The conjugate operator on variable harmonic Bergman spaces

Abstract

We prove that the harmonic conjugation operator on variable exponent harmonic Bergman spaces in the unit disc is bounded when the exponent has positive minimum and finite maximum and satisfies the log-Hölder condition. Under these conditions on the exponent, we also give a characterization of variable exponent Bergman spaces in terms of the derivatives of functions in them. Several key lemmas involve the boundedness of various maximal operators in these variable spaces.