[Paper Review] Weighted Hardy spaces on the unit disk

Evgeny A. Poletsky|arXiv (Cornell University)|Sep 4, 2014

Holomorphic and Operator Theory13 references3 citations

TL;DR

This paper establishes that for weighted Hardy spaces $H^p_u(\mathbb{D})$ on the unit disk defined via exhaustion functions, the dilations $f_t(z) = f(tz)$ converge to $f$ in the $H^p_u$-norm for $p > 0$, which implies that polynomials are dense in $H^p_u(\mathbb{D})$. It further shows that the $h^p_u$-norm of a pluriharmonic function equals the $L^p_u$-norm of its boundary values, and that the measures $\mu_{u,r}$ converge weak-* to $\mu_u$ in a stronger sense than previously known.

ABSTRACT

In this paper we mainly discuss three things. First, there is no canonical norm on the space $H^p_u(\mathbb{D})$. Second, we improve the weak-$*$ convergence of the measures $μ_{u,r}$. Third, the dilations $f_t$ of the function $f\in H^p_u(\mathbb{D})$ converge to $f$ in $H^p_u$-norm and hence the polynomials are dense in $H^p_u(\mathbb{D})$.

Motivation & Objective

• To investigate the convergence of dilations $f_t(z) = f(tz)$ in weighted Hardy spaces $H^p_u(\mathbb{D})$ for $p > 0$.
• To establish that polynomials are dense in $H^p_u(\mathbb{D})$ via dilation convergence.
• To improve the weak-* convergence of the measures $\mu_{u,r}$ to $\mu_u$ on the boundary $\mathbb{T}$.
• To clarify the relationship between the $h^p_u$-norm and the $L^p_u$-norm of boundary values.
• To demonstrate that no canonical norm exists for $H^p_u(\mathbb{D})$ due to non-uniqueness of exhaustion functions.

Proposed method

• The paper uses the Lelong–Jensen formula to express the $H^p_u$-norm as an integral involving $|f|^p$ and $\Delta u$.
• It applies the monotone convergence theorem and properties of subharmonic functions to replace $\varlimsup$ with $\lim$ in the norm definition.
• It proves weak-* convergence of $\mu_{u,r}$ to $\mu_u$ in $C^*(\overline{\mathbb{D}})$ by analyzing the limit of integrals over level sets $S_{u,r}$.
• It establishes norm convergence of dilations $f_t$ to $f$ in $H^p_u$ by showing $\|f_t - f\|_{H^p_u} \to 0$ using Egorov's theorem and $L^p$-convergence of boundary values.
• It uses Poisson integral representations and the subharmonicity of $|f|^p$ to compare $\|f_t\|_{H^p_u}$ with $\|f^*\|_{L^p_u}$.
• It proves that $\|f\|_{H^p_u}^p = \|f^*\|_{L^p_u}^p$ by showing equality of the limit of integrals over $S_{u,r}$ and the boundary $L^p$-norm.

Experimental results

Research questions

• RQ1 Does the dilation $f_t(z) = f(tz)$ converge to $f$ in the $H^p_u$-norm for $f \in H^p_u(\mathbb{D})$?
• RQ2 Are polynomials dense in $H^p_u(\mathbb{D})$ for $p > 0$?
• RQ3 Can the weak-* convergence of $\mu_{u,r}$ to $\mu_u$ be strengthened beyond Demailly's result?
• RQ4 Is there a canonical norm on $H^p_u(\mathbb{D})$ induced by a unique exhaustion function?
• RQ5 How does the $h^p_u$-norm relate to the $L^p_u$-norm of the boundary function?

Key findings

• The dilations $f_t(z) = f(tz)$ converge to $f$ in the $H^p_u$-norm for all $f \in H^p_u(\mathbb{D})$ and $p > 0$.
• Polynomials are dense in $H^p_u(\mathbb{D})$ for $p > 0$, as a consequence of dilation convergence.
• The measures $\mu_{u,r}$ converge weak-* to $\mu_u$ in $C^*(\overline{\mathbb{D}})$, and this convergence is stronger than previously known.
• For $f \in H^p_u(\mathbb{D})$, the $H^p_u$-norm equals the $L^p_u$-norm of its radial boundary function $f^*$.
• The $h^p_u$-norm of a pluriharmonic function equals the $L^p_u$-norm of its boundary value function.
• There is no canonical norm on $H^p_u(\mathbb{D})$, as different exhaustion functions in $\mathcal{E}_0$ yield the same space but different unit balls, whose intersection is the unit ball of $H^\infty(\mathbb{D})$.