Integral Equations and Operator Theory

, Volume 29, Issue 3, pp 261–268

On singular integrals along surfaces related to black spaces

  • Lung-Kee Chen
  • Dashan Fan
Article

Abstract

Leth(t) be an arbitrary bounded radial function and let Γ(x) be a real measurable and radial function defined onRn−1. Forx, yRn−1, we establish that the singular integral along surfacex → (x, Γ(x)):
$$Tf(x,x_n ) = p.\upsilon .\smallint h(y)\frac{{\Omega (y)}}{{|y|^{n - 1} }}f(x - y,x_n - \Gamma (y))dy,$$
and the associated maximal singular integral are bounded inLp(Rn) for 1<p<∞,n≥3, provided that the maximal operator
$${\rm M}_\Gamma g(x_n ) = \mathop {\sup }\limits_r \frac{1}{r}\smallint _{r/2< |t| \leqslant r} |f(x_n - \Gamma (t))|dt$$
is bounded onLp(R) for all 1<p.

1991 Mathematics Subject Classification

42B20 42B25 

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References

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Copyright information

© Birkhäuser Verlag 1997

Authors and Affiliations

  • Lung-Kee Chen
    • 1
  • Dashan Fan
    • 2
  1. 1.Department of MathematicsOregon State UniversityCorvallisUSA
  2. 2.Department of Mathematical SciencesUniversity of Wisconsin-MilwaukeeMilwaukeeUSA

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