Littlewood-Paley and multiplier theorems on weighted 𝐿^{𝑝} spaces

Kurtz, Douglas S.

doi:10.1090/S0002-9947-1980-0561835-X

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Littlewood-Paley and multiplier theorems on weighted $L\sp{p}$ spaces

Author: Douglas S. Kurtz
Journal: Trans. Amer. Math. Soc. 259 (1980), 235-254
MSC: Primary 42B25; Secondary 42B15
MathSciNet review: 561835
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Abstract: The Littlewood-Paley operator $\gamma (f)$ , for functions f defined on ${{\textbf{R}}^n}$ , is shown to be a bounded operator on certain weighted ${L^p}$ spaces. The weights satisfy an ${A_p}$ condition over the class of all n-dimensional rectangles with sides parallel to the coordinate axes. The necessity of this class of weights demonstrates the 1-dimensional nature of the operator. Results for multipliers are derived, including weighted versions of the Marcinkiewicz Multiplier Theorem and Hörmander's Multiplier Theorem.

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