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On singular values of Hankel operators on Bergman spaces

In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces $A^2_{ω_φ}$, where $ω_φ= e^{-φ}$ and $φ$ is a subharmonic function. We consider compact Hankel operators $H_{\overline ϕ}$, with anti-analytic symbols ${\overline ϕ}$, and give estimates of the trace of $h(|H_{\overline ϕ}|)$ for any convex function $h$. This allows us to give asymptotic estimates of the singular values $(s_n(H_{\overline ϕ}))_n$ in terms of decreasing rearrangement of $|ϕ'|/\sqrt{Δφ}$. For the radial weights, we first prove that the critical decay of $(s_n(H_{\overline ϕ}))_n$ is achieved by $(s_n (H_{\overline{z}}))_n$. Namely, we establish that if $s_n(H_{\overline ϕ})= o (s_n(H_{\overline {z}}))$, then $H_{\overline ϕ} = 0$. Then, we show that if $Δφ(z) \asymp \frac{1}{(1-|z|^2)^{2+β}}$ with $β\geq 0$, then $s_n(H_{\overline ϕ}) = O(s_n(H_{\overline {z}}))$ if and only if $ϕ'$ belongs to the Hardy space $H^p$, where $p= \frac{2(1+β)}{2+β}$. Finally, we compute the asymptotics of $s_n(H_{\overline ϕ})$ whenever $ ϕ' \in H^{p }$.