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Weighted Hardy spaces associated with elliptic operators. I: Weighted norm inequalities for conical square functions. (English) Zbl 1380.42019
The article is the first part of a series of three articles dealing with the study of different characterizations of weighted Hardy spaces related to a second order divergence form elliptic operator with bounded complex coefficients.
Precisely, let \(A\) be an \(n\times n\) matrix of complex and \(L^{\infty}\)-valued coefficients defined on \(\mathbb{R}^n\), and the divergence form elliptic operator \[ Lf=-\operatorname{div}(A \nabla f). \] The operator \(-L\) generates a \(C^0\)-semigroup, \(\{e^{-tL} \}_{t>0}\), called the Heat semigroup, which, together with the Poisson semigroup, \(\{e^{-t\sqrt{L}} \}_{t>0}\), allows to define different conical square functions. Weighted norm inequalities for these conical square functions are obtained, where the weights involved in the estimates are those belonging to the \(A_p\) class of Muckenhoupt. In the comparison of square functions in weighted spaces with cones having different apertures, an important tool introduced are the change-of-angle formulas. Another important tool is the introduction of a modified version of the Carleson condition for estimates below \(p=2\).

MSC:
42B30 \(H^p\)-spaces
42B25 Maximal functions, Littlewood-Paley theory
35J15 Second-order elliptic equations
47A60 Functional calculus for linear operators
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