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

Henri Martikainen, Mihalis Mourgoglou

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## Abstract

We continue the study of local Tb theorems for square functions defined in the upper half-space (R-+(n+1), mu x dt/t). Here mu is allowed to be a non-homogeneous measure in R-n. In this paper we prove a boundedness result assuming local L-q type testing conditions in the difficult range q is an element of (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.
Original language English Mathematical Research Letters 22 5 1417-1457 41 1073-2780 https://doi.org/10.4310/MRL.2015.v22.n5.a8 Published - 2015 A1 Journal article-refereed

## Fields of Science

• 111 Mathematics
• ### Multiparameter Dyadic Harmonic Analysis and Probabilistic Methods

Martikainen, H.