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Sharp estimates for singular values of Hankel operators

We consider compact Hankel operators realized in $\ell^2(\mathbb Z_+)$ as infinite matrices $Γ$ with matrix elements $h(j+k)$. Roughly speaking, we show that, for all $α>0$, the singular values $s_{n}$ of $Γ$ satisfy the bound $s_{n}= O(n^{-α})$ as $n\to \infty$ provided $h(j)= O(j^{-1}(\log j)^{-α})$ as $j\to \infty$. These estimates on $s_{n}$ are sharp in the power scale of $α$. Similar results are obtained for Hankel operators $\mathbfΓ$ realized in $L^2(\mathbb R_+)$ as integral operators with kernels $\mathbf h(t+s)$. In this case the estimates of singular values of $\mathbfΓ$ are determined by the behavior of $\mathbf h(t)$ as $t\to 0$ and as $t\to\infty$.