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Transactions of the American Mathematical Society

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



Weighted Sobolev spaces and pseudodifferential operators with smooth symbols

Author: Nicholas Miller
Journal: Trans. Amer. Math. Soc. 269 (1982), 91-109
MSC: Primary 47G05; Secondary 35S05, 46E35
MathSciNet review: 637030
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Abstract: Let $ {u^\char93 }$ 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)^\char93 }(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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Keywords: $ {A_p}$ weight, maximal function, pseudodifferential operator, Sobolev space
Article copyright: © Copyright 1982 American Mathematical Society