The asymptotic norm conjecture for Toeplitz matrices with multiple Fisher–Hartwig singularities

Let $R\geq 2$ and let

$$a(t)=b(t)\prod_{r=1}^R |t-t_r|^{-2\alpha_r}\varphi_{\beta_r,t_r}(t),$$

where $t_1,\ldots,t_R$ are distinct points on ${\bf T}$, $0<\operatorname{Re}\alpha_r<1/2$, $-1/2<\operatorname{Re}\beta_r\leq 1/2$, and $b\in L^\infty({\bf T})$ is continuous and nonzero at each $t_r$. Set

$$\operatorname{Re}\alpha=\max\{\operatorname{Re}\alpha_1,\ldots,\operatorname{Re}\alpha_R\},\qquad M=\{r:\operatorname{Re}\alpha_r=\operatorname{Re}\alpha\}.$$

For each $r$, let $K_r$ be the integral operator on $L^2(0,1)$ associated with $|t-t_r|^{-2\alpha_r}\varphi_{\beta_r,t_r}(t)$, with kernel $C_{\alpha_r,\beta_r}^+(x-y)^{2\alpha_r-1}$ for $x>y$ and $C_{\alpha_r,\beta_r}^-(y-x)^{2\alpha_r-1}$ for $x<y$. The multiple-singularity norm conjecture.

$$\|T_n(a)\|\sim \max_{r\in M}\|K_r\|\,|b(t_r)|\,n^{2\operatorname{Re}\alpha}.$$

When the maximum real part is attained at only one singularity, this asymptotic follows from the preceding theorem; the conjectural case is when several singularities attain the maximum. Establishing the formula requires controlling the interaction between equally dominant Fisher–Hartwig singularities.