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A characterization of two weight norm inequalities for fractional and Poisson integrals


Author: Eric T. Sawyer
Journal: Trans. Amer. Math. Soc. 308 (1988), 533-545
MSC: Primary 26A33; Secondary 26D10, 42B25, 47G05
DOI: https://doi.org/10.1090/S0002-9947-1988-0930072-6
MathSciNet review: 930072
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Abstract: For $ 1 < p \leqslant q < \infty $ and $ w(x)$, $ v(x)$ nonnegative functions on $ {{\mathbf{R}}^n}$, we show that the weighted inequality

$\displaystyle {\left( {\int {\vert Tf{\vert^q}w} } \right)^{1/q}} \leqslant C{\left( {\int {{f^p}v} } \right)^{1/p}}$

holds for all $ f \geqslant 0$ if and only if both

$\displaystyle \int {{{[T({\chi _Q}{v^{1 - p'}})]}^q}w \leqslant {C_1}{{\left( {\int_Q {{v^{1 - p'}}} } \right)}^{q/p}} < \infty } $

and

$\displaystyle {\int {{{[T({\chi _Q}w)]}^{p'}}{v^{1 - p'}} \leqslant {C_2}\left( {\int_Q w } \right)} ^{p'/q'}} < \infty $

hold for all dyadic cubes $ Q$. Here $ T$ denotes a fractional integral or, more generally, a convolution operator whose kernel $ K$ is a positive lower semicontinuous radial function decreasing in $ \vert x\vert$ and satisfying $ K(x) \leqslant CK(2x)$, $ x \in {{\mathbf{R}}^n}$. Applications to degenerate elliptic differential operators are indicated.

In addition, a corresponding characterization of those weights $ v$ on $ {{\mathbf{R}}^n}$ and $ w$ on $ {\mathbf{R}}_ + ^{n + 1}$ for which the Poisson operator is bounded from $ {L^p}(v)$ to $ {L^q}(w)$ is given.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0930072-6
Article copyright: © Copyright 1988 American Mathematical Society

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