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 linear operator in $$L^2(X)$$ which generates a semigroup $$e^{-tL}$$ whose kernels $$p_t(x,y)$$, $$t > 0$$, satisfy the Gaussian upper bound. In this article, we prove sharp$$L^p_w$$ norm inequalities for a number of square functions associated to L including the vertical square functions, the Lusin area integral square functions and the Littlewood–Paley g-functions. We note that our conditions on the heat kernels $$p_t(x,y)$$ are mild in the sense that the associated kernels of the square functions do not possess enough regularity for those operators to belong to the standard class of Calderon–Zygmund operators.