AN INTERPOLATION THEOREM RELATED TO THE A.E. CONVERGENCE OF INTEGRAL OPERATORS

ALEXANDER KISELEV

Abstract. We show that for integral operators of general form the norm bounds in Lorentz spaces imply certain norm bounds for the maximal function. As a consequence, the a.e. convergence for the integral operators on the Lorentz spaces follows from the appropriate norm estimates.

1. Introduction

In this paper, we study the question of a.e. convergence of integral operators satisfying certain norm estimates in most general setting. The main tool in this studies is an interpolation theorem yielding the norm bounds on the maximal function of the integral operator given norm estimates on the operator itself. The result we show looks so natural and is so general that we hope it may be useful in many situations. In particular, in the Appendix we sketch the application to the spectral analysis of Schrödinger operators. It was hard to believe that the result we prove here is not known, but discussions with many specialists in the field convinced me that at least the reference is not simple to find.

The main result we prove here is as follows.

Let us consider two measure spaces (X, μ) and (Y, ν) with μ, ν being positive measures. Let A(k, x) be a measurable function on M × N, integrable over the sets of finite measure in M for a.e. k ∈ N. Let an operator T be given by

(T f)(k) = ∫_{X} A(k, x)f(x) dμ(x).  (1)

The operator T may be defined originally for all simple functions or bounded functions of finite support. Consider an extending family of measurable regions Ω_{t} ⊂ X, depending on the real parameter t, so that Ω_{t_{1}} ⊂ Ω_{t_{2}} if t_{2} > t_{1}. Define a maximal function M_{T} , corresponding to the family {Ω_{t}}_{t∈R} and the operator T by

(M_{T} f)(k) = sup_{t} | ∫_{Ω_{t}} A(k, x)f(x) dx| .  (2)

We use the notation ∥ ∥_{pq} for the norm in the Lorentz space L_{pq} or for a conventional quasinorm when the norm does not exist.

Then the following theorem holds:

Theorem. Suppose that an operator T given by (1) satisfies the bounds

∥T f∥_{s_{i}∞} ≤ C_{i}∥f∥_{p_{i}1}, i = 1, 2,

where all s_{i}, p_{i} are more than or equal to 1, s_{1} > s_{2}, and p_{1} ≠ p_{2}. Then the maximal function M_{T} , given by (2), satisfies

∥(M_{T} f)∥_{s_{r}q} ≤ C_{s}(q)∥f∥_{p_{r}q}  (3)

where s_{r}^{−1} = (1 − r)s_{1}^{−1} + rs_{2}^{−1}, p_{r}^{−1} = (1 − r)p_{1}^{−1} + rp_{2}^{−1}, 0 < r < 1, and q is any number satisfying 1 ≤ q ≤ ∞.

Remark. The theorem above would turn into just generalized Marcinkiewicz interpolation theorem if we would change M_{T} to T in (3).

The motivation for studying this problem comes, in particular, from the applications of a.e. convergence results for certain integral operators to the study of the spectrum of Schrödinger operators (continuous as well as discrete) [5], [6]. We briefly sketch this relationship in the Appendix, referring to the above works for details.

We note that the above theorem generalizes some old and well-known results on the a.e. convergence of the orhtogonal series and Fourier integrals, in particular, the results of Paley [7] and Zygmund [9]. Of course the theorem does not touch on the question of a.e. convergence at the corner points (r = 0, 1). These questions are very subtle even in the orthogonal series setting. In the case of trigonometric series the question about a.e. convergence of the series with coefficients from l_{2}(Z) is equivalent to the celebrated Luzin’s conjecture. Although in this case the a.e. convergence holds by the result of Carleson [2], it does not hold in general for orthonormal systems of functions (see, e.g. [4]). In particular, in the case of the multiple Fourier series, C.Fefferman showed that one can choose the extending sets Ω_{t} so that there is no a.e. convergence on l_{2}(Z^{n}). [3].

2. Proof of the main result

Proof. Fix an arbitrary real number r̃, 0 < r̃ < 1, and take s_{3}, s_{4} so that s_{1} > s_{3} > sr̃ > s_{4} > s_{2}. Let p_{3}, p_{4} correspond to s_{3}, s_{4} in a usual way. Also let us fix some number q′ such that q′ > max{p_{3}, p_{4}}.

First, we prove the theorem under an additional assumption that μ(Ω_{t}) depends continuously on t. The first step is to decompose the support of f into dyadic pieces and estimate certain auxilliary maximal functions. Such approach was used already by Paley [7] in his work on a.e. convergence of the series of orthonormal functions. Let f be a measurable bounded function of finite support and choose n so that 2^{n−1} ≤ μ(supp(f)) ≤ 2^{n}. Let the measurable set E be the support of the function f : E = {x| |f(x)| > 0}. For every integer m < n, we consider a partition of the set E into the dyadic pieces of size 2^{m} in a following way:

E_{m,l} = (Ω_{t_{m,l+1}} \ Ω_{t_{m,l}}) ∩ E,

where t_{m,l} is defined by a condition

t_{m,l} = inf{t|μ(Ω_{t} ∩ E) = 2^{m}l.}

The value t_{m,l} is well-defined at least for l ≤ 2^{n−m−1} because μ(Ω_{t}) depends on t continuously. The number of the sets E_{m,l} is between 2^{n−m} and 2^{n−m−1}. For notational convenience, we will assume that this number is always 2^{n−m} and will define the missing E_{m,l} as empty sets.

Let us define the functions M_{m,l}f and M_{m}f by

M_{m,l}f(k) = ∫_{E_{m,l}} A(k, x)f(x) dx.

and

M_{m}f(k) = sup_{l} |M_{m,l}f(k)|

Considering a dyadic development of every real number N, it is easy to see that

(M_{T} f)(k) ≤ ∑_{m=−∞}^{n} M_{m}f(k).  (4)

Indeed, suppose that for a given value of k, the supremum in (2) is reached when the value of the parameter t is equal to N (clearly if f has finite support, the supremum is reached for some value of t). Define the real number a by a = μ(E ∩ Ω_{N}) and consider the dyadic development of a :

a = ∑_{j=−∞}^{n} a_{j}2^{j},

where a_{j} is equal to 0 or 1 for every j. Then by construction, we can find disjoint sets E_{m,l}, at most one for each value of m, so that

μ ((E ∩ Ω_{N}) \ (∪_{m}E_{m,l})) = 0.

In fact, for each m the set E_{m,l} belongs to the union iff a_{m} = 1. The corresponding value of l then may be found by the formula l = ∑_{j=m+1}^{n} a_{j}2^{j−m}. By the generalized Marcinkiewicz interpolation theorem (see, e.g., [1] or [8]) we have

∥T f∥_{s_{r}q′} ≤ C_{r}(q′)∥f∥_{p_{r}q′}, 0 < r < 1.

Recall that the quasinorm ∥ · ∥^{∗} in Lorentz spaces may be defined by

∥f∥_{pq}^{∗} = (\frac{q}{p} ∫_{0}^{∞}|f^{∗}(t)|^{q}t^{\frac{q}{p} −1} dt)^{\frac{1}{q}} .

Let us denote by χ(E) the characteristic function of the measurable set E. Using the equivalence of the quasinorm ∥ · ∥_{pq}^{∗} and norm ∥ · ∥_{pq} for p > 1 (note that p_{i} are automatically more than 1), we see that in particular

∥M_{m,l}f(k)∥_{q′}^{s_{i}q′} = ∥T (fχ(E_{m,l}))(k)∥_{p_{i}q′}^{q′} ≤ C_{i}(q′)^{q′} (\frac{q′}{p_{i}} ∫_{0}^{∞}|(fχ(E_{m,l}))^{∗}|^{q′}t^{\frac{q′}{p_{i}} −1} dt) ≤

≤ C_{i}(q′)^{q′}2^{m( \frac{q′}{p_{i}} −1)} \frac{q′}{p_{i}} ∥fχ(E_{m,l})(t)∥_{q′}^{q′}, i = 1, 2.

Note that

∥M_{m}f∥_{s_{i}q′}^{q′} ≤ ∑_{l=1}^{2^{n−m}} ∥M_{ml}f∥_{s_{i}q′}^{q′}.

Hence summing over l we obtain

∥M_{m}f∥_{s_{i}q′}^{q′} ≤ C_{i}(q′)^{q′} \frac{q′}{p_{i}} 2^{m( \frac{q′}{p_{i}} −1)}∥f∥_{q′}^{q′}.

By (4), we have that

∥M_{T} f∥_{s_{i}q′} ≤ C_{i}(q′) (\frac{q′}{p_{i}} )^{\frac{1}{q′}} ∥f∥_{q′} ∑_{m=−∞}^{n} 2^{m( \frac{1}{p_{i}} − \frac{1}{q′})} = B_{i}(q′)2^{n( \frac{1}{p_{i}} − \frac{1}{q′})}∥f∥_{q′}. (5)

In the last inequality we denoted by B_{i}(q′) a new constant, which only depends on i and q′. Now we note that in a particular case when f is a characteristic function of a set, f = χ(E), (5) means

∥M_{T} χ(E)∥_{s_{i}q′} ≤ B_{i}(q′)2^{n( \frac{1}{p_{i}} − \frac{1}{q′})}2^{\frac{n}{q′}} = B_{i}(q′)2^{\frac{n}{p_{i}}} ≤ 2^{\frac{1}{p_{i}}} B_{i}(q′)∥χ(E)∥_{p_{i}1}. (6)

It is easy to verify that the operator M_{T} f, defined originally on the measurable bounded functions of compact support, is a sublinear operator. We remind that it means

|(M_{T} (f_{1} + f_{2}))(k)| ≤ |(M_{T} f_{1})(k)| + |(M_{T} f_{2})(k)|, |(M_{T} (bf))(k)| = |b||(M_{T} f)(k)|

for every scalar b. It is a well-known and simple to check fact that from the inequality (6) for sublinear operator it follows that ∥Mg∥_{s_{i}q′} ≤ C∥g∥_{p_{i}1} holds for all simple functions (see [8]) and hence by simple limiting argument for all bounded functions of finite support. Interpolating, we obtain that ∥Mf∥_{sr̃q} ≤ Cr̃(q)∥f∥_{pr̃q} for every function f bounded and of compact support. It is straightforward to see that this relation is then extended to all functions f ∈ L_{pr̃q}. This completes the proof under the assumption that μ(Ω_{t}) is continuous.

Consider now the general case. The technical problem we face here is that now it is not so easy to apply the dyadic decomposition of the supp(f) = E into coherent pieces: the jumps of the monotone function μ(E ∩Ω_{t}) may in general pose obstacles to that. We will handle this problem by constructing an auxilliary measure space and an auxilliary operator. We prove the estimates for the maximal function of this auxilliary operator, and then show that from these estimates follows the result for the original problem.

At every value of the parameter t the function μ(Ω_{t}), as a monotone function, has limits from the left and from the right. Denote by t_{∓} the value of the jump on the left and on the right respectively: t_{+} = μ(Ω_{t+0} \ Ω_{t}) and t_{−} = μ(Ω_{t} \ Ω_{t−0}). The set of values of t where any jump may occur is clearly at most countable. Let the sequence {t_{n}}_{n=1}^{∞} denote these points. To each t_{n} corresponds the size of the jump of μ(Ω_{t}) at this point, y_{n}. If for some values of t we have both t_{−} and t_{+} nonzero, this value of t is encountered in the above sequence twice: say, t_{n} = t_{n+1} = t and y_{n} = t_{−}, y_{n+1} = t_{+}. Hence, for each member of the sequence t_{n} only one of the two possible jumps takes place. We denote by Γ_{t_{n}} the portion of the measure space corresponding to t_{n} : Γ_{t_{n}} = Ω_{t_{n}+0} \ Ω_{t_{n}} if we have a jump on the right at t_{n} and Γ_{t_{n}} = Ω_{t_{n}} \ Ω_{t_{n}−0} otherwise.

Let us consider the following auxilliary measure space X̃ which we biuld out of X. At each value of t_{n} we replace Γ_{t_{n}} ⊂ X by

B_{t_{n}} = Γ_{t_{n}} × [0, 1]

with the structure of the product measure space. Hence

X̃ = (X \ (∪_{n}Γ_{t_{n}})) ∪ (∪_{m}B_{t_{m}}) .

The measure μ̃ on X̃ coincides with measure μ on the measurable set

X_{0} = X \ (∪_{n}Γ_{t_{n}}) ,

while on B_{t_{n}} the measure μ̃ equals the product measure μ×dx (dx being a Lebesgue measure on [0, 1]). Furthermore, we let the kernel Ã(k, x), defined on Y × X̃, be equal to A(k, x) for all k when x ∈ M_{0} and Ã(k, x, y) = A(k, x) for all k ∈ Y and all y ∈ [0, 1] if x belongs to Γ_{t_{n}} for some n. Define an integral operator T̃ by

(T̃f)(k) = ∫_{X̃} Ã(k, x)f(x) dx.

Next, define a family Ω̃_{u} of the extending measurable sets in X̃. We construct Ω̃_{u} so that μ̃(Ω̃_{u}) = u. Let t_{0}(u) = sup_{t}{t|μ(Ω_{t}) ≤ u}. If t_{0}(u) ≠ t_{n} for any n, we let

Ω̃_{u} = (Ω_{t_{0}(u)} ∩ X_{0}) ⋂ (∪_{t_{m}<t_{0}(u)}B_{t_{m}}) .

In this case μ̃(Ω̃_{u}) = u, since μ(Ω_{t}) is continuous at every t ∉ {t_{n}}_{n=1}^{∞}. Suppose now that t_{0}(u) = t_{n}. If at t_{n} we have a jump on the left (μ(Ω_{t_{n}} \ Ω_{t_{n}−0}) > 0) we let

Ω̃_{u} = (Ω_{t_{n}−0} ∩ X_{0}) ∪ (∪_{t_{m}<t_{n}}Γ_{t_{m}} × [0, 1]) ∪ (Γ_{t_{n}} × [0, \frac{u − μ(Ω_{t_{n}−0})}{μ(Γ_{t_{n}})}]) .

Otherwise, if the jump is on the right (μ(Ω_{t_{n}+0} \ Ω_{t_{n}}) > 0), we define

Ω̃_{u} = (Ω_{t_{n}} ∩ X_{0}) ∪ (∪_{t_{m}<t_{n}}Γ_{t_{m}} × [0, 1]) ∪ (Γ_{t_{n}} × [0, \frac{u − μ(Ω_{t_{n}})}{μ(Γ_{t_{n}})}]) ,

essentially just changing t_{n} −0 to t_{n}. Defined this way, Ω̃_{u} constitutes a measurable, extending family of sets such that μ̃(Ω̃_{u}) = u and, in particular, is continuous. Define the maximal function X̃, corresponding to the operator T̃ and the family Ω̃_{u} :

(M̃T̃f)(k) = sup_{u} | ∫_{Ω̃_{u}} Ã(k, x)f(x) dμ(x)| .

We have Lemma. For every r, 1 > r > 0, the operator T̃ satisfies the norm bounds

∥T̃f∥_{s_{r}p_{r}} ≤ C(r)∥f∥_{p_{r}}.

Proof. Indeed,

T̃f(k) = ∫_{X_{0}} A(k, x)f(x) dμ(x) + ∑_{n} ∫_{Γ_{tn}} dμ(x) ∫_{0}^{1} Ã(k, x, y)f(x, y) dy =

= ∫_{X_{0}} A(k, x)f(x)dμ(x) + ∑_{n} ∫_{Γ_{tn}} dμ(x)A(k, x) ∫_{0}^{1} f(x, y) dy

by the definition of the kernel Ã(k, x, y). Because of the norm bounds on the operator T and interpolation between them, we obtain

∥T̃f∥_{L_{srpr} (Y)} ≤ C(r) ( ∫_{X_{0}} |f(x)|^{p_{r}}dμ(x) + ∑_{n} ∫_{Γ_{tn}} dμ(x) |∫_{0}^{1} f(x, y) dy|^{p_{r}} )^{\frac{1}{pr}} ≤

≤ C(r) (∫_{X_{0}} |f(x)|^{p_{r}}dμ(x) + ∑_{n} ∫_{Γ_{tn}} dμ(x) ∫_{0}^{1} |f(x, y)|^{p_{r}} dy)^{\frac{1}{pr}} ≤ C(r)∥f∥_{L_{pr} (X̃)}.

We used Jensen inequality in the second step. □

Now let us choose r_{1}, r_{2} so that s_{r_{1}} = s_{3} and s_{r_{2}} = s_{4}. By the lemma, we have the bounds

∥T̃f∥_{s_{i}∞} ≤ C(i)∥f∥_{p_{i}}, i = 3, 4.

Since the function μ̃(Ω̃_{u}) is continuous in u, we can infer from the first part of the proof that the bound

∥M̃T̃∥_{sr̃q} ≤ Cr̃(q)∥f∥_{pr̃q}

hold for every q ∈ [1, ∞]. But for every t, we have

∫_{Ω_{t}} A(k, x)f(x)dμ(x) = ∫_{Ω̃_{u(t)}} Ã(k, y)f̃(y) dμ̃(y).

Here u(t) is such that μ(Ω_{t}) = u(t) and f̃ is a function which we define in a clear way. Namely, if y = x ∈ X_{0} we let f̃(y) = f(x) and if y = (x_{1}, y_{1}) ∈ Γ_{t_{n}} × [0, 1], we let f̃(x_{1}, y_{1}) = f(x_{1}). It is easy to see that ∥f̃∥_{L_{pq}(X̃)} = ∥f∥_{L_{pq}(X)} for all p, q because the distribution functions of f̃ and f coincide. Therefore,

(M_{T} f)(k) = sup_{t} | ∫_{Ω_{t}} A(k, x)f(x)dμ(x)| ≤ sup_{u} | ∫_{Ω̃_{u}} Ã(k, y)f̃(y)dμ̃(y)| = M̃T̃f̃(k)

and hence

∥M_{T} f∥_{L_{sr̃q}(X)} ≤ ∥M̃T̃f̃∥_{L_{sr̃q}(X̃)} ≤ Cr̃(q)∥f̃∥_{L_{pr̃q}(X̃)} = Cr̃(q)∥f∥_{L_{pr̃q}(X)}.

Hence we have shown the bound (3) in the general case and since the number r̃ is arbitrary between 0 and 1, the proof of the theorem is now complete. □

We may obtain the Zygmund’s and Paley’s theorems by specifying the measure spaces and the kernels. Let X and Y to be real lines equipped with Lebesgue measure and A(k, x) = exp(ikx). The L_{1} − L_{∞} and L_{2} − L_{2} bounds are obvious and we get the Zygmunds theorem, which says that the Fourier integral converges a.e. for functions f from L_{p}, 1 ≤ p < 2 and claims the corresponding estimate (3) for the maximal function.

Next let X = Z^{1}, Y = (a, b) and A(k, n) = ϕ_{n}(k), where {ϕ_{n}(k)}_{n=1}^{∞} is an orthonormal uniformly bounded system in L_{2}(a, b). Again the same bounds as above hold and we get the Paley’s theorem which says that the series of orhtonormal functions ∑_{n} c_{n}ϕ_{n}(x) converges a.e. if c_{n} ∈ l_{p}, 1 ≤ p < 2.

The last example we would like to present here is that of a pseudodifferential operator in R^{n} with bounded symbol a(k, x) :

(T f)(k) = ∫_{R^{n}} exp(ikx)a(k, x)f(x) dx.  (7)

If the symbol a(k, x) is from the class for which an L_{2}−L_{2} estimate is valid, we have that the integral defining (7) convergres a.e. k no matter which system of extending regions Ω_{t} we take. This is the situation which appears (with n = 1) in some of the applications to Schrödinger operators [5].

Appendix

In [5], a new approach to the investigation of the stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators under slowly decreasing pertubations is developed. The absolutely continuous spectrum corresponds to the infinite motion of the quantum particle. The question of the preservation of the absolutely continuous spectrum under decaying perturbations means, roughly speaking, finding out which local, i.e. decaying, pertubations are not strong enough to destroy this infinite motion. It turns out that in many cases the problem of spectral analysis may be reduced to a problem of finding asymptotics of the solutions of a certain ODE system, depending on the parameter λ, for a.e. values of this parameter (λ has a meaning of energy for Schrödinger equation). The latter problem may be tackled with asymptotic integration methods. The first step is an introduction of a transformation of the system, containing, in a simplest case, a function

q(x, λ) = ∫_{x}^{∞}θ^{2}(y, λ)V (y) dy.  (8)

Here θ(x, λ) is a solution of the unperturbed Schrödinger equation, so that

(−\frac{d^{2}}{dx^{2}} + U) θ = λθ

and V (x) is a perturbation. It turns out that if the function q(x, λ) may be defined and decays sufficiently fast for the values of λ from the support of the absolutely continuous spectrum of the unperturbed operator (−\frac{d^{2}}{dx^{2}} + U), then this absolutely continuous spectrum is preserved under perurbation by V. For example, the simplest condition under which the a.c. spectrum is preserved is that q(x, λ)V (x) ∈ L^{1} for a.e. λ from the support of the a.c. spectrum of the unperturbed operator. The interesting case is when V is not absolutely integrable; otherwise the stability is known for a long time and may be proven in a very simple way. However, although there has been a considerable attention to the subject, until recently no other general classes preserving the a.c. spectrum, given in terms of the rate of decay, were known even in the simplest situations (such as U = 0).

Thus one is led to the question of studying a.e. convergence and rate of decay estimates for the integral operators like (8). The paper [5] contains a simpler and less general version of the theorem we proved here. That result suffices for the applications of that paper, in particular producing a new general class of potentials preserving the a.c. spectrum of the free (U = 0) and periodic (U periodic) Schrödinger operators: all potentials V satisfying |V (x)| ≤ C(1+x)^{− \frac{2}{3} −ϵ}. The more general version we present here, however, may also be applied to studying discrete Schrödinger operators [6].

Acknowledgement

I am very grateful to Prof. S. Semmes and Prof. G. Pisier for stimulating discussions. I gratefully acknowledge hospitality of IHES, where part of this work was done. Research at MSRI supported in part by NSF grant DMS 9022140.

References

[1] J. Bergh and J. Löfström, Interploation Spaces: An Introduction, Sprinder-Verlag, Berlin Heidelberg 1976.

[2] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157.

[3] C.Fefferman, On the divergence of multiple Fourier series, Bull. of the Amer. Math. Soc. 77 (1976), 87–88.

[4] A. Garsia, Topics in a.e. Convergence, Markham Pub. Company, Chicago 1970.

[5] A. Kiselev, Preservation of the absolutely continuous spectrum of Schrödinger equation under perturbations by slowly decreasing potentials and a.e. convergence of integral operators, preprint.

[6] A. Kiselev, Stability of the absolutely continuous spectrum of Jacobi matrices under slowly decaying perturbations, in preparation.

[7] R.E.A.C. Paley, Some theorems on orthonormal functions, Studia Math. 3 (1931), 226–245.

[8] E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton 1971.

[9] A. Zygmund, A remark on Fourier transforms, Proc. Camb. Phil. Soc. 32 (1936), 321–327.

Mathematical Sciences Research Institute, 1000 Centennial Dr., Berkeley, CA 94720, USA

 E-mail address: kiselev@msri.org