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Bilinear operators with non-smooth symbol
| Venue: | I, J. Fourier Anal. Appl |
| Citations: | 29 - 3 self |
BibTeX
@ARTICLE{Gilbert_bilinearoperators,
author = {John E. Gilbert and Andrea R. Nahmod},
title = {Bilinear operators with non-smooth symbol},
journal = {I, J. Fourier Anal. Appl},
year = {},
pages = {435--467}
}
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Abstract
� � � This paper proves the L p-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of Coifman-Meyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth. Using a Whitney decomposition in the Fourier plane a general bilinear operator is represented as infinite discrete sums of time-frequency paraproducts obtained by associating wave-packets with tiles in phase-plane. Boundedness for the general bilinear operator then follows once the corresponding L p-boundedness of time-frequency paraproducts has been established. The latter result is the main theorem proved in Part II, our subsequent paper [11], using phase-plane analysis. 1.
Keyphrases
non-smooth symbol general bilinear operator time-frequency paraproducts main theorem infinite discrete sum phase-plane analysis whitney decomposition fourier plane latter result smooth multiplier bilinear hilbert transform general result corresponding p-boundedness part ii subsequent paper
