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

Authors: Douglas S. Kurtz and Richard L. Wheeden
Journal: Trans. Amer. Math. Soc. 255 (1979), 343-362
MSC: Primary 42A45; Secondary 42B20
DOI: https://doi.org/10.1090/S0002-9947-1979-0542885-8
MathSciNet review: 542885
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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 $\left\vert x \right\vert$ , sharp results are obtained which indicate such restrictions are necessary. The method of proof is based on the function ${f^\char93 }$ 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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