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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
DOI: https://doi.org/10.1090/S0002-9939-1985-0792281-1
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].


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

  • [1] R. Coifman and I. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315-331. MR 0380244 (52:1144)
  • [2] A. El Kohen, On the hyperbolic Riesz means, Proc. Amer. Math. Soc. 89 (1983), 113-116. MR 706521 (85c:42017)
  • [3] -, $ L_{{\text{loc}}}^2$-boundedness of a class of singular Fourier integral operators, Proc. Amer. Math. Soc. 91 (1984), 389-394. MR 744636 (85j:47053)
  • [4] A. Erdelyi et al., Higher transcendental function. Vol. I, McGraw-Hill, New York, 1953.
  • [5] A. Nagel and S. Wainger, $ {L^2}$ boundedness of Hilbert transforms along surfaces and convolution operators homogeneous with respect to a multiple parameter group, Amer. J. Math. 99 (1977), 761-785. MR 0450901 (56:9192)
  • [6] E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), 1239-1295. MR 508453 (80k:42023)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0792281-1
Article copyright: © Copyright 1985 American Mathematical Society

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