New uniform bounds for a Walsh model of the bilinear Hilbert transform

2010/04/22 by Oberlin, Richard, Thiele, Christoph
#42B20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1004.4019

Abstract

We prove old and new L^p bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, uniformly in the parameter that models degeneration of the bilinear Hilbert transform. We obtain the full range of exponents that can be expected from known bounds in the degenerate and non-degenerate cases. For the new estimates with exponents p close to 1 the argument relies on a multi-frequency Calderon-Zygmund decomposition.

Related

• Endpoint bounds for the quartile operator
• L^p estimates for the biest I. The Walsh case
• L^p estimates for the bilinear Hilbert transform for 1/2
• L^p estimates for the biest II. The Fourier case
• Endpoint bounds for the bilinear Hilbert transform