Characterizations of the exponential square class on spaces ofhomogeneous type and applications

Abstract

Let $(X,d,\mu)$ be a space of homogeneous type.We establish three equivalentcharacterizations of the exponential square class associated to the classical dyadic square function on $(X,d,\mu)$: exponential square integrability, improved good-$\lambda$ inequalities, and the sharp $L^p$ lower bound for the classical dyadic square function. This result generalizes the famous Chang-Wislon-Wolff theorem on $\mathbb{R}^n$.We then apply this result to obtain sharp $L^p$ estimates for the square function associated to the Laplace-Beltrami operator on Riemannian manifolds with nonnegative Ricci curvature.