a sum of multilinear transforms of classes M_{j} (composed from the kernels k_{1}(λ, x) and k_{2}(λ, x) and with all arguments equal to V_{x}) and is defined for almost every λ ∈ S. Moreover,

q^{(n)}(x, λ) − q^{(n−1)}(x, λ) = ∑_{i} T_{j_{i}}(V_{x}, V, … V),

where the sum is taken over finitely many orders j_{i}, each of which satisfies 2n − 1 ≤ j_{i} ≤ 2^{n+1} − 1.

Proof. We use induction. For n = 0, all statements are obvious (defining q^{(−1)}(x, λ) to be 0). Suppose they are also true for m ≤ n − 1. The fact that q^{(n)}(x, λ) is a sum of multilinear transforms of classes M_{j} with some j follows immediately from the induction hypothesis and formula (35). The fact that q^{(n)} is well-defined for a.e. λ ∈ S is then a consequence of Theorem 5.3. Note that

q^{(n)}(x, λ) − q^{(n−1)}(x, λ)

= \frac{−i}{2ℑ(θ\overline{θ})} ∫_{x}^{∞}θ^{2} exp(−2ip)V (q^{(n−1)}(t, λ) − q^{(n−2)}(t, λ))(q^{(n−1)}(t, λ) + q^{(n−2)}(t, λ)) dt.

By the induction hypothesis, every term on the right-hand side is a multilinear transform of the class M_{j}, where the order j is no less then (2n − 3) + 2 = 2n − 1 and no higher than (2^{n} − 1)2 + 1 = 2^{n+1} − 1.

Now we can complete the proof of Theorem 1.3. If x^{ϵ}V (x) ∈ L^{p}, p ≤ 2, Lemma 5.5 implies that for any multilinear operator of order j_{i}

|T_{j_{i}}^{∞,...∞}(V_{x}, ...V)(λ)| ≤ C(λ)(1 + x)^{−δj_{i}}

for almost every λ ∈ S and any δ < ϵ. Notice that

(q^{(n)})′(x, λ) + \frac{i}{2ℑ(θ\overline{θ})}V (x)\overline{θ}^{2}(x, λ) exp(2ip(x, λ))

+\frac{i}{2ℑ(θ\overline{θ})}V (x)θ^{2}(x, λ) exp(−2ip(x, λ))(q^{(n)})^{2}(x, λ)

= \frac{i}{2ℑ(θ\overline{θ})}V (x)θ^{2}(x, λ) exp(−2ip(x, λ)) ((q^{(n)})^{2}(x, λ) − (q^{(n−1)})^{2}(x, λ)) .

Pick n so that (2n − 1)δ > \frac{1}{2}. Then by Lemma 6.1 we find that the expression on the right hand side is absolutely integrable for almost every λ, since it is the product of some L^{p} function V (x) with a factor which for almost every λ is O((1 + x)^{− \frac{1}{2} −η}) for some η > 0, and hence belongs to L^{q} where q^{−1} = 1 − p^{−1}.

Hence q^{(n)}(x, λ) satisfies the condition (9) and therefore we can take q(x, λ) = q^{(n)}(x, λ) for a.e. λ ∈ 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}(λ, x), k_{2}(λ, 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

exp (\frac{i}{2ℑ(θ\overline{θ})} ∫_{0}^{x}(1 − |q|^{2})^{−1}V ℜ(θ^{2}q exp(−2ip)) dt) (36)

in the asymptotic formula for the solutions in Theorem 2.2. But note that the limit of the integral

∫_{0}^{N}(1 − |q(t, λ)|^{2})^{−1}V (t)ℜ(θ^{2}(t, λ)q(t, λ) exp(−2ip(t, λ))) dt

as N → ∞ exists for a.e. λ. 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,∞,...∞}(V_{x}, ...V)(λ). Starting from some l, the integrand will become absolutely integrable over the whole axis for a.e. λ by Lemma 5.5; in the remaining finite sum every term is convergent by Theorem 5.3. Therefore, for a.e. λ, the expression (36) can be written as C(λ)(1 + o(1)) and hence can be omitted in the asymptotic expression for the solutions. This completes the proof.

We now prove Theorems 1.1 and 1.2.

Proof. It remains only to verify the L^{2} − L^{2} bounds for the corresponding operator (4). It is convenient to choose θ(x, λ) = exp(i\sqrt{λ}x) in the free case and θ(x, λ) Bloch functions in the periodic case. The corresponding L^{2}−L^{2} bounds were already shown in [13]. 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 [13] for details.

Without loss of generality, we can restrict our attention to some interval (a, b), 0 < a < b < ∞. Consider ϕ(λ) ∈ C_{0}^{∞}(0, ∞), 1 ≥ ϕ(λ) ≥ 0 such that ϕ(λ) = 1 when λ ∈ (a, b). Clearly it suffices to show L^{2} − L^{2} bound on functions of compact support for an operator

Kf(λ) = ϕ(λ) ∫_{0}^{∞}exp (2i\sqrt{λ}x − \frac{i}{\sqrt{λ}} ∫_{0}^{x} V (t) dt) f(x) dx.

We have

∥Kf∥_{L^{2}(a,b)}^{2} = ∫_{0}^{∞} ∫_{0}^{∞}dxdyf(x)f(y) ∫ dλϕ(λ) exp (2i\sqrt{λ}(x − y) − \frac{i}{\sqrt{λ}} ∫_{y}^{x} V (t) dt) .

Let us denote by Z(x, y) the kernel

Z(x, y) = ∫ dλϕ(λ) exp (2i\sqrt{λ}(x − y) − \frac{i}{\sqrt{λ}} ∫_{y}^{x} V (t) dt) .

Let us integrate by parts in λ in the expression for Z(x, y) N times, integrating exp(2i\sqrt{λ}(x − y)) and differentiating the rest. We obtain

(37)  |Z(x, y)| ≤ C(ϕ, a, b, N)min(1, |x − y|^{−N}) |∫_{y}^{x} V (t) dt|^{N}

≤ C(ϕ, a, b, N)min(1, |x − y|^{−N(1− \frac{1}{q})})∥V ∥_{p}.

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∥_{L^{2}(a,b)}^{2} ≤ C(ϕ, a, b)∥f∥_{2}^{2}.

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).

Theorem 6.2 Fix a potential V (x). Suppose that there exists a monotone differentiable function d(x), d(x) > 0, d(x) ⟶^{x→∞} 0, d′(x) ≤ 0, such that V (x)d(x)^{−1} ∈ L^{p} with some p < 2 and V (x)d(x)^{N} ∈ 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).

Proof. Going through the proofs of Lemma 5.5 and Theorem 1.3, we substitute x^{−ϵ} with d(x). Given that d(x)^{−1}V (x) ∈ L^{p}, p < 2, all proofs go unchanged, and in the final step instead of V (x)(1 + x)^{−δ(2n−1)} being absolutely integrable we need to check that for some N,

V (x)d(x)^{N} ∈ L^{1}.

This is exactly what we assumed in the statement of Theorem 6.2.

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 Ṽ (x) be a potential obtained from V by adding such intervals I_{n} where Ṽ 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 [12] (where it was derived in a slightly different context).

Appendix: Singular potentials.

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 Schrödinger operators have purely singular spectrum [23] 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 ≤ p ≤ 2, with the norm given by

∥f∥_{l^{p}(L^{1})} = (∑_{n=0}^{∞} ( ∫_{n}^{n+1}|f(x)| dx)^{p})^{\frac{1}{p}} .

The main result we show is

Theorem A.1. 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 ≤ 2 and ϵ > 0 with V (x)x^{−ϵ} ∈ 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 λ ∈ 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)).

As before, Theorem A.1 is a corollary of the following general criterion (an analogue of Theorem 1.3):

Theorem A.2. Suppose that the potential V (x) is such that there exist ϵ > 0 and p ≤ 2 so that x^{ϵ}V (x) ∈ l^{p}(L^{1}). Assume that an operator

(Kf)(λ) = χ(S) ∫_{0}^{∞}θ(x, λ)^{2} exp (\frac{i}{ℑ(θ\overline{θ}′)} ∫_{0}^{x} V (t)|θ(t, λ)|^{2} dt) f(x) dx

satisfies the bound

∥Kf(λ)∥_{L^{2}(S)} ≤ 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 λ ∈ S, there exist solutions ψ_{λ}(x), \overline{ψ}_{λ}(x) of the equation (1) with the asymptotic behavior (2).

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.

Lemma A.3. Suppose that 1 ≤ p < ∞, 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)(λ) = sup_{N}|K(fχ_{R^{+}}(N − x))(λ)|

satisfies

∥M_{K}f∥_{q} ≤ C_{q}∥f∥_{l^{p}(L^{1})}

for the same pair of exponents p, q. Moreover the integral defining (M_{K}f)(λ) converges for almost every λ, for any f ∈ l^{p}(L^{1}).

Proof. 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

∥fχ_{E(1,1)}∥_{l^{p}(L^{1})}^{p} = ∥fχ_{E(1,2)}∥_{l^{p}(L^{1})}^{p}.

We note that in contrast to the L^{p} case, we cannot in general say that ∥fχ_{E(1,i)}∥_{l^{p}(L^{1})}^{p} = 2^{−1}. However, it is easy to see that

∥fχ_{E(1,i)}∥_{l^{p}(L^{1})}^{p} ≤ 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} ∪ A_{2} = (m, m + 1) for some m. In this case

(∫ |fχ_{A_{1}}| dx)^{p} + (∫ |fχ_{A_{2}}| dx)^{p} ≤ ( ∫_{m}^{m+1}|f(x)| dx)^{p}

(since p ≥ 1), and hence

∥fχ_{E(1,2)}∥_{l^{p}(L^{1})}^{p} + ∥fχ_{E(1,1)}∥_{l^{p}(L^{1})}^{p} ≤ 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 ∪_{l}E(m, l) = R^{+}, E(m, l_{1}) lies to the left of E(m, l_{2}) if l_{1} ≤ l_{2}. We choose these intervals so that the total number of the intervals of generation m is 2^{m} and ∥fχ_{E(m,l)}∥_{l^{p}(L^{1})}^{p} ≤ 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)(λ) = sup_{l}|M_{K}(fχ_{E(m,l)})(λ)|.

Then

(M_{K}f)(λ) ≤ ∑_{m=1}^{∞}(M_{K,m}f)(λ).

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} ≤ ∑_{m=1}^{∞} ∥M_{K,m}f∥_{q}.

On the other hand,

∥M_{K,m}f∥_{q}^{q} ≤ ∑_{l=1}^{2^{m}} ∥K(fχ_{E(m,l)})∥_{q}^{q} ≤ ∑_{l=1}^{2^{m}} ∥fχ_{E(m,l)}∥_{l^{p}(L^{1})}^{q} ≤ 2^{m(1− \frac{q}{p})}.

Therefore,

∥M_{K}f∥_{q} ≤ ∑_{m=1}^{∞} 2^{m(\frac{1}{q} − \frac{1}{p})} ≤ 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.

Now we prove Theorem A.2.

Proof. 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) = (∑_{i=1}^{n} |f_{i}(x)|^{p})^{\frac{1}{p}} ,

used to construct the family E(m, l) in the proof may be replaced by

f(x) = ∑_{i=1}^{n} |f_{i}(x)|.

The only other change we need to make in the proof is to change throughout ∥ · ∥_{p} to ∥ · ∥_{l^{p}(L^{1})}.

To prove Theorem A.1, we need to show the l^{2}(L^{1}) − L^{2}(S) bounds.

Proof. 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 [13] 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)(λ) = ϕ(λ)^{2} ∫_{0}^{∞}exp (2i\sqrt{λ}x − \frac{i}{\sqrt{λ}} ∫_{0}^{x} V (t) dt) f(x) dx

satisifes l^{2}(L^{1}) − L^{2}(I) bound on functions of compact support, where ϕ ∈ C_{0}^{∞}(R^{+}) and ϕ(λ) = 1 for λ ∈ I. Note that

(Kf)(λ) = ∑_{l=0}^{∞} ϕ(λ) exp (2i\sqrt{λ}l − \frac{i}{\sqrt{λ}} ∫_{0}^{l} V (t) dt)

· ϕ(λ) ∫_{0}^{1} exp (2i\sqrt{λ}y − \frac{i}{\sqrt{λ}} ∫_{l}^{y+l}V (t) dt) f(y + l) dy. (38)

We can write Kf(λ) as follows:

Kf(λ) = ∑_{l=0}^{∞} ϕ(λ) exp (2i\sqrt{λ}l − \frac{i}{2\sqrt{λ}} ∫_{0}^{l} V (t) dt) f(λ, l),

where the expression f(λ, l) has the following property: for every m ≥ 0,

|∂_{λ}^{m}f(λ, l)| ≤ C_{m}(I, ϕ)sup_{n} ( ∫_{n}^{n+1}|V (x)| dx)^{m} f_{l}

where f_{l} = ∫_{l}^{l+1} |f(x)| dx. This property of f(λ, l) is evident from (38).

We compute

∥Kf∥_{L^{2}(I)}^{2} ≤ ∑_{l=0}^{∞} ∑_{m=0}^{∞} ∫ dλϕ^{2}(λ) exp (2i\sqrt{λ}(l − m) − \frac{i}{2\sqrt{λ}} ∫_{m}^{l} V (t) dt) f(λ, l)\overline{f}(λ, m).

Let us integrate by parts, differentiating f(λ, l)\overline{f}(λ, m) and integrating the rest. By virtually the same computation as one which led us to (37) we obtain

|∫ dλϕ(λ) exp (2i\sqrt{λ}(l − m) − \frac{1}{2\sqrt{λ}} ∫_{m}^{l} V (t) dt)| ≤ C_{r}min(1, |l − m|)^{− \frac{r}{p}}

for every positive integer r. Therefore, taking into account the properties of f(λ, n), we obtain

∥Kf∥_{L^{2}(I)}^{2} ≤ C_{r} ∑_{l=0}^{∞} ∑_{m=0}^{∞} 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.

Acknowledgment.

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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Michael Christ, Department of Mathematics, University of California, Berkeley CA 94720

  E-mail address: mchrist@math.berkeley.edu

Alexander Kiselev, Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley CA 94720

  E-mail address: kiselev@msri.org