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Register a new userDuality for large Bergman-Orlicz spaces and Boundedness of Hankel Operators
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For $mathbb B^n$ the unit ball of $mathbb C^n$, we consider Bergman-Orlicz spaces of holomorphic functions in $L^Phi_alpha$, which are generalizations of classical Bergman spaces. We characterize the dual space of large Bergman-Orlicz space, and bounded Hankel operators between some Bergman-Orlicz spaces $A_alpha^{Phi_1}(mathbb B^n)$ and $A_alpha^{Phi_2}(mathbb B^n)$ where $Phi_1$ and $Phi_2$ are either convex or concave growth functions.
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In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces $A^2_{omega _varphi}$, where $omega _varphi= e^{-varphi}$ and $varphi$ is a subharmonic function. We consider compact Hankel operators $H_{overline {phi}}$, with anti-analytic symbols ${overline {phi}}$, and give estimates of the trace of $h(|H_{overline phi}|)$ for any convex function $h$. This allows us to give asymptotic estimates of the singular values $(s_n(H_{overline {phi}}))_n$ in terms of decreasing rearrangement of $|phi |/sqrt{Delta varphi}$. For the radial weights, we first prove that the critical decay of $(s_n(H_{overline {phi}}))_n$ is achieved by $(s_n (H_{overline{z}}))_n$. Namely, we establish that if $s_n(H_{overline {phi}})= o (s_n(H_{overline {z}}))$, then $H_{overline {phi}} = 0$. Then, we show that if $Delta varphi (z) asymp frac{1}{(1-|z|^2)^{2+beta}}$ with $beta geq 0$, then $s_n(H_{overline {phi}}) = O(s_n(H_{overline {z}}))$ if and only if $phi $ belongs to the Hardy space $H^p$, where $p= frac{2(1+beta)}{2+beta}$. Finally, we compute the asymptotics of $s_n(H_{overline {phi}})$ whenever $ phi in H^{p }$.
In this paper we characterize off-diagonal Carleson embeddings for both Hardy-Orlicz spaces and Bergman-Orlicz spaces of the upper-half plane. We use these results to obtain embedding relations and pointwise multipliers between these spaces.
We completely characterize the simultaneous membership in the Schatten ideals $S_ p$, $0<p<infty$ of the Hankel operators $H_ f$ and $H_{bar{f}}$ on the Bergman space, in terms of the behaviour of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.
For $mathbb B^n$ the unit ball of $mathbb C^n$, we consider Bergman-Orlicz spaces of holomorphic functions in $L^Phi_alpha(mathbb B^n)$, which are generalizations of classical Bergman spaces. We obtain atomic decomposition for functions in the Bergman-Orlicz space $mathcal A^Phi_alpha (mathbb B^n)$ where $Phi$ is either convex or concave growth function. We then prove weak factorization theorems involving the Bloch space and a Bergman-Orlicz space and also weak factorization theorems involving two Bergman-Orlicz spaces.
We prove Carleson embeddings for Bergman spaces of tube domains over symmetric cones, we apply them to characterize symbols of bounded Ces`aro-type operators from weighted Bergman spaces to weighted Besov spaces. We also obtain Schatten class criteria of Toeplitz operators and Ces`aro-type operators on weighted Hilbert-Bergman spaces.
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