Integral equations of the first kind of Sonine type

Source abstract

A Volterra integral equation of the first kind with a locally integrable kernel is called Sonine equation if there exists another locally integrable kernel ℓ ( x ) such that (locally integrable divisors of the unit, with respect to the operation of convolution). The formal inversion is well known, but it does not work, for example, on solutions in the spaces X = L p ( ℝ 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 L p ( ℝ 1 ), in Marchaud form: with the interpretation of the convergence of this hypersingular integral in L p ‐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 L p ( ℝ 1 ).