Averages over starlike sets, starlike maximal functions, and homogeneous singular integrals

Watson, David; Wheeden, Richard

doi:10.1090/S0002-9947-2011-05135-4

Averages over starlike sets, starlike maximal functions, and homogeneous singular integrals
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by David K. Watson and Richard L. Wheeden

Trans. Amer. Math. Soc. 363 (2011), 5179-5206

DOI: https://doi.org/10.1090/S0002-9947-2011-05135-4

Published electronically: May 18, 2011

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Abstract:

We improve some of the results in our 1999 paper concerning weighted norm estimates for homogeneous singular integrals with rough kernels. Using a representation of such integrals in terms of averages over starlike sets, we prove a two-weight $L^{p}$ inequality for $1 < p < 2$ which we were previously able to obtain only for $p \geq 2$. We also construct examples of weights that satisfy conditions which were shown in our earlier paper to be sufficient for one-weight inequalities when $1<p<\infty$.

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