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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
DOI: https://doi.org/10.1090/S0002-9947-1982-0637030-4
MathSciNet review: 637030
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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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Keywords: <IMG WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${A_p}$"> weight, maximal function, pseudodifferential operator, Sobolev space
Article copyright: © Copyright 1982 American Mathematical Society