[Paper Review] Is the Dirichlet Space a Quotient of $DA_n$?

Richard Rochberg|arXiv (Cornell University)|May 9, 2016

Holomorphic and Operator Theory6 references3 citations

TL;DR

This paper proves that the Dirichlet space on the unit disk is not isometrically representable as a quotient of the Drury-Arveson space $DA_n$ for any finite $n$. Using the metric $\delta$ induced by the angle between normalized reproducing kernels, the authors derive a contradiction via volume comparison in high-dimensional balls, showing that the required isometric embedding cannot exist due to geometric incompatibility between the metrics of the two spaces.

ABSTRACT

We show that the Dirichlet space is not a quotient of the Drury-Arveson space on the n-ball for any finite n. The proof is based a quantitative comparison of the metrics induced by the Hilbert spaces

Motivation & Objective

• To determine whether the Dirichlet space $\mathcal{D}$ can be realized as a quotient of the Drury-Arveson space $DA_n$ for any finite $n$.
• To investigate the geometric and metric incompatibility between the Dirichlet space and $DA_n$ using the $\delta$-metric derived from kernel angles.
• To establish that the Hilbert space quotient structure of $\mathcal{D}$ via $DA_n$ fails for finite $n$, despite being possible in infinite dimensions.
• To provide a geometric obstruction based on volume growth and ball packing in $\delta$-metric spaces.

Proposed method

• Define the metric $\delta_K(x,w) = \sqrt{1 - |\langle \widehat{k_x}, \widehat{k_w} \rangle|^2}$, which measures the sine of the angle between normalized kernel functions in a reproducing kernel Hilbert space.
• Use the identity $\delta_{DA_1}(z,w) = \left| \frac{z-w}{1-\bar{z}w} \right|$, linking $\delta_{DA_n}$ to the pseudohyperbolic metric on the disk.
• Construct a finite set of points $\{z_i\}$ in the unit disk such that their images under a hypothetical map $\Phi: \mathbb{D} \to \mathbb{B}^n$ would require disjoint small balls in $\delta_{DA_n}$-metric around each image point.
• Estimate the number $N \geq e^K$ of such disjoint balls and their individual volumes $V_S \geq (A/K)^n$, while the large ball containing them has volume $V_L \leq (BK)^n$, leading to a contradiction for large $K$.
• Apply the strengthened triangle inequality for $\delta_{DA_n}$ to bound the distance from the origin to any point in the small balls, ensuring they lie within a ball of radius $1 - 1/(3K)$.

Experimental results

Research questions

• RQ1 Can the Dirichlet space $\mathcal{D}$ be represented as a quotient of $DA_n$ for some finite $n$?
• RQ2 Is there a map $\Phi: \mathbb{D} \to \mathbb{B}^n$ and a positive function $\lambda$ such that the kernel of $\mathcal{D}$ satisfies $h(w,z) = \lambda(z)\lambda(w)j^{(n)}(\Phi(w),\Phi(z))$?
• RQ3 What geometric obstruction prevents such a quotient structure for finite $n$, despite its existence in infinite dimensions?
• RQ4 How does the $\delta$-metric reveal fundamental differences between the Dirichlet space and $DA_n$ in terms of metric geometry?

Key findings

• The Dirichlet space $\mathcal{D}$ is not a quotient of $DA_n$ for any finite $n$, resolving a question stemming from discussions with Ken Davidson and Orr Shalit.
• The number of disjoint $\delta_{DA_n}$-balls of radius $\sim \sqrt{C^2/K}$ centered at $\Phi(z_i)$ exceeds $e^K$ for large $K$, implying exponential growth in the number of such balls.
• The volume of each small ball is bounded below by $\left(A/K\right)^n$, while the volume of the large ball containing them is bounded above by $\left(BK\right)^n$, making the volume inequality $NV_S \leq V_L$ impossible to satisfy for large $K$.
• The contradiction arises from the geometric incompatibility between the metric structures of $\mathcal{D}$ and $DA_n$, as captured by the $\delta$-metric and its behavior under isometric embedding.
• The argument fails for $\mathcal{D}_\alpha$ with $0 < \alpha < 1$, indicating that the Dirichlet space is fundamentally different in metric structure from these related spaces.
• An infinitesimal version of the argument may involve curvature obstructions in the Riemannian metric derived from $\delta_K$, suggesting a potential link to sectional curvature in the kernel space geometry.