Sharp endpoint for the quadratic Carleson operators

Determine whether the sharp endpoint space for the full quadratic Carleson operator and its lacunary counterpart is the Orlicz space L log_2 L.

Background

The paper studies the near-L^1 behavior of the maximally quadratically modulated Hilbert transform, namely the full quadratic Carleson operator C_2, and its lacunary version C_{2,lac}, whose modulation parameters are restricted to powers of two.

The authors prove negative results showing that neither operator satisfies a corresponding modular estimate for Young functions growing more slowly than t log_2 t, and positive results establishing an L log_1 L estimate for C_2 and an L (log_2 L)^2 log_4 L estimate for C_{2,lac}. These bounds leave a gap between the known lower and upper endpoint behavior. The unresolved issue is whether the natural candidate L log_2 L gives the sharp endpoint space.