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On weighted norm inequalities for positive linear operators. (English) Zbl 0673.47030
Let (X,m,\(\mu)\) be a \(\sigma\)-finite measure space and let K(x,y) be a nonnegative and measurable function on \(X\times X\). Set \[ Tf(x)=\int_{X}K(x,y)f(y)d\mu (y) \] for nonnegative functions f. Given \(1<p<\infty\) and a nonnegative weight function w on X, the authors show that there exists a nonnegative weight function v, finite \(\mu\)-almost everywhere on X, such that \[ \int_{X}(Tf)^ Pwd \mu <\int_{X}f^ Pvd \mu,\quad for\quad all\quad f\geq 0, \] if and only if there exists a \(\mu\)-almost everywhere positive function \(\phi\) on X with \(\int_{X}(T\phi)^ Pwd \mu <\infty\). Some applications of this result are given.
Reviewer: Ju.V.Egorov

MSC:
47B38 Linear operators on function spaces (general)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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