[Paper Review] Generalized Hilbert Operator Acting on Bloch Type Spaces

Shanli Ye, Zhihui Zhou|arXiv (Cornell University)|Jul 22, 2022

Holomorphic and Operator Theory4 citations

TL;DR

This paper characterizes the positive Borel measures μ on [0,1) for which the generalized Hilbert operator H_{μ,α} is bounded or compact from Bloch type spaces B_β into B_{α−1}, establishing sharp Carleson measure conditions depending on α and β. For α ≥ 2, boundedness and compactness are equivalent to μ being a 2-Carleson measure when 0 < β < 1, a (β+1)-Carleson measure when 1 < β < ∞, and a 1-logarithmic 2-Carleson measure (or its vanishing version) when β = 1.

ABSTRACT

Let $μ$ be a positive Borel measure on the interval [0,1). For $α&gt;0$, the Hankel matrix $\mathcal{H}_{μ,α}=(μ_{n,k,α})_{n,k\geq 0}$ with entries $μ_{n,k,α}=\int_{[0,1)}\frac{Γ(n+α)}{n!Γ(α)}t^{n+k}dμ(t)$ formally induces the operator $$\mathcal{H}_{μ,α}(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n, k,α} a_{k} ight)z^{n} $$ on the space of all analytic functions $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ in the unit disc $\mathbb{D}$. In this paper, we characterize the measures $μ$ for which $\mathcal{H}_{μ,α}$ ($α\geq 2$) is a bounded (resp., compact) operator from the Bloch type space $\mathscr{B}_β$ ($0&lt;β&lt;\infty$) into $\mathscr{B}_{α-1}$. We also give a necessary condition for which $\mathcal{H}_{μ,α}$ is a bounded operator by acting on Bloch type spaces for general cases.

Motivation & Objective

• To characterize the positive Borel measures μ on [0,1) for which the generalized Hilbert operator H_{μ,α} is bounded or compact from Bloch type spaces B_β into B_{α−1}.
• To establish sharp measure-theoretic conditions (Carleson-type) that determine boundedness and compactness of H_{μ,α} for α ≥ 2.
• To extend known results on classical and derivative Hilbert operators to the generalized case with parameter α > 0.
• To provide necessary conditions for boundedness of H_{μ,α} when acting on B_β for general α > 0, particularly for 0 < α ≤ 1.

Proposed method

• The generalized Hilbert operator H_{μ,α} is defined via a Hankel matrix with entries μ_{n,k,α} = ∫_{[0,1)} Γ(n+α)/(n!Γ(α)) t^{n+k} dμ(t), acting on Taylor coefficients of analytic functions in the unit disk.
• The operator is analyzed through integral representations: H_{μ,α}(f)(z) = ∫_{[0,1)} f(t)/(1−tz)^α dμ(t), linking it to the generalized integral-Hilbert operator I_{μ,α}.
• The study uses test functions, such as f(z) = 1 and f_λ(z) = (1−λz)^{-β}, to derive necessary conditions on μ via norm estimates in Bloch and Q_p spaces.
• Carleson and logarithmic Carleson measure theory is applied, with the key idea of testing measure decay via μ([λ,1)) and relating it to (1−λ)^s or (1−λ)^s (log(2π/|I|))^α.
• The proofs rely on asymptotic analysis using Stirling’s formula to estimate Γ(n+α)/(n!Γ(α)) ∼ n^{α−1} as n → ∞.
• Techniques from Bergman and Hardy space theory are adapted, particularly integral inequalities and embedding theorems for Bloch and Q_p spaces.

Experimental results

Research questions

• RQ1 For α ≥ 2, what measure condition on μ ensures that H_{μ,α} is bounded from B_β into B_{α−1}?
• RQ2 For α ≥ 2, what measure condition on μ ensures that H_{μ,α} is compact from B_β into B_{α−1}?
• RQ3 For 0 < α ≤ 1 and 0 < β < 1, what necessary condition on μ is implied if H_{μ,α} is bounded from B_β into Q_p or BMOA?
• RQ4 How do the necessary conditions for boundedness differ across different ranges of β and α, particularly in relation to Carleson measure strength?
• RQ5 What is the precise relationship between the operator norm of H_{μ,α} and the Carleson or logarithmic Carleson norm of μ?

Key findings

• For α ≥ 2 and 0 < β < 1, H_{μ,α} is bounded (resp. compact) from B_β into B_{α−1} if and only if μ is a 2-Carleson measure (resp. vanishing 2-Carleson measure).
• For α ≥ 2 and β = 1, H_{μ,α} is bounded (resp. compact) from B into B_{α−1} if and only if μ is a 1-logarithmic 2-Carleson measure (resp. vanishing 1-logarithmic 2-Carleson measure).
• For α ≥ 2 and 1 < β < ∞, H_{μ,α} is bounded (resp. compact) from B_β into B_{α−1} if and only if μ is a (β+1)-Carleson measure (resp. vanishing (β+1)-Carleson measure).
• For 0 < α ≤ 1 and 0 < β < 1, if H_{μ,α} is bounded from B_β into Q_p, then μ must be an α-Carleson measure, providing a necessary condition.
• For 0 < α ≤ 1 and 0 < β < 1, if H_{μ,α} is bounded from B_β into BMOA or the Bloch space B, then μ must be an α-Carleson measure.
• The results show that the strength of the required Carleson condition on μ increases with β and depends critically on α, with the threshold α ≥ 2 being essential for the equivalence between boundedness and compactness.