Integral equations of the first kind of Sonine type

Abstract

A Volterra integral equation of the first kind Kφ (x):=∫ -∞x k (x-t)φ (t)dt=f (x) with a locally integrable kernel k (x)∈L1 loc (□+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)∫0x ℓ (x-t)f (t)dt is well known, but it does not work, for example, on solutions in the spaces X=Lp (□1) 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 (□1), in Marchaud form: K -1f (x)=ℓ (∞)f (x) + ∫ (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 (□1). © 2003 Hindawi Publishing Corporation. All rights reserved.