On the Existence of Certain Singular Integrals

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


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
$$ \tilde{f}(x) = \frac{1}{\pi }\int\limits_{{ - \infty }}^{{ + \infty }} {\frac{{f(t)}}{{x - t}}dt} $$


Singular Integral Finite Measure Logarithmic Potential Preceding Chapter Finite Sphere 
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© Kluwer Academic Publishers 1989

Authors and Affiliations

  • A. P. Calderon
  • A. Zygmund

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