Results on weighted norm inequalities for multipliers

Kurtz, Douglas S.; Wheeden, Richard L.

doi:10.1090/S0002-9947-1979-0542885-8

Results on weighted norm inequalities for multipliers
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by Douglas S. Kurtz and Richard L. Wheeden PDF

Trans. Amer. Math. Soc. 255 (1979), 343-362 Request permission

Abstract:

Weighted ${L^p}$-norm inequalities are derived for multiplier operators on Euclidean space. The multipliers are assumed to satisfy conditions of the Hörmander-Mikhlin type, and the weight functions are generally required to satisfy conditions more restrictive than ${A_p}$ which depend on the degree of differentiability of the multiplier. For weights which are powers of $\left | x \right |$, sharp results are obtained which indicate such restrictions are necessary. The method of proof is based on the function ${f^\# }$ of C. Fefferman and E. Stein rather than on Littlewood-Paley theory. The method also yields results for singular integral operators.

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Littlewood-Paley and multiplier theorems on weighted

spaces

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