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On the $ L\sp{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$.


References [Enhancements On Off] (What's this?)

  • [1] A. P. Calderón and Rémi Vaillancourt, On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23 (1971), 374-378. MR 44 #2096. MR 0284872 (44:2096)
  • [2] H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudo-differential operators, J. Functional Analysis 18 (1975), 115-131. MR 0377599 (51:13770)
  • [3] A. G. Childs, On $ {L^2}$-boundedness of pseudo-differential operators, Ph.D. thesis, Univ. of Calif. at Berkeley, 1975. MR 0442755 (56:1135)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0442755-2
Keywords: $ {L^2}$-boundedness, pseudo-differential operator, symbol, Fourier transform, modified Hankel function
Article copyright: © Copyright 1976 American Mathematical Society

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