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

We continue the study of local Tb theorems for square functions defined in the upper half-space (ℝn+1+ , μ × dt/t). Here μ is allowed to be a non-homogeneous measure in ℝn. In this paper we prove a boundedness result assuming local Lq type testing conditions in the difficult range q ϵ (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 L2 testing conditions have been considered.