Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ L\sp{p}$ behavior of certain second order partial differential operators

Authors: Carlos E. Kenig and Peter A. Tomas
Journal: Trans. Amer. Math. Soc. 262 (1980), 521-531
MSC: Primary 42B15; Secondary 35E20, 42A45
MathSciNet review: 586732
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Abstract: We give examples of bounded inverses of polynomials in $ {{\textbf{R}}^n}$, $ n\, > \,1$, which are not Fourier multipliers of $ {L^p}\,({{\textbf{R}}^n})$ for any $ p\, \ne \,2$. Our main tool is the Kakeya set construction of C. Fefferman. Using these results, we relate the invertibility on $ {L^p}$ of a linear second order constant coefficient differential operator D on $ {{\textbf{R}}^n}$ to the geometric structure of quadratic surfaces associated to its symbol d. This work was motivated by multiplier conjectures of N. Rivière and R. Strichartz.

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Article copyright: © Copyright 1980 American Mathematical Society