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On the Existence of Certain Singular Integrals

  • A. P. Calderon
  • A. Zygmund
Part of the Mathematics and Its Applications book series (MAEE, volume 41)

Abstract

Let f (x) and K (x) be two functions integrable over the interval (-∞,+∞). It is very well known that their composition
$$ \int\limits_{{ - \infty }}^{{ + \infty }} {f(t)K\left( {x - t} \right)dt} $$
exists, as an absolutely convergent integral, for almost every x. The integral can, however, exist almost everywhere even if K is not absolutely integrable. The mostinteresting special case is that of K (x) = 1/x. Let us set
$$ \tilde{f}(x) = \frac{1}{\pi }\int\limits_{{ - \infty }}^{{ + \infty }} {\frac{{f(t)}}{{x - t}}dt} $$
.

Keywords

Singular Integral Finite Measure Logarithmic Potential Preceding Chapter Finite Sphere 
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

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • A. P. Calderon
  • A. Zygmund

There are no affiliations available

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