Loukas Grafakos

Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces

We obtain sharp weighted L p estimates in the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 r it is bounded on L p (v) by the same increasing function of the Ap characteristic constant of v, and for p < r it is bounded on L p (v) by the same increasing function of the r 1 p 1 power of the Ap characteristic constant of v. For some operators these bounds are sharp, but not always. In particular, we show that they are sharp for the Hilbert, Beurling, and martingale transforms.

Some remarks on multilinear maps and interpolation

Abstract. A multilinear version of the Boyd interpolation theorem is proved in the context of quasi-normed rearrangement-invariant spaces. A multilinear Marcinkiewicz interpolation theorem is obtained as a corollary. Several applications are given, including estimates for bilinear fractional integrals.

The Kato-Ponce Inequality

In this article we revisit the inequalities of Kato and Ponce concerning the L r norm of the Bessel potential J s = (1 − Δ) s/2 (or Riesz potential D s = (− Δ) s/2) of the product of two functions in terms of the product of the L p norm of one function and the L q norm of the Bessel potential J s (resp. Riesz potential D s ) of the other function. Here the indices p, q, and r are related as in Hölders inequality 1/p + 1/q = 1/r and they satisfy 1 ≤ p, q ≤ ∞ and 1/2 ≤ r < ∞ and . Also the estimate is of weak-type when either p or q is equal to 1. In the case r < 1 we indicate via an example that when the inequality fails. Furthermore, we extend these results to the multi-parameter case.

Extrapolation of Weighted norm inequalities for multivariable operators and applications

Two versions of Rubio de Francia’s extrapolation theorem for multivariable operators of functions are obtained. One version assumes an initial estimate with different weights in each space and implies boundedness on all products of Lebesgue spaces. Another version assumes an initial estimate with the same weight but yields boundedness on a product of Lebesgue spaces whose indices lie on a line. Applications are given in the context of multilinear Calderón-Zygmund operators. Vector-valued inequalities are automatically obtained for them without developing a multilinear Banach-valued theory. A multilinear extension of the Marcinkiewicz and Zygmund theorem on ℓ2-valued extensions of bounded linear operators is also obtained.

Multilinear operators with non-smooth kernels and commutators of singular integrals

We obtain endpoint estimates for multilinear singular integral operators whose kernels satisfy regularity conditions significantly weaker than those of the standard Calder6n-Zygmund kernels. As a consequence, we deduce endpoint L 1 × ··· x L 1 to weak L 1/m estimates for the mth-order commutator of Calderon. Our results reproduce known estimates for m = 1, 2 but are new for m > 3. We also explore connections between the 2nd-order higher-dimensional commutator and the bilinear Hilbert transform and deduce some new off-diagonal estimates for the former.

On Multilinear Fourier Multipliers of Limited Smoothness

In this paper, we prove the L2-boundedness of multilinear Fourier multiplier operators with multipliers of limited smoothness. As a result, we can extend Calderón and Torchinsky’s result in the linear theory to the multilinear case. The sharpness of our results is also discussed.

The Hormander multiplier theorem for multilinear operators

Abstract In this paper, we provide a version of the Mikhlin–Hörmander multiplier theorem for multilinear operators in the case where the target space is Lp for p ≦ 1. This extends a recent result of Tomita [15] who proved an analogous result for p > 1.

Discrete decompositions for bilinear operators and almost diagonal conditions

This is the published version, also available here: http://dx.doi.org/10.1090/S0002-9947-01-02912-9. First published in Transaction of American Mathematical Society in 2002, published by the American Mathematical Society.

The Multilinear Strong Maximal Function

A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Bilinear interpolation results between distributional estimates, such as those satisfied by the multivariable strong maximal function, are also proved.

Bilinear operators on Herz-type Hardy spaces

The authors prove that bilinear operators given by finite sums of products of Calderon-Zygmund operators on Rn are bounded from HK11 q1 × HK22 q2 into HK α,p q if and only if they have vanishing moments up to a certain order dictated by the target space. HereHK q are homogeneous Herztype Hardy spaces with 1/p = 1/p1 + 1/p2, 0 < pi ≤ ∞, 1/q = 1/q1 + 1/q2, 1 < q1, q2 < ∞, 1 ≤ q < ∞, α = α1 + α2 and −n/qi < αi < ∞. As an application they obtain that the commutator of a Calderon-Zygmund operator with a BMO function maps a Herz space into itself.