The Hilbert matrix on analytic tent spaces

TL;DR

This paper investigates the boundedness of the Hilbert matrix operator on analytic tent spaces, extending known results from Bergman spaces and providing norm estimates within a new functional framework.

Contribution

It is the first study of the Hilbert matrix acting on analytic tent spaces, establishing boundedness conditions and norm estimates, thus generalizing previous Bergman space results.

Findings

Hilbert operator is bounded on AT_p^q when 1/p + 1/q < 1 and p > 2

Extension of boundedness results from Bergman spaces to analytic tent spaces

Provided estimates for the norm of the Hilbert operator

Abstract

We study for the first time the action of the Hilbert matrix $$\mathcal H=(c_{n,k})_{n,k\geq 0}, \quad c_{n,k}=\frac{1}{n+k+1}$$ on the analytic tent spaces $AT^q_p, 1<p,q <\infty,$ of the unit disc $\mathbb D$ of the complex plane. They were proposed by Triebel as the natural analytic version of the tent spaces of measurable functions defined by Coifman, Meyer and Stein. The $AT_p^q$ spaces are consisted of those analytic functions $f$ in $\mathbb D$ such that $$ \|f\|_{AT_{p}^{q}}= \left\{\int_{\mathbb T} \left(\int_{\Gamma_{1/2}(\xi)} |f(z)|^p \ \frac{dA(z)}{1-|z|^2} \right)^{q/p}\ |d\xi|\right \}^{1/q}<+\infty, $$ where $$ \Gamma_{1/2}(\xi) =\bigl\{ z\in \mathbb{D} : |z|< 1/2 \bigr\} \cup \bigcup_{|z|<1/2}[z,\xi), $$ $dA(z)$ is the normalized area Lebesgue measure in $\mathbb D$ and $|d\xi|$ is the arc length in the unit circle $\mathbb T$. The Bergman spaces $A^p, p>1,$ stand among the $AT_{p}^{q}$ and correspond to the case $p=q$. The multiplication of the Hilbert matrix with the column matrix with entries the Taylor coefficients of an $f(z)=\sum_{k\geq 0} a_k z^k $ analytic in $\mathbb D$ introduces the series $$ \mathcal H (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \frac{a_k}{n+k+1}\right)z^n\,, \quad z\in \mathbb D\,\, $$ known in the literature as Hilbert operator. We prove that it is a bounded operator on the $AT_{p}^{q}$ when $1/p + 1/q <1,\, p>2$. This is a natural range for the values of the indices $p,q$ compared to what is known in the special case of the Bergman spaces. We confront the question under discussion through a more general point of view by studying an associated integral operator defined with respect to a positive Borel measure $\mu$ on $[0,1)$. Finally, we provide an estimation of the norm of the Hilbert operator. Our work extends in a non-trivially way previous results on the Bergman spaces to the analytic tent spaces.