New uniform bounds for a Walsh model of the bilinear Hilbert transform Richard OberlinChristoph Thiele 42B20bilinear Hilbert transformuniform estimates 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 Calder\'on-Zygmund decomposition. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4445 10.1512/iumj.2011.60.4445 en Indiana Univ. Math. J. 60 (2011) 1693 - 1712 state-of-the-art mathematics http://iumj.org/access/