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

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.

arXiv comment29 pages