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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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Weighted nonlinear potential theory
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by David R. Adams PDF
Trans. Amer. Math. Soc. 297 (1986), 73-94 Request permission


The potential theoretic idea of the "thinness of a set at a given point" is extended to the weighted nonlinear potential theoretic setting—the weights representing in general singularities/degeneracies—and conditions on these weights are given that guarantee when two such notions are equivalent at the given point. When applied to questions of boundary regularity for solutions to (degenerate) elliptic second-order partial differential equations in bounded domains, this result relates the boundary Wiener criterion for one operator to that of another, and in the linear case gives conditions for boundary regular points to be the same for various operators. The methods also yield two weight norm inequalities for Riesz potentials \[ {\left ( {\int {{{({I_\alpha }{\ast }f)}^q}v dx} } \right )^{1/q}} \leqslant {\left ( {\int {{f^p}w dx} } \right )^{1/p}},\] $1 < p \leqslant q < \infty$, which at least in the first-order case $(\alpha = 1)$ have found some use in a number of places in analysis.
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 297 (1986), 73-94
  • MSC: Primary 31B25; Secondary 26D10, 31C15, 46E35
  • DOI:
  • MathSciNet review: 849468