Sparse and weighted estimates for generalized Hörmander operators and commutators

Abstract

In this paper a pointwise sparse domination for generalized Hörmander and also for iterated commutators with those operators is provided generalizing the sparse domination result in [24]. 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. Other results that are obtained in this work are some local exponential decay estimates for generalized Hörmander operators in the spirit of [34] and some negative results concerning Coifman-Fefferman estimates for a certain class of kernels satisfying particular generalized Hörmander conditions.2000 Mathematics Subject Classification. 42B20, 42B25.

Commutators of singular integrals revisited

“…To settle the case , we hinge upon the counterpart for iterated commutators obtained in , The idea of the proof consists in combining that sparse domination with . However, in contrast with the case , the sparse terms that arise after applying may be raised to a power greater than 1.…”

Vector-valued operators, optimal weighted estimates and the Cp condition

“…In the following Theorems we gather some estimates in the spirit of [29], some of them already contained there, that can be settled combining sparse domination results with ideas in [16,32]. We will finish this section with a similar type result for rough singular integrals.…”

On two weight estimates for iterated commutators

“…Consider the m-th iterated commutator T m b . Recall the following pointwise sparse bound established for m = 1 in [27] and for m ≥ 1 in [17]: for every bounded and compactly supported f , there exist 3 n sparse families S j ⊂ D j such that for a.e.…”