[Paper Review] The Riesz transform of codimension smaller than one and the Wolff energy

Benjamin Jaye, Fëdor Nazarov+2 more|arXiv (Cornell University)|Feb 8, 2016

Advanced Harmonic Analysis Research36 references4 citations

TL;DR

This paper establishes a characterization of non-negative, locally finite, non-atomic Borel measures $μ$ in $\mathbb{R}^d$ for which the $s$-Riesz transform is bounded in $L^2(\mu)$, when $s \in (d-1,d)$, by showing that this boundedness is equivalent to a uniform control of the Wolff energy $\mathcal{W}_2(\mu,Q) \leq \widetilde{C}\mu(Q)$ for all cubes $Q$. The result extends the Mateu-Prat-Verdera theorem to higher codimensions and provides a metric characterization of removable sets for locally Lipschitz solutions of the fractional Laplacian $(-\Delta)^{\alpha/2}$ with $\alpha \in (1,2)$, contrasting with classical removability for harmonic functions.

ABSTRACT

Fix $d\geq 2$, and $s\in (d-1,d)$. We characterize the non-negative locally finite non-atomic Borel measures $μ$ in $\mathbb{R}^d$ for which the associated $s$-Riesz transform is bounded in $L^2(μ)$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, we give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-Δ)^{α/2}$, $α\in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.

Motivation & Objective

• To extend the Mateu-Prat-Verdera characterization of $s$-Riesz transform boundedness from $s \in (0,1)$ to $s \in (d-1,d)$, where $d \geq 2$.
• To establish a necessary and sufficient condition for $L^2(\mu)$-boundedness of the $s$-Riesz transform in terms of the Wolff energy condition $\mathcal{W}_2(\mu,Q) \leq \widetilde{C}\mu(Q)$ for all cubes $Q$.
• To apply the characterization to give a metric description of removable sets for locally Lipschitz solutions of the fractional Laplacian $(-\Delta)^{\alpha/2}$ with $\alpha \in (1,2)$, using a capacity from nonlinear potential theory.

Proposed method

• The proof relies on a refined analysis of the Riesz transform's behavior through energy estimates and blow-up techniques, particularly focusing on density drops in dyadic cubes.
• A key step involves constructing a 'shell' around a cube where the measure density drops significantly, enabling improved energy estimates in that region.
• The authors use a modified Calderón-Zygmund decomposition and a variational argument to control the $L^2$ norm of the truncated Riesz transform.
• They employ a smoothing operation and localization around the shell to derive uniform bounds, leveraging the maximum principle and properties of the $\alpha$-Poisson kernel.
• The argument involves a contradiction scheme: assuming the Wolff energy condition fails leads to a contradiction with the $L^2(\mu)$-boundedness of the Riesz transform.
• The proof draws on techniques from potential theory, including the use of reflectionless measures and the structure of the Wolff energy as a non-linear positive functional.

Experimental results

Research questions

• RQ1 Is the boundedness of the $s$-Riesz transform on $L^2(\mu)$ equivalent to the Wolff energy condition $\mathcal{W}_2(\mu,Q) \leq \widetilde{C}\mu(Q)$ for all cubes $Q$ when $s \in (d-1,d)$?
• RQ2 Can the Mateu-Prat-Verdera characterization, originally valid for $s \in (0,1)$, be extended to the range $s \in (d-1,d)$?
• RQ3 What is the metric characterization of removable sets for locally Lipschitz solutions of the fractional Laplacian $(-\Delta)^{\alpha/2}$ with $\alpha \in (1,2)$?
• RQ4 How does the structure of measures with bounded $s$-Riesz transform differ in the regime $s \in (d-1,d)$ compared to $s \in (0,1)$?
• RQ5 What role does the Wolff energy play in capturing the cancellation structure of the Riesz kernel for non-integer $s$?

Key findings

• The $s$-Riesz transform is bounded in $L^2(\mu)$ if and only if the Wolff energy condition $\mathcal{W}_2(\mu,Q) \leq \widetilde{C}\mu(Q)$ holds for all cubes $Q \subset \mathbb{R}^d$, for $s \in (d-1,d)$.
• The 'if' direction of the characterization holds for all $s \in (0,d)$, integer or not, and is established via a standard energy estimate argument.
• The 'only if' direction is proven via a blow-up argument involving a density drop in a carefully chosen dyadic shell, leading to a contradiction if the Wolff energy condition fails.
• The result implies that a compact set $E \subset \mathbb{R}^d$ is removable for locally Lipschitz solutions of $(-\Delta)^{\alpha/2}u = 0$ if and only if the capacity $\gamma_s(E) = 0$, where $s = d - \alpha$, and this capacity is defined via the Wolff energy.
• The paper shows that the classical removability theory for Lipschitz harmonic functions does not extend to the fractional Laplacian setting, as the condition for removability is fundamentally different.
• The authors establish that for $s \in (d-1,d)$, any measure $\mu$ with $\mathcal{H}^s(\operatorname{supp}(\mu)) < \infty$ and bounded $s$-Riesz transform must be the zero measure, a result that strengthens earlier findings in the field.