Weighted $L^p$ estimates for the area integral associated to self-adjoint operators

This article is concerned with some weighted norm inequalities for the so-called horizontal (i.e. involving time derivatives) area integrals associated to a non-negative self-adjoint operator satisfying a pointwise Gaussian estimate for its heat kernel, as well as the corresponding vertical (i.e. involving space derivatives) area integrals associated to a non-negative self-adjoint operator satisfying in addition a pointwise upper bounds for the gradient of the heat kernel. As applications, we obtain sharp estimates for the operator norm of the area integrals on $L^p(\text{RN})$ as $p$ becomes large, and the growth of the $A_p$ constant on estimates of the area integrals on the weighted $L^p$ spaces.