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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Weighted inequalities for the disc multiplier

Author: Kenneth F. Andersen
Journal: Proc. Amer. Math. Soc. 83 (1981), 269-275
MSC: Primary 42B15
MathSciNet review: 624912
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Abstract: Charles Fefferman has shown that the disc multiplier is not a bounded operator on $ {L^p}({{\mathbf{R}}^n})$, $ n > 1$, $ p \ne 2$. On the other hand, Carl Herz has shown that when this operator is restricted to radial functions in $ {{\mathbf{R}}^n}$, it is bounded in $ {L^p}({{\mathbf{R}}^n})$ provided $ p$ satisfies $ 2n/(n + 1) < p < 2n/(n - 1)$. In this paper, sufficient conditions on the weight function $ \omega $ are given in order that the disc multiplier restricted to radial functions should be bounded in $ {L^p}({{\mathbf{R}}^n},\omega (\left\vert x \right\vert))$. When applied to power weights $ \omega (r) = {r^\alpha }$ these conditions are also necessary.

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Keywords: Disc multiplier, radial functions, spherical harmonics, weighted inequalities
Article copyright: © Copyright 1981 American Mathematical Society