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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the Riesz-transforms along surfaces in $ {\bf R}\sp 3$


Author: A. El Kohen
Journal: Proc. Amer. Math. Soc. 94 (1985), 672-674
MSC: Primary 42B20; Secondary 42B15, 47G05
MathSciNet review: 792281
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Abstract: For $ x = ({x_1},{x_2},{x_3})$ in $ {R^3}$, $ t = ({t_1},{t_2})$ in $ {R^2}$, $ \vert t\vert = {(t_1^2 + t_2^2)^{1/2}}$ and $ a > 0$, we define

$\displaystyle {R_a}f(x) = \int_{{R^2}} {f({x_1} - {t_1},{x_2} - {t_2},{x_3} - \vert t{\vert^a})\frac{{{t_1}}}{{\vert t{\vert^3}}}dt.} $

The transformation $ {R_a}$ can be thought of as a Riesz-transform along the surface $ ({t_1},{t_2},\vert t{\vert^a})$ in $ {R^3}$. Our purpose here is to show that, for all $ a > 0$, the operator $ {R_a}$ is bounded on $ {L^p}$ for $ 6/5 < p < 6$.

Singular integrals along lower dimensional nonaffine varieties have been studied by several authors. See for example [3, 5, 6]. In the proof of our result, we use techniques similar to the ones used in [1, 2].


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0792281-1
PII: S 0002-9939(1985)0792281-1
Article copyright: © Copyright 1985 American Mathematical Society