Eigenvalue asymptotics for a class of multi-variable Hankel matrices

A one-variable Hankel matrix $H_a$ is an infinite matrix $H_a=[a(i+j){]}_{i,j\ge 0}$. Similarly, for any $d\ge 2$, a $d$-variable Hankel matrix is defined as $H_{\mathbf{a}}=[\mathbf{a}(\mathbf{i}+\mathbf{j})]$, where $\mathbf{i}=(i_1,\dots ,i_d)$ and $\mathbf{j}=(j_1,\dots ,j_d)$, with $i_1,\dots ,i_d,j_1,\dots ,j_d\ge 0$. For $\gamma >0$, A. Pushnitski and D. Yafaev proved that the eigenvalues of the compact one-variable Hankel matrices $H_a$ with $a(j)=j^{-1}(\log j{)}^{-\gamma }$, for $j\ge 2$, obey the asymptotics ${\lambda }_n(H_a)\sim C_{\gamma }{n}^{-\gamma }$, as $n\to +\mathrm{\infty }$, where the constant $C_{\gamma }$ is calculated explicitly. This paper presents the following $d$-variable analogue. Let $\gamma >0$ and $a(j)=j^{-d}(\log j{)}^{-\gamma }$, for $j\ge 2$. If $\mathbf{a}(j_1,\dots ,j_d)=a(j_1+\cdots +j_d)$, then $H_{\mathbf{a}}$ is compact and its eigenvalues follow the asymptotics ${\lambda }_n(H_{\mathbf{a}})\sim C_{d,\gamma }{n}^{-\gamma }$, as $n\to +\mathrm{\infty }$, where the constant $C_{d,\gamma }$ is calculated explicitly.