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).
