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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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$L^{p}$ behavior of certain second order partial differential operators
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by Carlos E. Kenig and Peter A. Tomas PDF
Trans. Amer. Math. Soc. 262 (1980), 521-531 Request permission

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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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 262 (1980), 521-531
  • MSC: Primary 42B15; Secondary 35E20, 42A45
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0586732-5
  • MathSciNet review: 586732