Sharp weighted estimates for square functions associated to operators on spaces of homogeneous type

Let X be a metric space with a doubling measure and let L be a linear operator in L² (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 _w^p 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 Calderón–Zygmund operators.