Validity of Theorem 1.3 in the L∞ case

Determine whether the Sobolev-type inequality for the ∂̄-operator stated in Theorem 1.3 holds when p = ∞; specifically, ascertain whether there exists a constant δ(Ω,n,∞) > 0 such that δ ||f||_{L∞(Ω)} ≤ ||∂̄f||_{L∞(Ω)} + ||f||_{L∞(∂Ω)} for every function f ∈ C^1(Ω) ∩ C^0(closure Ω) on a bounded domain Ω ⊂ C^n with Lipschitz boundary, where ∂̄f admits a continuous continuation to the closure of Ω.

Background

Theorem 1.3 establishes a ∂̄-version of a Sobolev-type inequality for Lp norms on bounded complex domains with Lipschitz boundary, covering all finite p (1 ≤ p < ∞). It generalizes the Poincaré inequality to complex analysis settings and relates the Lp norm of a function to the Lp norms of its ∂̄ derivative and its boundary trace.

The authors explicitly note uncertainty about whether this inequality extends to the case p = ∞, i.e., whether a corresponding bound in terms of L∞ norms exists with a domain-dependent constant. Resolving this would clarify the endpoint behavior of their ∂̄ Sobolev-type framework and its parallels with classical real-variable inequalities.