Square functions with general measures II

TL;DR

This paper advances the theory of conical and vertical square functions with general measures, providing new boundedness criteria, counterexamples, and exploring the dependence on measure properties and aperture size.

Contribution

It introduces a local $Tb$ theorem for non-doubling measures, constructs counterexamples for $L^{p}$ boundedness, and analyzes aperture dependence in non-homogeneous settings.

Findings

Conical square functions are generally bounded on $L^{2}$ and $L^{p}$, but vertical ones may not be.

A non-doubling Cantor measure affects the boundedness depending on cone aperture.

Sharp weighted bounds are established for doubling measures.

Abstract

We continue developing the theory of conical and vertical square functions on $R^{n}$, where $\mu$ is a power bounded measure, possibly non-doubling. We provide new boundedness criteria and construct various counterexamples. First, we prove a general local $Tb$ theorem with tent space $T^{2,\infty}$ type testing conditions to characterise the $L^{2}$ boundedness. Second, we completely answer the question, whether the boundedness of our operators on $L^{2}$ implies boundedness on other $L^{p}$ spaces, including the endpoints. For the conical square function, the answers are generally affirmative, but the vertical square function can be unbounded on $L^{p}$ for $p > 2$, even if $\mu = dx$. For this, we present a counterexample. Our kernels $s_t$, $t > 0$, do not necessarily satisfy any continuity in the first variable -- a point of technical importance throughout the paper. Third, we construct a non-doubling Cantor-type measure and an associated conical square function operator, whose $L^{2}$ boundedness depends on the exact aperture of the cone used in the definition. Thus, in the non-homogeneous world, the 'change of aperture' technique -- widely used in classical tent space literature -- is not available. Fourth, we establish the sharp $A_{p}$-weighted bound for the conical square function under the assumption that $\mu$ is doubling.