[Paper Review] Hankel matrices acting on the Hardy space $H^1$ and on Dirichlet spaces

Daniel Girela, Noel Merchán|arXiv (Cornell University)|Apr 6, 2018

Holomorphic and Operator Theory18 references5 citations

TL;DR

This paper investigates Hankel matrices induced by finite positive Borel measures on [0,1), analyzing their action on Hardy spaces $H^1$ and Dirichlet-type spaces. It proves that if the measure is a 1-logarithmic 1-Carleson measure, then the Hankel operator maps $H^1$ into the Dirichlet space $\mathcal{D}^1_0$, extending known results on boundedness and range properties of generalized Hilbert operators.

ABSTRACT

If $\,μ\,$ is a finite positive Borel measure on the interval $\,[0,1)$, we let $\,\mathcal H_μ\,$ be the Hankel matrix $\,(μ_{n, k})_{n,k\ge 0}\,$ with entries $\,μ_{n, k}=μ_{n+k}$, where, for $\,n\,=\,0, 1, 2, \dots $, $μ_n\,$ denotes the moment of order $\,n\,$ of $\,μ$. This matrix induces formally the operator $\,\mathcal{H}_μ(f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n,k}{a_k} ight)z^n\,$ on the space of all analytic functions $\,f(z)=\sum_{k=0}^\infty a_kz^k\,$, in the unit disc $\,\mathbb D $. When $\,μ\,$ is the Lebesgue measure on $\,[0,1)\,$ the operator $\,\mathcal H_μ\,$ is the classical Hilbert operator $\,\mathcal H\,$ which is bounded on $\,H^p\,$ if $\,1

Motivation & Objective

• To characterize the action of Hankel matrices $\mathcal{H}_\mu$ on the Hardy space $H^1$ when $\mu$ is a finite positive Borel measure on $[0,1)$.
• To determine the precise range of $\mathcal{H}_\mu$ on $H^1$ when $\mu$ is an $\alpha$-logarithmic 1-Carleson measure for $0 < \alpha < 1$.
• To study the boundedness and mapping properties of $\mathcal{H}_\mu$ on Bergman and Dirichlet-type spaces, particularly $\mathcal{D}^p_\alpha$.
• To extend known results on the classical Hilbert operator $\mathcal{H}$, which is bounded on $H^p$ for $1 < p < \infty$ but not on $H^1$, by analyzing its action into the space of Cauchy transforms and Dirichlet spaces.

Proposed method

• The authors define the Hankel matrix $\mathcal{H}_\mu$ via moments $\mu_{n,k} = \mu_{n+k}$, inducing an operator on analytic functions through formal matrix multiplication.
• They use the duality between $H^1$ and the Bloch space, and the identification of $A^1_\alpha$ with the dual of the little Bloch space, to analyze boundedness via integral pairings.
• The key technical tool is the use of integral representations, such as $\mathcal{I}_\mu f(z) = \int_0^1 \frac{f(t)}{1 - tz} dt$, which coincides with $\mathcal{H}_\mu f$ for $f \in H^1$.
• They apply Hölder’s inequality and estimates involving the maximal function $M_\infty(t,f)$ to bound the $L^p$-norms of derivatives in Dirichlet spaces.
• The proof relies on known results on Carleson measures and $\alpha$-Carleson measures, particularly the fact that $\nu(t) = (1-t)^\alpha \mu(t)$ is a Carleson measure if $\mu$ is a $1$-logarithmic $1$-Carleson measure.
• They use the boundedness of the operator $T\phi(\xi) = (1-|\xi|^2)^{-\alpha} \int_\mathbb{D} \frac{(1-|z|^2)^\alpha \phi(z)}{(1 - \xi \bar z)^2} dA(z)$ from the Bloch space to $L^\infty$ to control integral pairings.

Experimental results

Research questions

• RQ1 Under what conditions on the measure $\mu$ is the Hankel operator $\mathcal{H}_\mu$ bounded from $H^1$ into the Dirichlet space $\mathcal{D}^1_0$?
• RQ2 What is the precise range of $\mathcal{H}_\mu$ on $H^1$ when $\mu$ is an $\alpha$-logarithmic 1-Carleson measure for $0 < \alpha < 1$?
• RQ3 Is the operator $\mathcal{H}_\mu$ bounded from $\mathcal{D}^1_\alpha$ to itself when $\mu$ is a Carleson measure and $-1 < \alpha < 0$?
• RQ4 Does the converse of Theorem 6 hold, i.e., is $\mu$ necessarily a Carleson measure if $\mathcal{H}_\mu$ is bounded on $\mathcal{D}^1_\alpha$?

Key findings

• If $\mu$ is a 1-logarithmic 1-Carleson measure, then $\mathcal{H}_\mu$ maps $H^1$ into the Dirichlet space $\mathcal{D}^1_0$.
• For $\mu$ an $\alpha$-logarithmic 1-Carleson measure with $0 < \alpha < 1$, the range of $\mathcal{H}_\mu$ on $H^1$ is contained in $\mathcal{D}^1_0$.
• When $\mu$ is a Carleson measure, $\mathcal{H}_\mu$ is a bounded operator from $\mathcal{D}^1_\alpha$ to itself for $-1 < \alpha < 0$.
• The duality pairing $\langle h, \mathcal{I}_\mu f \rangle$ is bounded by $\|h\|_{\mathcal{B}} \|f\|_{\mathcal{D}^1_\alpha}$ when $\mu$ is a Carleson measure, establishing boundedness via duality.
• The inequality $\sum_{n=0}^\infty |a_n| (n+1)^{-(1+\alpha)} \lesssim \|f\|_{\mathcal{D}^1_\alpha}$ holds for all $f \in \mathcal{D}^1_\alpha$, but the reverse inequality is not established for $p=1$.