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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


$ L\sp{2}$-boundedness for pseudo-differential operators with unbounded symbols

Author: Gary Childs
Journal: Proc. Amer. Math. Soc. 72 (1978), 77-81
MSC: Primary 47G05; Secondary 35S05
MathSciNet review: 0500300
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Abstract: Kato has proven $ {L^2}$-boundedness if the symbol $ a(x,z)$ is such that $ \vert D_x^\beta D_z^\alpha a(x,z)\vert\; \leqslant ({\text{constant}}){(1 + \vert z\vert)^{(\vert\beta \vert - \vert\alpha \vert)\rho }}$ for $ \vert\alpha \vert \leqslant [n/2] + 1,\vert\beta \vert \leqslant [n/2] + 2$ and $ 0 < \rho < 1$. In this paper, $ {L^2}$-boundedness is shown for a corresponding Hölder continuity condition which requires slightly less smoothness for $ a(x,z)$.

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PII: S 0002-9939(1978)0500300-9
Keywords: $ {L^2}$-boundedness, pseudo-differential operator, symbol, Fourier transform, modified Hankel function
Article copyright: © Copyright 1978 American Mathematical Society