Harmonic conjugates on Bergman spaces induced by doubling weights

Abstract

A radial weight $$\omega $$ belongs to the class $$\widehat{\mathcal {D}}$$ if there exists $$C=C(\omega )\ge 1$$ such that $$\int _r^1 \omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds$$ for all $$0\le r<1$$ . Write $$\omega \in \check{\mathcal {D}}$$ if there exist constants $$K=K(\omega )>1$$ and $$C=C(\omega )>1$$ such that $${\widehat{\omega }}(r)\ge C{\widehat{\omega }}\left( 1-\frac{1-r}{K}\right) $$ for all $$0\le r<1$$ . These classes of radial weights arise naturally in the operator theory of Bergman spaces induced by radial weights (Pelaez and Rattya in Bergman projection induced by radial weight, 2019. arXiv:1902.09837 ). Classical results by Hardy and Littlewood (J Reine Angew Math 167:405–423, 1932) and Shields and Williams (Mich Math J 29(1):3–25, 1982) show that the weighted Bergman space of harmonic functions is not closed by harmonic conjugation if $${\omega \in \widehat{\mathcal {D}}\setminus \check{\mathcal {D}}}$$ and $$0<p\le 1$$ . In this paper we establish sharp estimates for the norm of the analytic Bergman space $$A^p_\omega $$ , with $${\omega \in \widehat{\mathcal {D}}\setminus \check{\mathcal {D}}}$$ and $$0<p<\infty $$ , in terms of quantities depending on the real part of the function. It is also shown that these quantities result equivalent norms for certain classes of radial weights.