The norm of the Hilbert matrix operator on Bergman spaces

TL;DR

The paper addresses the norm of the Hilbert matrix operator on Bergman spaces $A^p_\alpha$ and verifies Karapetrović's conjecture that the norm equals $\frac{\pi}{\sin((2+\alpha)\pi/p)}$ when $1<\alpha+2<p$. It develops two analytic tools based on $\alpha(p)$-curves and $p(\alpha)$-curves to identify parameter regimes where the conjectured value holds. The main result proves the conjecture for $0\le\alpha\le \frac{6p^3-29p^2+17p-2+2p\sqrt{6p^2-11p+4}}{(3p-1)^2}$, improving the known range beyond previous bounds, especially for $\alpha>1/47$ and $\alpha\neq 1$. This advances understanding of operator norms on analytic function spaces and provides explicit, sharp conditions under which the Hilbert matrix operator attains the conjectured norm.

Abstract

Karapetrović conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_α$ is equal to $π/\sin((2+α)π/p)$ when $-1<α<p-2$. In this paper, we provide a proof of this conjecture for $0\leq α\leq \frac{6p^3-29p^2+17p-2+2p\sqrt{6p^2-11p+4}}{(3p-1)^2}$, and this range of $α$ improves the best known result when $α>\frac{1}{47}$ and $α\not=1$.

Figures (2)

• Figure 1:
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Theorems & Definitions (9)

• Proposition 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• proof : Proof of Proposition \ref{['1main']}
• Proposition 3.1
• proof
• Theorem 4.1
• proof