A theorem in complex and harmonic analysis: on the weighted harmonic Bergman space a^p over the unit disc (0<p<infinity), the operator sending a harmonic function to its harmonic conjugate is bounded whenever the weight is a Bekolle-Bonami weight for some q and additionally satisfies a simple p-dependent condition. This extends the classical Hardy-Littlewood result (harmonic conjugation is bounded on unweighted Bergman spaces for all 0<p<infinity, in contrast to Hardy spaces where it fails for 0<p<=1) to the weighted setting, using among other tools a good-lambda inequality.
Proves a boundedness theorem for the harmonic-conjugation operator on weighted harmonic Bergman spaces over the unit disc. For the harmonic Hardy space h^p, a function's harmonic conjugate stays in t…