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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ L\sp p$-boundedness of pseudo-differential operators of class $ S\sb {0,0}$


Authors: I. L. Hwang and R. B. Lee
Journal: Trans. Amer. Math. Soc. 346 (1994), 489-510
MSC: Primary 35S05; Secondary 47G30
MathSciNet review: 1264147
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Abstract: We study the $ {L^p}$-boundedness of pseudo-differential operators with the support of their symbols being contained in $ E \times {{\mathbf{R}}^n}$, where $ E$ is a compact subset of $ {{\mathbf{R}}^n}$, and their symbols have derivatives with respect to $ x$ only up to order $ k$, in the Hölder continuous sense, where $ k > n/2$ (the case $ 1 < p \leqslant 2$) and $ k > n/p$ (the case $ 2 < p < \infty $). We also give a new proof of the $ {L^p}$-boundedness, $ 1 < p < \infty $, of pseudo-differential operators of class $ S_{0,0}^m$, where $ m = m(p) = - n\vert 1/p - 1/2\vert$, and $ a \in S_{0,0}^m$ satisfies $ \vert\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )\vert \leqslant {C_{\alpha ,\beta }}{\langle \xi \rangle ^m}$ for $ (x,\xi ) \in {{\mathbf{R}}^n} \times {{\mathbf{R}}^n},\vert\alpha \vert \leqslant k$ and $ \vert\beta \vert \leqslant k'$, in the Hölder continuous sense, where $ k > n/2,k' > n/p$ (the case $ 1 < p \leqslant 2$) and $ k > n/p,k' > n/2$ (the case $ 2 < p < \infty $).


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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1264147-4
PII: S 0002-9947(1994)1264147-4
Article copyright: © Copyright 1994 American Mathematical Society