Boundedness of non-homogeneous square functions and Lq 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 (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.

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.

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.