  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.
