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



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

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


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.


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

2010 Mathematics Subject Classification


Accepted 21 January 2015

Published 13 April 2016