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A Geometric Condition that Implies the Existence of Certain Singular Integrals of Banach-Space-Valued Functions

  • Burgess Davis
  • Renming Song
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

Consider first the special case of the Hilbert transform
$$Hf\left( x \right) = \frac{1}{\Pi }\int_{ - \infty }^\infty {\frac{{f(t)}}{{x - t}}dt.}$$
If 1 < p < ∞ and fL p (R), then for almost all x, the above integral exists in the principal value sense, satisfies the M. Riesz inequality
$$||Hf|{|_p} \leq {c_p}||f|{|_{p^{\prime}}}\,1 < p < \infty, $$
(1)
and has many other remarkable properties.

Keywords

Banach Space Singular Integral Geometric Condition Convex Space Convex Banach Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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