Integral equations of the first kind of Sonine type

A Volterra integral equation of the first kind K φ ( x ) : ≡ ∫ − ∞ x k ( x − t ) φ ( t ) d t = f ( x ) with a locally integrable kernel k ( x ) ∈ L 1 loc ( ℝ + 1 ) is called Sonine equation if there exists another locally integrable kernel ℓ ( x ) such that ∫ 0 x k ( x − t ) ℓ ( t ) d t ≡ 1 (locally integrable divisors of the unit, with respect to the operation of convolution). The formal inversion φ ( x ) = ( d / d x ) ∫ 0 x ℓ ( x − t ) f ( t ) d t 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: K − 1 f ( x ) = ℓ ( ∞ ) f ( x ) + ∫ 0 ∞ ℓ ′ ( t ) [ f ( x − t ) − f ( x ) ] d t 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 ) .