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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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