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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A characterization of two weight norm inequalities for fractional and Poisson integrals
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by Eric T. Sawyer PDF
Trans. Amer. Math. Soc. 308 (1988), 533-545 Request permission

Abstract:

For $1 < p \leqslant q < \infty$ and $w(x)$, $v(x)$ nonnegative functions on ${{\mathbf {R}}^n}$, we show that the weighted inequality \[ {\left ( {\int {|Tf{|^q}w} } \right )^{1/q}} \leqslant C{\left ( {\int {{f^p}v} } \right )^{1/p}}\] holds for all $f \geqslant 0$ if and only if both \[ \int {{{[T({\chi _Q}{v^{1 - p’}})]}^q}w \leqslant {C_1}{{\left ( {\int _Q {{v^{1 - p’}}} } \right )}^{q/p}} < \infty } \] and \[ {\int {{{[T({\chi _Q}w)]}^{p’}}{v^{1 - p’}} \leqslant {C_2}\left ( {\int _Q w } \right )} ^{p’/q’}} < \infty \] 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 $|x|$ 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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Additional Information
  • © Copyright 1988 American Mathematical Society
  • 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