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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the $L^{2}$-boundedness of pseudo-differential operators


Author: A. G. Childs
Journal: Proc. Amer. Math. Soc. 61 (1976), 252-254
MSC: Primary 47G05; Secondary 35S05
DOI: https://doi.org/10.1090/S0002-9939-1976-0442755-2
MathSciNet review: 0442755
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Abstract: A. P. Calderón and R. Vaillancourt have established ${L^2}$-boundedness in case the symbol has bounded partial derivatives to order $3$ in any variable. H. O. Cordes has proved ${L^2}$-boundedness if this order is $1$. It is shown here that it suffices for the symbol to satisfy a uniform Hölder continuity condition of order $\tfrac {1} {2} + \delta ,\;\delta > 0$.


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Keywords: <IMG WIDTH="28" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${L^2}$">-boundedness, pseudo-differential operator, symbol, Fourier transform, modified Hankel function
Article copyright: © Copyright 1976 American Mathematical Society