## Annals of Functional Analysis

### Embedding theorems and integration operators on Bergman spaces with exponential weights

Xiaofen Lv

#### Abstract

In this article, given some positive Borel measure $\mu$, we define two integration operators to be

$$I_{\mu}(f)(z)=\int_{\mathbf{D}}f(w)K(z,w)e^{-2\varphi(w)}\,d\mu(w)$$ and

$$J_{\mu}(f)(z)=\int_{\mathbf{D}}\vert f(w)K(z,w)\vert e^{-2\varphi(w)}\,d\mu(w).$$ We characterize the boundedness and compactness of these operators from the Bergman space $A^{p}_{\varphi}$ to $L^{q}_{\varphi}$ for $1\lt p,q\lt \infty$, where $\varphi$ belongs to a large class ${\mathcal{W}}_{0}$, which covers those defined by Borichev, Dhuez, and Kellay in 2007. We also completely describe those $\mu$’s such that the embedding operator is bounded or compact from $A^{p}_{\varphi}$ to $L^{q}_{\varphi}(d\mu)$, $0\lt p,q\lt \infty$.

#### Article information

Source
Ann. Funct. Anal., Volume 10, Number 1 (2019), 122-134.

Dates
Accepted: 23 May 2018
First available in Project Euclid: 16 January 2019

https://projecteuclid.org/euclid.afa/1547629228

Digital Object Identifier
doi:10.1215/20088752-2018-0013

Mathematical Reviews number (MathSciNet)
MR3899961

Zentralblatt MATH identifier
07045490

Subjects
Primary: 30H20: Bergman spaces, Fock spaces
Secondary: 47B34: Kernel operators

#### Citation

Lv, Xiaofen. Embedding theorems and integration operators on Bergman spaces with exponential weights. Ann. Funct. Anal. 10 (2019), no. 1, 122--134. doi:10.1215/20088752-2018-0013. https://projecteuclid.org/euclid.afa/1547629228

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