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 $L^{p,q}_\omega$ consists of complex-valued measurable functions $f$ on the unit disk such that $$ \| f\|_{L^{p,q}_\omega}^q = \int_0^1 \left (\frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^pd\theta \right )^{\frac{q}{p}}r\omega(r)\,dr, $$ and the mixed norm space $A^{p,q}_\omega$ is the subset of $L^{p,q}_\omega$ consisting of analytic functions. We say that a radial weight $\omega$ belongs to $\widehat{\mathcal{D}}$ if there exists $C=C(\omega)>0$ such that $$\int_r^1\omega(s)ds \leq C \int_{\frac{1+r}{2}}^1\omega(s)\,ds \,\, \text{for every}\,\, 0\leq r <1.$$ We describe the dual space of $A^{p,q}_\omega$ for every $0<p,q<\infty$ and $\omega\in\widehat{\mathcal{D}}$. Later on, we apply the obtained description of the dual space of $A^{p,q}_\omega$ to prove that the Bergman projection induced by $\omega$, $P_\omega$, is bounded on $L^{p,q}_\omega$ for $1<p,q<\infty$ and $\omega\in \widehat{\mathcal{D}}$. Besides, we also prove that $P_\omega$ and the corresponding maximal Bergman projection $P_\omega^+$ are not simultaneously bounded on $L^{p,q}_\omega$ for $1<p,q<\infty$ and $\omega\in \widehat{\mathcal{D}}$.