Mathematical Research Letters

Volume 22 (2015)

Number 5

Boundedness of non-homogeneous square functions and $L^q$ type testing conditions with $q \in (1, 2)$

Pages: 1417 – 1457

DOI: https://dx.doi.org/10.4310/MRL.2015.v22.n5.a8

Authors

Henri Martikainen (Department of Mathematics and Statistics, University of Helsinki, Finland)

Mihalis Mourgoglou (Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain)

Abstract

We continue the study of local $Tb$ theorems for square functions defined in the upper half-space $(\mathbb{R}^{n+1}_{+} , \mu \times dt / t)$. Here $\mu$ is allowed to be a non-homogeneous measure in $\mathbb{R}^n$. In this paper we prove a boundedness result assuming local $L^q$ type testing conditions in the difficult range $q \in (1, 2)$. Our theorem is a non-homogeneous version of a result of S. Hofmann valid for the Lebesgue measure. It is also an extension of the recent results of M. Lacey and the first named author where non-homogeneous local $L^2$ testing conditions have been considered.

Keywords

square function, non-homogeneous analysis, local $Tb, L^q$ test functions

2010 Mathematics Subject Classification

42B20

Full Text (PDF format)

Accepted 21 January 2015

Published 13 April 2016