TY - JOUR T1 - Direct and Reverse Carleson Conditions on Generalized Weighted Bergman-Orlicz Spaces JO - Analysis in Theory and Applications VL - 3 SP - 287 EP - 300 PY - 2017 DA - 2017/08 SN - 33 DO - http://doi.org/10.4208/ata.2017.v33.n3.8 UR - https://global-sci.org/intro/article_detail/ata/10518.html KW - Orlicz function, global $\Delta_2$-condition, reverse Carleson condition, Direct Carleson condition, closed range, Pseudo-hyperbolic disks, Orlicz spaces, weighted Bergman spaces, generalized weighted Bergman-Orlicz spaces. AB -

Let $\mathbb{D}$ be the open unit disk in the complex plane $\mathbb{C}$. For $\alpha>-1$, let $dA_{\alpha}(z)= (1+\alpha)\left(1-|z|^2\right)^{\alpha}dA(z)$ be the weighted Lebesgue measure on $\mathbb{D}$. For a positive function $\omega\in L^1(\mathbb{D}, dA_{\alpha})$, the generalized weighted Bergman-Orlicz space $\mathcal{A}_{\omega}^{\psi}(\mathbb{D}, dA_{\alpha})$ is the space of all analytic functions such that $$\|f\|_{\omega}^{\psi}=\int_{\mathbb{D}} \psi(|f(z)|) \omega(z) dA_{\alpha}(z) <\infty,$$ where $\psi$ is a strictly convex Orlicz function that satisfies other technical hypotheses. Let $G$ be a measurable subset of $\mathbb{D}$, we say $G$ satisfies the reverse Carleson condition for $\mathcal{A}_{\omega}^{\psi}(\mathbb{D}, dA_{\alpha})$ if there exists a positive constant $C$ such that$$\int_{G} \psi(|f(z)|) \omega(z) dA_{\alpha}(z) \geq C \int_{\mathbb{D}} \psi(|f(z)|) \omega(z) dA_{\alpha}(z),$$for all $f\in \mathcal{A}_{\omega}^{\psi}(\mathbb{D}, dA_{\alpha})$. Let $\mu$ be a positive Borel measure, we say $\mu$ satisfies the direct Carleson condition if there exists a positive constant $M$ such that for all $f\in \mathcal{A}_{\omega}^{\psi}(\mathbb{D}, dA_{\alpha})$,$$\int_{\mathbb{D}} \psi(|f(z)|) d\mu(z) \leq M \int_{\mathbb{D}} \psi(|f(z)|) \omega(z) dA_{\alpha}(z).$$ In this paper, we study the direct and reverse Carleson condition on the generalized weighted Bergman-Orlicz space $\mathcal{A}_{\omega}^{\psi}(\mathbb{D}, dA_{\alpha})$. We present conditions on the set $G$ such that the reverse Carleson condition holds. Moreover, we  give a sufficient condition for the finite positive Borel measure $\mu$ to satisfy the direct carleson condition on the generalized weighted Bergman-Orlicz spaces.