Duality of mixed norm spaces induced by radial one-sided doubling weight

Abstract

For $0<p,q<\infty$ and $\omega$ a radial weight, the space $L_{\omega }^{p,q}$ consists of complex-valued measurable functions $f$ on the unit disk such that $$\Vert f{\Vert }_{L_{\omega }^{p,q}}^q={\int }_0^1{\left(\frac{1}{2\pi }{\int }_0^{2\pi }|f(r{e}^{i\theta }){|}^{p}d\theta \right)}^{\frac{q}{p}}r\omega (r)dr,$$ and the mixed norm space $A_{\omega }^{p,q}$ is the subset of $L_{\omega }^{p,q}$ consisting of analytic functions. We say that a radial weight $\omega$ belongs to $\hat{\mathcal{D}}$ if there exists $C=C(\omega )>0$ such that $${\int }_r^1\omega (s)ds\le C{\int }_{\frac{1+r}{2}}^1\omega (s)ds\text{for every}0\le r<1.$$ We describe the dual space of $A_{\omega }^{p,q}$ for every $0<p,q<\infty$ and $\omega \in \hat{\mathcal{D}}$. Later on, we apply the obtained description of the dual space of $A_{\omega }^{p,q}$ to prove that the Bergman projection induced by $\omega$, $P_{\omega }$, is bounded on $L_{\omega }^{p,q}$ for $1<p,q<\infty$ and $\omega \in \hat{\mathcal{D}}$. Besides, we also prove that $P_{\omega }$ and the corresponding maximal Bergman projection $P_{\omega }^{+}$ are not simultaneously bounded on $L_{\omega }^{p,q}$ for $1<p,q<\infty$ and $\omega \in \hat{\mathcal{D}}$.