Douglas question for functions of the form z̄+h

The problem asks whether the relevant operator is invertible for symbols of the form $\bar z+h$, with $h$ in the disk algebra, under a positive lower-bound hypothesis. The result concerns this explicit class rather than every version of Douglas's question.

Known results

• McDonald and Sundberg proved the positive result for real harmonic symbols.
• Analytic symbols reduce to pointwise multipliers.
• Zhao and Zheng produced a counterexample to the general question using a second-order real polynomial in $|z|$.
• Earlier work established partial results for harmonic symbols under quantitative lower-bound assumptions.

August 2026 preprint

The preprint claims an affirmative answer for symbols $\bar z+h$ satisfying the stated positive lower-bound condition. This gives a claimed solution of the tracked case, but the result remains unverified; the source explicitly cautions that it does not cover every formulation of Douglas's question.

Current status (as of August 2026): The stated positive-lower-bound case for symbols $\bar z+h$ is claimed solved by the August 2026 preprint, but broader formulations remain open.