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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 Sobolev spaces and pseudodifferential operators with smooth symbols
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by Nicholas Miller PDF
Trans. Amer. Math. Soc. 269 (1982), 91-109 Request permission

Abstract:

Let ${u^\# }$ be the Fefferman-Stein sharp function of $u$, and for $1 < r < \infty$, let ${M_r}u$ be an appropriate version of the Hardy-Littlewood maximal function of $u$. If $A$ is a (not necessarily homogeneous) pseudodifferential operator of order $0$, then there is a constant $c > 0$ such that the pointwise estimate ${(Au)^\# }(x) \leqslant c{M_r}u(x)$ holds for all $x \in {R^n}$ and all Schwartz functions $u$. This estimate implies the boundedness of $0$-order pseudodifferential operators on weighted ${L^p}$ spaces whenever the weight function belongs to Muckenhoupt’s class ${A_p}$. Having established this, we construct weighted Sobolev spaces of fractional order in ${R^n}$ and on a compact manifold, prove a version of Sobolev’s theorem, and exhibit coercive weighted estimates for elliptic pseudodifferential operators.
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 269 (1982), 91-109
  • MSC: Primary 47G05; Secondary 35S05, 46E35
  • DOI: https://doi.org/10.1090/S0002-9947-1982-0637030-4
  • MathSciNet review: 637030