ABSTRACT KERNEL MANIPULATION

TERENCE TAO

Abstract. We show various standard ways to freely manipulate the kernel K(x, y) of a linear operator without significantly affecting its mapping properties.

1. Introduction

Let K(x, y) be a distribution on R × R. We say that K is of type (p, q) for some 1 ≤ p, q ≤ ∞ if the linear operator T_{K}f(x) = ∫ K(x, y)f(y) dy maps L^{p}(R) to L^{q}(R). (We could take the underlying space to be something other than R, and could deal with other spaces or with sub-linear operators, but we shall not do so here for the sake of simplicity). The objective of this note is to use a single kernel K of type (p, q) to generate other kernels which are also of type (p, q).

Of course, the set of all kernels of type (p, q) is a Banach space, and we have the composition law that if K is of type (p, q) and K′ is of type (q, r), then

K′ ∘ K(x, z) = ∫ K′(x, y)K(y, z) dy

is of type (p, r). Also, if K(x, y) is of type (p, q), then K(y, x) and \overline{K(y, x)} are both of type (q, p). As a consequence, complexification is trivial: a kernel K(x, y) is of type (p, q) if and only if its real and imaginary parts are.

If a(x), b(y) are bounded functions, and K(x, y) is of type (p, q), then K(x, y)a(x)b(y) is also of type (p, q). In particular one can restrict K(x, y) to a product set E × F without destroying the type (p, q) property.

As an easy corollary of the above we have

Lemma 1.1. (see e.g. [8], [9], [2], [5], [1], [7]) Let K(x, y) be a kernel of type (p, q), and let c(x, y) be a function which is sufficiently smooth and compactly supported in the x variable, uniformly in y. Then K(x, y)c(x, y) is also of type (p, q).

Proof Write c(x, y) as a Fourier series in x, thus

c(x, y) = ∑_{k} e^{ikx}c_{k}(y)

where k ranges over a discrete lattice and the c_{k} are bounded functions with a bound that decreases rapidly in k. The claim then follows from the previous remarks and the triangle inequality.

Here is a typical application of the lemma: if K(x, y) is compactly supported and of type (p, q), and ϕ(x, y) is a smooth real-valued function, then e^{iϕ(x,y)}K(x, y) is also of type (p, q) and has a comparable operator norm. (This can be viewed as a generalization of the uncertainty principle: smooth perturbations of phase (such as ΔxΔξ) which are O(1) are negligible).

1991 Mathematics Subject Classification. 42B.

Suppose K_{i}(x, y) are a sequence of kernels which are uniformly of type (p, q) for some q ≥ p, and such that the x-supports of the K_{i} are finitely overlapping, and the y-supports of the K_{i} are also finitely overlapping. Then the sum ∑_{i} K_{i}(x, y) is also of type (p, q); this is clear from some applications of Hölder’s inequality. (In fact, even when q < p one can obtain a fairly precise computation of the operator norm of (p, q) in terms of that of the K_{i}, though without some decay in i this norm is going to be infinite).

By combining this observation with the previous lemma we obtain a handy truncation result:

Corollary 1.2. Let K(x, y) be a kernel of type (p, q) for some q ≥ p, and let ψ be a Schwarz function. Then K(x, y)ψ(x − y) is also of type (p, q).

For instance, (the higher dimensional version of) this can be used to show that the L^{p} mapping property for Bochner-Riesz kernels is equivalent (modulo epsilons) to the L^{p} mapping property for the same kernels restricted to a single annulus (see e.g. [?]).

2. Changing the measure

Up to now, our considerations have been with respect to Lebesgue measure, but we can easily generalize. If dμ, dν are positive measures on R, we say that K is of type (p; dμ, q; dν) if the operator

T_{K}^{ν}f(x) = ∫ K(x, y)f(y) dν(y)

maps L^{p}(dμ) to L^{q}(dν). All the above observations continue to hold in the general measure setting, with the obvious modifications.

If Φ : R → R and Ψ : R → R are bijections, then K(x, y) is of type (p; dμ, q; dν) if and only if K(Φ(x), Ψ(y)) is of type (p; Φ_{∗}(dμ), q; Φ_{∗}(dν)).

If dμ, dν are absolutely bi-continuous with respect to Lebesgue measure, then K(x, y) is of type (p; dμ, q; dν) if and only if

K(x, y)(\frac{dμ(x)}{dx})^{1/p′}(\frac{dν(x)}{dy})^{1/q}

is of type (p, q). So, in principle, nothing new is gained by altering measure. Certainly, modifying a measure by a factor bounded above and below has essentially no effect.

One can combine the above observations to get some fairly exotic kernel truncations. For instance, if ψ is a Schwarz function, Φ, Ψ are bijections and K(x, y) is of type (p, q) for some q ≥ p, then so is K(x, y)ψ(Φ(x) − Ψ(y)).

When q > p we can generalize this to the case when ψ merely has bounded variation; this is an observation of Christ and Kiselev. More precisely:

Lemma 2.1. [3], [4] If ψ is a bounded variation function, Φ, Ψ : R → R are arbitrary functions, and K(x, y) is of type (p; dμ, q; dν) for some q > p, then the kernel

K(x, y)ψ(Φ(x) − Ψ(y))

is also of type (p; dμ, q; dν).

Proof (Informal) By decomposing ψ into Heaviside functions we may assume that ψ = χ_{(0,∞)}.

By duality, it suffices to show that

| ∫ ∫_{Φ(x)>Ψ(y)} K(x, y)g(x)f(y) dμ(x)dν(y)| ≲ (∫ |g(x)|^{q′} dμ(x))^{1/q′}(∫ |f(y)|^{p} dν(y))^{1/p}

for all f, g. We may assume that f, g are non-negative and normalized ∥f∥_{p} = ∥g∥_{q′}.

Fix f, g. We define d̃μ = g(x)^{q′} dμ, d̃ν = f(y)^{p} dν, and

K̃(x, y) = K(x, y)g(x)^{1−q′}f(y)^{1−p}.

The measures d̃μ, d̃ν are then probability measures, and it suffices to show that

| ∫ ∫_{Φ(x)>Ψ(y)} K̃(x, y) d̃μ(x)d̃ν(y)| ≲ 1.

On the other hand, from the previous observations we know that K̃(x, y) is of type (p; d̃μ, q; d̃ν).

By a re-arrangement of the x variable (adjusting K̃ if necessary) we may assume that Φ(x) = x and d̃μ is Lebesgue measure on [0, 1]. Similarly we may assume that Ψ(y) = y and d̃ν is Lebesgue measure on [0, 1]. We then have to show that

| ∫ ∫_{1≥x>y≥0} K̃(x, y) dxdy| ≲ 1.

But this follows from a Whitney decomposition of the region of integration, the estimate

| ∫ ∫_{IxJ} K̃(x, y) dxdy| ≲ 2^{−j/q′}2^{−j/p}

for all squares IxJ of side-length 2^{−j} (this follows from the hypothesis on K̃), and the assumption q > p.

As a special case of this lemma, we can truncate any kernel K(x, y) to the halfspace {(x, y) : x > y} without destroying the (p, q) type for q > p. This has a number of useful consequences, for instance it allows one to obtain inhomogeneous Strichartz estimates from homogeneous ones as long as one is mapping a lower time exponent to a higher time exponent. See [10], [6]. Unfortunately it fails at the most interesting case of p = q, as one can see by attempting to truncate the Hilbert transform.

Another application of this lemma is to maximal functions. For instance, one can start with the Hausdorff-Young inequality

∥ ∫ e^{ixy}f(y) dy∥_{p′} ≤ ∥f∥_{p}

for 1 ≤ p ≤ 2, and conclude the Menshov-Paley-Zygmund inequality

∥ sup_{w} | ∫_{y<w} e^{ixy}f(y) dy|∥_{p′} ≤ ∥f∥_{p}

if p is strictly less than 2. (The claim also holds for p = 2, but this is substantially more difficult, being equivalent to Carleson’s theorem. For further discussion on this, see [4]).

References

[1] J. Bourgain, Some new estimates on oscillatory integrals, Essays in Fourier Analysis in honor of E. M. Stein, Princeton University Press (1995), 83–112.

[2] M. Christ, On the regularity of inverses of singular integral operators, Duke Math. J., 57 (1988): 459–484.

[3] M. Christ, A. Kiselev, A maximal inequality, preprint.

[4] M. Christ, A. Kiselev, Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: some optimal results, J. Amer. Math. Soc., to appear.

[5] I. L. Hwang, The L^{2}-boundedness of pseudodifferential operators, Trans. Amer. Math. Soc. 302 (1987), 55–76.

[6] H. Smith, C. D. Sogge, Global Strichartz estimates for non-trapping perturbations of the Laplacian, preprint.

[7] C. D. Sogge, E. M. Stein, Averages over hypersurfaces II, Invent. Math. 86 (1986), 233–242.

[8] T. Tao, The Bochner-Riesz conjecture implies the Restriction conjecture, to appear, Duke Math J.

[9] T. Tao, The weak-type endpoint Bochner-Riesz conjecture and related topics , Indiana Math. J. 47 (1998), 1097–1124.

[10] T. Tao, On the maximal Bochner-Riesz conjecture in the plane, for p < 2, to appear, Trans. Amer. Math. Soc.

[11] T. Tao, Radial endpoint Strichartz estimates for the two-dimensional Schrödinger equation, to appear.

Department of Mathematics, UCLA, Los Angeles, CA 90024

  E-mail address: tao@math.ucla.edu