Generalized Hörmander conditions and weighted endpoint estimates

Abstract

Abstract. We consider two-weight estimates for singular integral operators and their commutators with bounded mean oscillation functions. Hörmander type conditions in the scale of Orlicz spaces are assumed on the kernels. We prove weighted weak-type estimates for pairs of weights (u, Su) where u is an arbitrary nonnegative function and S is a maximal operator depending on the smoothness of the kernel. We also obtain sufficient conditions on a pair of weights (u, v) for the operators to be bounded from L p (v) to L p,∞ (u). One-sided singular integrals, like the differential transform operator, are considered as well. We also provide applications to Fourier multipliers and homogeneous singular integrals.1. Introduction. The Calderón-Zygmund decomposition is a powerful tool in harmonic analysis. Since its discovery in [6], many authors have used it to derive boundedness properties of singular integral operators. For instance, using the fact that the Hilbert and Riesz transforms are bounded on L 2 , and by means of the Calderón-Zygmund decomposition, one proves that these classical operators are of weak type (1, 1). From this starting point, in the literature one can find many boundedness results for the Hilbert and Riesz transforms: estimates on L p , one-weight and two-weight norm inequalities, etc.The Calderón-Zygmund theory generalizes these ideas to provide a general framework allowing one to deal with singular integral operators. A typical Calderón-Zygmund convolution operator T is bounded on L 2 (R n ) and has a kernel K on which various conditions are assumed. In the easiest case, K behaves as the kernel of the Hilbert or Riesz transform. That is, K decays as |x| −n and its gradient as |x| −n−1 . It was already proved in [15] that these assumptions can be relaxed if we want to show that T is of weak type (1, 1):

Sparse and weighted estimates for generalized Hörmander operators and commutators

“…M. Lorente, M. S. Riveros and A. de la Torre obtained Coifman-Fefferman estimates suited for those operators [30], the same authors in a joint work with J. M. Martell established Coifman-Fefferman inequalities and also weighted endpoint estimates in the case w ∈ A ∞ for commutators in [29]. Later on, M. Lorente, M. S. Riveros, J. M. Martell and C. Pérez proved some interesting endpoint estimates for arbitrary weights in [28]. The purpose of this work is to update and improve results in those works using sparse domination techniques.…”

“…Relying upon that sparse domination a number of quantitative estimates are derived. Some of them are improvements and complementary results to those contained in a series of papers due to M. Lorente, J. M. Martell, C. Pérez, S. Riveros and A. de la Torre [30,29,28]. Also the quantitative endpoint estimates in [24] are extended to iterated commutators.…”

Mixed Inequalities of Fefferman-Stein Type for Singular Integral Operators

“…When (t) = t r , r ≥ 1 , we write H = H r . In [12], the authors prove certain Fefferman-Stein inequalities for these types of operators. Concretely, if Φ is a Young function and there exists 1 < p < ∞ and Young functions , such that ∈ B p � and 𝜂 −1 (z)𝜑 −1 (z) ≲ Φ−1 (z) for z ≥ z 0 ≥ 0 , then the inequality holds with p (z) = (z 1∕p ) , and where Φ is the complementary Young function of Φ (see the "Preliminaries and basic definitions" section).…”

$A_p$-$A_\infty$ estimates for general multilinear sparse operators

“…To observe the other application, we first recall the class of multilinear integral operator which is bounded on certain products of Lebesgue spaces on R n where associated kernel satisfies some mild regularity condition which is weaker than the usual Hölder continuity of those in the class of multilinear Calderón-Zygmund singular integral operators. This class of the operators motivated from the recent works [3,10,14,22,25,26,27] and weighted bounds for such operators studied in [2] very recently. The main example of such operators is Multilinear Fourier multipliers.…”