[Paper Review] Norm of the Bergman projection

David Kalaj, Marijan Marković|arXiv (Cornell University)|Mar 27, 2012

Holomorphic and Operator Theory6 references3 citations

TL;DR

This paper determines the exact operator norm of the weighted Bergman projection $ P_{eta} : L^∞(\mathbb{B}) \to \mathcal{B} $ on the unit ball $ \mathbb{B} \subset \mathbb{C}^n $, where $ \mathcal{B} $ is the Bloch space. It establishes sharp bounds for both the standard and Möbius-invariant Bloch norms using integral estimates and special functions, extending prior results in the literature.

ABSTRACT

This paper deals with the the norm of the weighted Bergman projection operator $P_α:L^\infty o \mathcal{B}$ where $α>-1$ and $\mathcal{B}$ is the Bloch space of the unit ball of the complex space $\mathbf{C}^n$. We consider two Bloch norms, the standard Bloch norm and invariant norm w.r.t. automorphisms of the unit ball. Our work contains as a special case the main result of the recent paper \cite{Perala}.

Motivation & Objective

• To determine the exact operator norm of the weighted Bergman projection $ P_{\alpha} $ from $ L^\infty(\mathbb{B}) $ into the Bloch space $ \mathcal{B} $ of the unit ball in $ \mathbb{C}^n $.
• To compare and contrast the standard Bloch norm with the Möbius-invariant Bloch norm in the context of Bergman projection operator norms.
• To extend and generalize the main result of a prior paper [4] by providing sharp estimates for the operator norm in higher dimensions.
• To analyze the behavior of the Bergman projection on continuous functions and its image in the little Bloch space $ \mathcal{B}_0 $.
• To establish the sharpness of the norm estimate via a sequence of continuous functions achieving the supremum.

Proposed method

• The authors define the Bergman projection $ P_{\alpha}g(z) = \int_{\mathbb{B}} \mathcal{K}_{\alpha}(z,w) g(w) dv_{\alpha}(w) $, where $ \mathcal{K}_{\alpha}(z,w) = (1 - \langle z,w \rangle)^{-(n+1+\alpha)} $ is the weighted Bergman kernel.
• They analyze the Bloch semi-norm $ \|f\|_{\beta} = \sup_{z \in \mathbb{B}} (1 - |z|^2) |\nabla f(z)| $ and the invariant Bloch semi-norm $ \|f\|_{\tilde{\beta}} = \sup_{z \in \mathbb{B}} (1 - |z|^2) |\tilde{\nabla} f(z)| $, where $ \tilde{\nabla} f $ is the invariant gradient.
• Using polar coordinates and the automorphism $ \varphi_a $, they derive the transformation rule $ dv_{\alpha}(\varphi_a(\omega)) = \left( \frac{1 - |a|^2}{|1 - \langle \omega, a \rangle|^2} \right)^{n+1+\alpha} dv_{\alpha}(\omega) $.
• They compute the integral $ \ell(t) = \int_{\mathbb{B}} \frac{|(1 - w_1)\cos t + w_2 \sin t|}{|1 - w_1|^{n+1}} dv_{\alpha}(w) $, which captures the norm behavior along a one-parameter family.
• By substituting $ a_1 = w_2 / (1 - w_1) $, $ a_2 = w_1 $, they transform the integral into $ I(t) = \int_{B'} |\cos t + a_1 \sin t| dv(a) $ over a domain $ B' $ depending on $ a_2 $.
• They express the angular integral $ h(t) = \int_0^{2\pi} |\cos t + a_1 \sin t| d\sigma $ as the circumference of an ellipse, using elliptic integrals of the first and second kind.

Experimental results

Research questions

• RQ1 What is the exact operator norm of the Bergman projection $ P_{\alpha} : L^\infty(\mathbb{B}) \to \mathcal{B} $, where $ \mathcal{B} $ is the Bloch space on the unit ball in $ \mathbb{C}^n $?
• RQ2 How do the standard Bloch norm and the Möbius-invariant Bloch norm compare in the context of Bergman projection operator norms?
• RQ3 Can the sharp constant $ \tilde{C}_{\alpha,n} $ for the invariant norm be computed explicitly, and is it achieved by a sequence of continuous functions?
• RQ4 What is the role of the limit behavior of the invariant gradient $ \tilde{\nabla} f(z) $ as $ |z| \to 1 $, and how does it relate to the image of $ C(\overline{\mathbb{B}}) $ under $ P_{\alpha} $?
• RQ5 Is the function $ I(t) $, representing the norm along a one-parameter family, minimized or maximized at $ t = 0 $ or $ t = \pi/2 $, and what does this imply for the operator norm?

Key findings

• The operator norm of $ P_{\alpha} $ with respect to the standard Bloch semi-norm satisfies $ \|P_{\alpha}\|_{\beta} \leq \frac{\Gamma(2 + n + \alpha)}{\Gamma^2((2 + n + \alpha)/2)} $, and this bound is sharp.
• For the invariant Bloch semi-norm, the norm satisfies $ \|P_{\alpha}\|_{\tilde{\beta}} = \tilde{C}_{\alpha,n} $, and this value is achieved by a sequence of continuous functions $ g_k $, proving sharpness.
• The integral $ \ell(\pi/2) = \int_{\mathbb{B}} \frac{|w_2|}{|1 - w_1|^{n+1}} dv_{\alpha}(w) $ is computed as $ \frac{\pi \Gamma(2 + n + \alpha)}{2 \Gamma^2((2 + n + \alpha)/2)} $, which exceeds the standard norm bound.
• The function $ I(t) $, representing the norm along a direction, has stationary points at $ t = 0 $ and $ t = \pi/2 $, as shown by asymptotic analysis of elliptic integrals.
• The image of $ C(\overline{\mathbb{B}}) $ under $ P_{\alpha} $ lies in the little Bloch space $ \mathcal{B}_0 $, and the operator norm from $ C(\overline{\mathbb{B}}) \to \mathcal{B}_0 $ equals $ \tilde{C}_{\alpha,n} $.
• The authors confirm that $ \|P_{\alpha}\|_{\tilde{\beta}} = \tilde{C}_{\alpha,n} $, and this value is strictly greater than the standard norm, showing that the invariant norm yields a strictly larger operator norm.