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 Lerner et al. (Adv Math 319:153–181, 2017). 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 Lorente et al. (Stud Math 195(2):157–192, 2009, J Math Anal Appl 342(2):1399–1425, 2008, J Fourier Anal Appl 11(5):497–509, 2005). Also the quantitative endpoint estimates in Lerner et al. (Adv Math 319:153–181, 2017) 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 Ortiz-Caraballo et al. (Math Ann 357(4):1217–1243, 2013) and some negative results concerning Coifman–Fefferman estimates for a certain class of kernels satisfying particular generalized Hörmander conditions.