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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ L\sp 2$ boundedness of highly oscillatory integrals on product domains


Author: Elena Prestini
Journal: Proc. Amer. Math. Soc. 104 (1988), 493-497
MSC: Primary 47G05; Secondary 42B20, 42B25
MathSciNet review: 962818
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Abstract: We prove $ {L^2}$ boundedness of the oscillatory singular integral

$\displaystyle Tf(x,y) = \iint\limits_{{D_y}} {\frac{{\operatorname{exp} (2\pi iN(y)x')}}{{x'y'}}}f(x - x',y - y')dx'dy'$

where $ N(y)$ is an arbitrary integer-valued $ {L^\infty }$ function and where nothing is assumed on the dependency upon $ y$ of the domain of integration $ {D_y}$. We also prove $ {L^2}$ boundedness of the corresponding maximal opertaor. Operators of this kind appear in a problem of a.e. convergence of double Fourier series.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0962818-0
PII: S 0002-9939(1988)0962818-0
Article copyright: © Copyright 1988 American Mathematical Society