W1 vanishing on bounded Lipschitz pseudoconvex domains in CP^n

Determine whether the Sobolev cohomology group H^{0,1}_{W^1}(Ω) vanishes for every bounded pseudoconvex domain Ω ⊂ CP^n with Lipschitz boundary.

Background

Kohn’s global regularity theorem ensures W^s estimates for s > 0 on smooth bounded pseudoconvex domains, and W^1 estimates are known for C^2 boundaries. However, extending such estimates to Lipschitz boundaries would generalize these results significantly.

The authors explicitly pose the vanishing of H^{0,1}_{W^1}(Ω) as a problem and state that, although partial results exist, the general Lipschitz case remains unsolved.