Sharp estimates for singular values of Hankel operators

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