On the existence of certain singular integrals

1952/01/01 by A. P. Calderon, A. P. Calderón, A. Zygmund · 26 citations
Mathematics · #Nonlinear Differential Equations Analysis #Functional Equations Stability Results #Differential Equations and Boundary Problems

paper · pdf · doi:10.1007/bf02392130

Abstract

Let f (x) and K (x) be two functions integrable over the interval (-∞,+∞). It is very well known that their composition ∫_ - ∞ ^ + ∞ f(t)K( x - t )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 f(x) = (1)/(π )∫_ - ∞ ^ + ∞ \fracf(t)x - tdt .

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