Stefan G. Samko¹ and Rogério P. Cardoso²

Copyright © 2003 Stefan G. Samko and Rogério P. Cardoso. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A Volterra integral equation of the first kind Kφ(x):≡∫−∞xk(x−t)φ(t)dt=f(x) with a locally integrable kernel k(x)∈L1loc(ℝ+1) is called Sonine equation if there exists another locally integrable kernel ℓ(x) such that ∫0xk(x−t)ℓ(t)dt≡1 (locally integrable divisors of the unit, with respect to the operation of convolution). The formal inversion φ(x)=(d/dx)∫0xℓ(x−t)f(t)dt is well known, but it does not work, for example, on solutions in the spaces X=Lp(ℝ1) and is not defined on the whole range K(X). We develop many properties of Sonine kernels which allow us—in a very general case—to construct the real inverse operator, within the framework of the spaces Lp(ℝ1), in Marchaud form: K−1f(x)=ℓ(∞)f(x)+∫0∞ℓ′(t)[f(x−t)−f(x)]dt with the interpretation of the convergence of this “hypersingular” integral in Lp-norm. The description of the range K(X) is given; it already requires the language of Orlicz spaces even in the case when X is the Lebesgue space Lp(ℝ1).