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A weighted inequality for the maximal Bochner-Riesz operator on $ {\bf R}\sp 2$


Author: Anthony Carbery
Journal: Trans. Amer. Math. Soc. 287 (1985), 673-680
MSC: Primary 42B10
DOI: https://doi.org/10.1090/S0002-9947-1985-0768732-X
MathSciNet review: 768732
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Abstract: For $ f \in \mathcal{S}({{\mathbf{R}}^2})$, let $ (T_R^\alpha f)\hat \emptyset (\xi ) = (1 - \vert\xi {\vert^2}{R^2})_ + ^\alpha \hat f(\xi )$. It is a well-known theorem of Carleson and Sjölin that $ T_1^\alpha $ defines a bounded operator on $ {L^4}$ if $ \alpha > 0$. In this paper we obtain an explicit weighted inequality of the form

$\displaystyle \int {\mathop {\sup }\limits_{0 < R < \infty } \vert T_R^\alpha f(x){\vert^2}w(x)\;dx \leqslant \int {\vert f{\vert^2}{P_\alpha }w(x)\;dx,} } $

with $ {P_\alpha }$ bounded on $ {L^2}$ if $ \alpha > 0$. This strengthens the above theorem of Carleson and Sjölin. The method gives information on the maximal operator associated to general suitably smooth radial Fourier multipliers of $ {{\mathbf{R}}^2}$.

References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9947-1985-0768732-X
Article copyright: © Copyright 1985 American Mathematical Society