Weighted inequalities for a class of Volterra convolution operators. (English) Zbl 0703.42011

Stepanov, Vladimir D.

J. Lond. Math. Soc., II. Ser. 45, No. 2, 232-242 (1992).

Necessary and sufficient conditions for the boundedness from \(L_ v^ p(R^+)\) to \(L_ u^ q(R^+)\) of Volterra convolution operators of the form \(Kf(x)\equiv \int^{x}_{0}k(x-y)f(y)dy,\) where k(x) is a nonnegative nondecreasing kernel satisfying \(k(x+y)\leq D(k(x)+k(y))\) for all \(x,y\in R^+\) are obtained. The cases \(1<p,q<\infty\) and \(0<q<1<p<\infty\) are considered. Also the criteria for the compactness of K for \(1<p,q<\infty\) are given.

Reviewer: V.D.Stepanov

MSC:

42A85

Convolution, factorization for one variable harmonic analysis

Keywords:

weighted inequality; Lebesgue space; Volterra convolution operators