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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
##### Keywords:
Muckenhoupt condition; weight function
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##### References:
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