Comments on: “Sharp Weighted Estimates for Square Functions Associated to Operators on Spaces of Homogeneous Type ”

Abstract

Let X be a metric space with a doubling measure, and let L be a nonnegative self-adjoint operator in $L^2(X)$ which generates a semigroup $e^{-t L}$ whose kernels $p_t (x, y), t > 0$, satisfy the Gaussian upper bound. Inspired by Fefferman’s paper (Fefferman in Acta Math 124:9–36), in this note, we give sufficient conditions for which the square function $g_{L,\psi ,\alpha }^{*}$ is unbounded from $L^p(X)$ to $L^p(X)$. As an application, we discuss the sharpness of the exponent of aperture $\alpha$ in the (Bui and Duong in J Geom Anal 30:874–900, Theorem 1.6).