[Paper Review] Derivatives characterization of Bergman-Orlicz spaces and applications

Benoît F. Sehba|arXiv (Cornell University)|Oct 6, 2016

Holomorphic and Operator Theory17 references3 citations

TL;DR

This paper extends classical Hardy-type characterizations of Bergman spaces to Bergman-Orlicz spaces by proving that membership of a holomorphic function in a Bergman-Orlicz space is equivalent to integrability of its invariant gradient, Euclidean gradient, or radial derivative, weighted by $(1-|z|^2)$, in the corresponding Orlicz norm. The key contribution is a full derivatives characterization for Bergman-Orlicz spaces under natural convexity and growth conditions on the Orlicz function, enabling applications to interpolation and Cesàro operator theory.

ABSTRACT

It is well known that a function is in a Bergman space of the unit ball if and only if it satisfies some Hardy-type inequalities. We extend this fact to Bergman-Orlicz spaces. As applications, we obtain Gustavsson-Peetre interpolation of two Bergman-Orlicz spaces and we completely characterize symbols of bounded or compact Cesàro-type operators on Bergman-Orlicz spaces, extending known results for classical weighted Bergman spaces.

Motivation & Objective

• To generalize the classical derivatives characterization of Bergman spaces to Bergman-Orlicz spaces.
• To establish equivalent conditions for a holomorphic function to belong to a Bergman-Orlicz space using invariant, Euclidean, and radial derivatives.
• To apply the characterization to solve problems in interpolation theory and operator theory on Bergman-Orlicz spaces.
• To fully characterize symbols of bounded and compact Cesàro-type operators on Bergman-Orlicz spaces.

Proposed method

• Define Bergman-Orlicz spaces $\mathcal{A}_{\alpha}^{\Phi}(\mathbb{B}^n)$ using Luxemburg-type norms associated with a growth function $\Phi$.
• Introduce the classes $\mathscr{U}^q$ and $\mathscr{L}_p$ of Orlicz functions with specific upper and lower type conditions.
• Establish equivalence between $f \in \mathcal{A}_{\alpha}^{\Phi}(\mathbb{B}^n)$ and integrability of $|\widehat{\nabla}f(z)|$, $(1-|z|^2)|\nabla f(z)|$, and $(1-|z|^2)|\mathcal{R}f(z)|$ in $L^\Phi(\mathbb{B}^n, d\nu_\alpha)$.
• Use pointwise estimates and properties of complementary functions to control the growth of derivatives near the boundary.
• Apply the characterization to prove boundedness and compactness of Cesàro-type operators via norm estimates and convergence on compact subsets.
• Use the theory of Gustavsson-Peetre interpolation to characterize the interpolation space between two Bergman-Orlicz spaces.

Experimental results

Research questions

• RQ1 Under what conditions on the Orlicz function $\Phi$ is the membership of a holomorphic function in $\mathcal{A}_{\alpha}^{\Phi}(\mathbb{B}^n)$ equivalent to the integrability of its invariant gradient in $L^\Phi(\mathbb{B}^n, d\nu_\alpha)$?
• RQ2 Can the classical derivatives characterization of Bergman spaces be extended to Bergman-Orlicz spaces using radial or Euclidean derivatives?
• RQ3 What is the precise condition on the symbol $g$ for the Cesàro-type operator $T_g$ to be bounded or compact on $\mathcal{A}_{\alpha}^{\Phi}(\mathbb{B}^n)$?
• RQ4 How does the Gustavsson-Peetre interpolation method apply to Bergman-Orlicz spaces, and what is the resulting interpolation space?
• RQ5 What role do the upper and lower type indices of $\Phi$ play in the equivalence of the various derivative characterizations?

Key findings

• The paper establishes the equivalence of four conditions: $f \in \mathcal{A}_{\alpha}^{\Phi}(\mathbb{B}^n)$, $|\widehat{\nabla}f| \in L^\Phi(\mathbb{B}^n, d\nu_\alpha)$, $(1-|z|^2)|\nabla f(z)| \in L^\Phi(\mathbb{B}^n, d\nu_\alpha)$, and $(1-|z|^2)|\mathcal{R}f(z)| \in L^\Phi(\mathbb{B}^n, d\nu_\alpha)$, for $\Phi \in \mathscr{U}^q \cup \mathscr{L}_p$.
• The boundedness of the Cesàro-type operator $T_g$ on $\mathcal{A}_{\alpha}^{\Phi}(\mathbb{B}^n)$ is characterized by the condition $\int_{\mathbb{B}^n} \Phi\left( \frac{(1-|z|^2)|\mathcal{R}g(z)|}{\|g\|_{\Phi,\alpha}^{lux}} \right) d\nu_\alpha(z) \lesssim 1$, which is equivalent to $g \in \mathcal{A}_{\alpha}^{\Phi}(\mathbb{B}^n)$ under the given assumptions.
• The compactness of $T_g$ is characterized by the stronger condition that $\|T_g f_j\|_{\Phi,\alpha}^{lux} \to 0$ as $j \to \infty$ for any sequence $\{f_j\}$ converging to zero uniformly on compact subsets.
• The interpolation space between two Bergman-Orlicz spaces $\mathcal{A}_{\alpha}^{\Phi_0}$ and $\mathcal{A}_{\alpha}^{\Phi_1}$ via the Gustavsson-Peetre method is shown to be $\mathcal{A}_{\alpha}^{\Phi}$, where $\Phi$ is the associated interpolation function.
• The proof relies on the use of the complementary function $\Psi$ and the $\Delta_2$-condition to control the growth of $\Phi$, ensuring that the Luxemburg norm is well-behaved under the derivative conditions.
• The result generalizes known characterizations for classical weighted Bergman spaces ($\Phi(t) = t^p$) to the full class of Orlicz functions in $\mathscr{U}^q \cup \mathscr{L}_p$, extending the scope of derivative-based characterizations.