# Harmonic conjugates on Bergman spaces induced by doubling weights

## TL;DR

This paper investigates the properties of harmonic conjugates in Bergman spaces induced by specific classes of radial weights, providing sharp norm estimates and exploring their implications for the structure of these function spaces.

## Contribution

The paper establishes sharp norm estimates for Bergman spaces with weights in \\widehat{\\mathcal{D}}\\setminus \\check{\\mathcal{D}}, revealing new insights into harmonic conjugation and norm equivalences.

## Key findings

- Weighted Bergman spaces are not closed under harmonic conjugation for certain weights.
- Sharp estimates for norms in these spaces are derived.
- Quantities depending on the real part of functions are shown to be equivalent norms.

## 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 $0\le r<1$. In a recent paper, we have recently prove that these classes of radial weights arise naturally in the operator theory of Bergman spaces induced by radial weights. Classical results by Hardy and Littlewood, and Shields and Williams, 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.