Pavlović’s sharp Hardy-space problem

Determine the sharp exponent p_0(K) such that every K-quasiconformal harmonic mapping of the unit disk belongs to the harmonic Hardy space h^p for all p<p_0(K).

Background

For arbitrary K-quasiconformal mappings of the disk, membership in the Hardy space is known for p<1/(2K), and that range is sharp. Because K-quasiconformal harmonic mappings possess the additional structure of harmonicity and univalence, the paper discusses the possibility of improving this exponent.

Pavlović posed the problem of identifying the exact threshold p_0(K). The paper’s Theorem 1 gives significant progress by proving membership for p<1/α_K, but the exact sharp value is not established.