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Weighted norm inequalities for vector-valued singular integrals on homogeneous spaces

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Tozoni S.

Studia Mathematica

2004

tom 161

nr 1

71-97

Let X be a homogeneous space and let E be a UMD Banach space with a normalized unconditional basis $(e_{j})_{j≥1}$. Given an operator T from $L^{∞}_{c}(X)$ to L¹(X), we consider the vector-valued extension T̃ of T given by $T̃(∑_{j} f_{j}e_{j}) = ∑_{j} T(f_{j})e_{j}$. We prove a weighted integral inequality for the vector-valued extension of the Hardy-Littlewood maximal operator and a weighted Fefferman-Stein inequality between the vector-valued extensions of the Hardy-Littlewood and the sharp maximal operators, in the context of Orlicz spaces. We give sufficient conditions on the kernel of a singular integral operator to have the boundedness of the vector-valued extension of this operator on $L^{p}(X,Wdμ;E)$ for 1 < p < ∞ and for a weight W in the Muckenhoupt class $A_{p}(X)$. Applications to singular integral operators on the unit sphere Sⁿ and on a finite product of local fields 𝕂ⁿ are given. The versions of all these results for vector-valued extensions of operators on functions defined on a homogeneous space X and with values in a UMD Banach lattice are also given.

Ograniczanie wyników

1 Studia Mathematica

1 Tozoni S.

1 2004

Wyniki wyszukiwania

Weighted norm inequalities for vector-valued singular integrals on homogeneous spaces