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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 \({L^p(\mathbb{R}^N)}\) as p becomes large, and the growth of the A  p  constant on estimates of the area integrals on the weighted L  p  spaces.

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