Projections for harmonic Bergman spaces and applications

On the setting of general bounded smooth domains in Rn, we construct L1-bounded nonorthogonal projections and obtain related reproducing formulas for the harmonic Bergman spaces. In addition, we show that those projections satisfy Sobolev Lp-estimates of any order even for p=1. Among applications are Gleasonʹs problems for the harmonic Bergman–Sobolev and (little) Bloch functions on star-shaped domains with strong reference points.