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Transactions of the American Mathematical Society

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

 

 

Weighted estimates for fractional powers of partial differential operators


Author: Raymond Johnson
Journal: Trans. Amer. Math. Soc. 265 (1981), 511-525
MSC: Primary 46E35; Secondary 42B99, 46F12
DOI: https://doi.org/10.1090/S0002-9947-1981-0610962-8
MathSciNet review: 610962
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Abstract: It is shown that fractional powers defined by the wave polynomial $ P(\xi ) = \xi _{^1}^2 + \cdots + \xi _p^2 - \xi _{p + 1}^2 - \cdots - \xi _n^2$, defined in terms of Fourier transforms by $ \widehat{{T^\lambda }f} = {\left\vert {P(\xi )} \right\vert^\lambda }\hat f$, are in the Bernstein subalgebra of functions with integrable Fourier transforms for $ \lambda > (n - 1)/2$, provided $ f \in C_c^m$ with $ m$ large enough. The proof uses embedding theorems for Besov spaces and Stein's theorem on interpolation of analytic families of operators.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0610962-8
Article copyright: © Copyright 1981 American Mathematical Society