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

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
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