On the exact value of the norm of the Hilbert matrix operator on\n weighted Bergman spaces

2020/01/28 by Mikael Lindström, Lindström, Mikael, Santeri Miihkinen +3 · 2 citations
Mathematics · #30H20 (Secondary) #47B38 (Primary) #Advanced Topics in Algebra #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2001.10476

openalex publication_date 2020/01/28 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

In this article, the open problem of finding the exact value of the norm of\nthe Hilbert matrix operator on weighted Bergman spaces A^p_\α is\nadressed. The norm was conjectured to be \(\π)/(\sin\n\((2+\α)\π)/(p)) by Karapetrovi 'c. We obtain a complete solution to\nthe conjecture for \α \≥ 0 and\n2+\α+\√(\α²+\(7)/(2)\α+3) \≤ p < 2(2+\α) and a\npartial solution for 2+2\α < p <\n2+\α+\√(\α²+\(7)/(2)\α+3). Moreover, we also show that the\nconjecture is valid for small values of \α when 2+2\α < p \≤\n3+2\α. Finally, the case \α = 1 is considered.\n

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• Hilbert matrix norms on weighted Bergman spaces: even exponents and a counterexample to the beta formula
• Norm of the Hilbert matrix operator on logarithmically weighted Bloch and Hardy spaces

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