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 (Peláez and Rättyä 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.