New solution of a problem of Kolmogorov on width asymptotics in holomorphic function spaces

Abstract

Given a domain $D$ in $\\mathbb{C}^n$ and a compact subset $K$ of $D$, the set $\\mathcal{A}K^D$ of all restrictions of functions holomorphic on $D$ the modulus of which is bounded by $1$ is a compact subset of the Banach space $C(K)$ of continuous functions on $K$. The sequence $(d_m(\\mathcal{A}\_K^D)){m\\in \\mathbb{N}}$ of Kolmogorov $m$-widths of $\\mathcal{A}\_K^D$ provides a measure of the degree of compactness of the set $\\mathcal{A}\_K^D$ in $C(K)$ and the study of its asymptotics has a long history, essentially going back to Kolmogorov’s work on $\\epsilon$-entropy of compact sets in the 1950s. In the 1980s Zakharyuta showed that for suitable $D$ and $K$ the asymptotics $$ \\lim\_{m\\to \\infty}\\frac{- \\log d_m(\\mathcal{A}\_K^D)}{m^{1/n}} = 2\\pi \\bigg( \\frac{n!}{C(K,D)} \\bigg) ^{1/n}, $$ where $C(K,D)$ is the Bedford–Taylor relative capacity of $K$ in $D$, is implied by a conjecture, now known as Zakharyuta's Conjecture, concerning the approximability of the regularised relative extremal function of $K$ and $D$ by certain pluricomplex Green functions. Zakharyuta’s Conjecture was proved by Nivoche in 2004 thus settling (1) at the same time. We shall give a new proof of the asymptotics (1) for $D$ strictly hyperconvex and $K$ non-pluripolar which does not rely on Zakharyuta’s Conjecture. Instead we proceed more directly by a two-pronged approach establishing sharp upper and lower bounds for the Kolmogorov widths. The lower bounds follow from concentration results of independent interest for the eigenvalues of a certain family of Toeplitz operators, while the upper bounds follow from an application of the Bergman–Weil formula together with an exhaustion procedure by special holomorphic polyhedra.