INTEGRAL EQUATIONS OF THE FIRST KIND OF SONINE TYPE

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) ∈ L loc (R 1) is called Sonine equation if there exists another locally integrable kernel �(x) such thatx 0 k(x − t)�(t)dt ≡ 1( lo- cally integrable divisors of the unit, with respect to the operation of convolu- tion). The formal inversion ϕ(x) = (d/dx) � x 0 �(x − t)f (t)dt is well known, but it does not work, for example, on solutions in the spaces X = Lp(R 1 ) and is not defined on the whole range K(X). We develop many properties of Sonine ker- nels which allow us—in a very general case—to construct the real inverse oper- ator, within the framework of the spaces Lp(R 1 ), in Marchaud form: K −1 f( x)= �( ∞)f (x)+ � ∞ 0 � � (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