Norm estimates for a class of operators related to the Bergman projection

Abstract

Suppose $0<\alpha <\mathrm{\infty }$, ${\mathfrak{B}}_{\alpha }$ is an integral operator of $L^p(\text{ID},\mathrm{d}A)$ which is defined as follows: ${\mathfrak{B}}_{\alpha }\left[f\right]\left(z\right)={\int }_{\text{ID}}\frac{1}{(1-\overline{w}z{)}^{\alpha }}f\left(w\right)\mathrm{d}A\left(w\right)$, where $\mathbb{D}$ is the unit disk and $\mathrm{d}A\left(w\right)$ is the normalized area measure. For $0<\alpha <2$, we obtain the norm estimates of $\parallel {\mathfrak{B}}_{\alpha }{\parallel }_p$, where $1\le p\le \mathrm{\infty }$. Our results are sharp for $p=1,2,\mathrm{\infty }$. Moreover, we show that ${\mathfrak{B}}_{\alpha }$ is a compact operator and of Schatten p-class, where $p>\frac{1}{2-\alpha }$. For $2<\alpha <3$, we show that ${\mathfrak{B}}_{\alpha }\left(L^p\right(\text{ID}\left)\right)\subseteq L^q\left(\text{ID}\right)$, where $p>\frac{3}{3-\alpha }$ and q is the conjugate exponent of p. This result is sharp (see Remark 1.2) and the norm estimate of $\parallel {\mathfrak{B}}_{\alpha }{\parallel }_{L^p(\text{ID},\mathrm{d}A)\to L^q(\text{ID},\mathrm{d}A)}$ is also obtained.

Keywords:

• Bergman projection
• hypergeometric function
• compact operator
• adjoint operator
• Schatten p-class

AMS SUBJECT CLASSIFICATIONS:

• Primary 30H20
• 32A36
• secondary 47B38

Acknowledgments

The authors of this paper would like to thank Professor Kehe Zhu for his help on Theorems 1.2 and 1.3.

Disclosure statement

No potential conflict of interest was reported by the author(s).