IUMJ

Title: Square functions with general measures II

Authors: Henri Martikainen, Mihalis Mourgoglou and Tuomas Orponen

Issue: Volume 63 (2014), Issue 5, 1249-1279

Abstract:

We continue developing the theory of conical and vertical square functions on $\mathbb{R}^n,\mu)$, where $\mu$ can be 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 whether or not 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 $\mu=\mathrm{d}x$. 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.