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Two weight $ \Phi$-inequalities for the Hardy operator, Hardy-Littlewood maximal operator, and fractional integrals


Author: Qinsheng Lai
Journal: Proc. Amer. Math. Soc. 118 (1993), 129-142
MSC: Primary 42B25; Secondary 47B38, 47G10
DOI: https://doi.org/10.1090/S0002-9939-1993-1123665-4
MathSciNet review: 1123665
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Abstract: Suppose $ \Phi $ is an appropriate Young's function and $ w(x),v(x)$ are nonnegative locally integrable functions. Let $ T$ denote one of three linear operators of special importance that map suitable functions on $ {R^n}$ into functions on $ {R^n}$.

For the Hardy operator $ T$, we study the inequality

$\displaystyle \int_0^\infty {\Phi (\vert Tf(x)\vert)w(x)\,dx \leqslant C\int_0^\infty {\Phi (\vert f(x)\vert)v(x)\,dx} } $

and for the Hardy-Littlewood maximal operator or fractional integrals $ T$, we discuss the inequalities

$\displaystyle \int_{{R^n}} {\Phi (\vert T(fv)(x)\vert)w(x)\,dx \leqslant C\int_{{R^n}} {\Phi (\vert f(x)\vert)v(x)\,dx.} } $

In all cases we obtain the necessary and sufficient conditions.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1123665-4
Keywords: Young's function, Hardy operator, maximal operator, fractional integral
Article copyright: © Copyright 1993 American Mathematical Society

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