Boundedness of the Bergman projection on $L^p$ spaces with exponential weights

Abstract

Let $v(r)=\exp \left(-\frac{\alpha }{1-r}\right)$ with $\alpha >0$, and let $\mathbb{D}$ be the unit disc in the complex plane. Denote by $A_v^p$ the subspace of analytic functions of $L^p(\mathbb{D},v)$ and let $P_v$ be the orthogonal projection from $L^2(\mathbb{D},v)$ onto $A_v^2$. In 2004, Dostanic revealed the intriguing fact that $P_v$ is bounded from $L^p(\mathbb{D},v)$ to $A_v^p$ only for $p=2$, and he posed the related problem of identifying the duals of $A_v^p$ for $p\ge 1$, $p\ne 2$. In this paper we propose a solution to this problem by proving that $P_v$ is bounded from $\,L^p(\D,v^{p/2})$ to $A_{v^{p/2}}^p$ whenever $1\le p<\infty$, and, consequently, the dual of $A_{v^{p/2}}^p$ for $p\ge 1$ can be identified with $A_{v^{q/2}}^q$, where $1/p+1/q=1$. In addition, we also address a similar question on some classes of weighted Fock spaces.

References in corpus (1)

• Pointwise estimates for the Bergman kernel of the weighted Fock space

Cited by in corpus (1)

• $L^p$ Regularity of Weighted Szegö Projections on the Unit Disc