H\"ormander's conditions for vector-valued kernels of singular integrals and its commutators

TL;DR

This paper extends H"ormander's conditions to vector-valued kernels of singular integrals, establishing new weighted inequalities and commutator estimates under a generalized framework involving Young functions.

Contribution

It introduces a weaker H"ormander type condition for vector-valued kernels and derives corresponding Coifman type estimates and weighted norm inequalities.

Findings

Established a new H"ormander condition for vector-valued kernels.

Proved weighted norm inequalities for singular integrals with these kernels.

Applied results to the square operator and its commutator.

Abstract

In this paper we study Coifman type estimates and weighted norm inequalities for singular integral operators $T$ and its commutators, given by the convolution with a vector valued kernel $K$. We define a weaker H\"ormander type condition associated with Young functions for the vector valued kernels. With this general framework we obtain as an example the result for the square operator and its commutator given in [ M. Lorente, M.S. Riveros, A. de la Torre \emph{On the Coifman type inequality for the oscillation of the one-sided averages }, Journal of Mathematical Analysis and Applications, Vol 336, Issue 1,\ (2007) 577-592.]