[Paper Review] Non existence of principal values of signed Riesz transforms of non integer dimension

Aleix Ruiz de Villa, Xavier Tolsa|ArXiv.org|Dec 12, 2008

Advanced Harmonic Analysis Research1 references3 citations

TL;DR

This paper proves that the principal value of the signed Riesz transform of non-integer dimension $ s $ cannot exist for a non-zero Borel measure $ \mu $ in $ \mathbb{R}^m $ if the upper $ s $-density is bounded and the principal value exists $ \mu $-almost everywhere. The key result establishes that such existence forces $ s $ to be an integer, resolving a long-standing question in geometric measure theory using a novel analytic approach based on a specially constructed cutoff function and density estimates.

ABSTRACT

In this paper we prove that, given s> 0, if E is a subset of R^m with positive and bounded s-dimensional Hausdorff measure H^s and the principal values of the s-dimensional signed Riesz transform of H^s|E exist H^s-almost everywhere in E, then s is integer. Other more general variants of this result are also proven.

Motivation & Objective

• To determine whether the principal value of the signed Riesz transform can exist for non-integer dimensions $ s \in (n, n+1) $, where $ n \in \mathbb{Z} $, under bounded upper $ s $-density conditions.
• To establish that the existence of the principal value of the Riesz transform for $ \mu $-a.e. $ x \in \mathbb{R}^m $, with $ 0 < \theta^{s,*}_{\mu}(x) < \infty $, implies $ s \in \mathbb{Z} $.
• To extend previous results relying on tangent measures by introducing a new method that avoids the need for lower density assumptions, using a $ C^2 $ cutoff function and energy-type estimates.
• To prove that for a set $ E \subset \mathbb{R}^m $ with $ 0 < H^s(E) < \infty $, the existence of the principal value of the Riesz transform $ H^s|_E $-a.e. on $ E $ implies $ s \in \mathbb{Z} $.

Proposed method

• Construct a smooth, compactly supported cutoff function $ \varphi \in C^2(0,\infty) $ with specific behavior: $ \varphi(r) = r^{(s+1)/2} $ for small $ r $, linear decay for intermediate $ r $, and compact support, to control the Riesz kernel near the origin.
• Define a modified Riesz transform $ R^{s}_{\varphi,\varepsilon}\mu(x) = \int \varphi\left(\frac{|x-y|^2}{\varepsilon^2}\right) \frac{x-y}{|x-y|^{s+1}} d\mu(y) $, which regularizes the singular kernel and allows for $ \varepsilon $-dependent energy estimates.
• Use a density point argument: for $ x_0 \in F_\delta $, a set of $ \mu $-density points with bounded upper density, construct $ n+2 $ points in a ball around $ x_0 $ with controlled separation from lower-dimensional affine spans.
• Establish a lower bound on the $ L^1 $-norm of the modified Riesz transform at the origin, $ |U^\varepsilon(0)| \gtrsim \frac{\theta^s(\varepsilon)}{\varepsilon} $, using the geometry of the point configuration and the properties of $ \varphi $.
• Derive an upper bound on $ |U^\varepsilon(0)| $ via the Lipschitz continuity of $ R^{s}_{\varphi,\varepsilon}\mu $ on $ F_\delta $, controlled by $ \delta $, and relate it to the density $ \theta^s(\varepsilon) $.
• Derive a contradiction by showing $ \theta^s(\varepsilon)r \lesssim \tau \theta^s(\varepsilon)r $ for arbitrarily small $ \tau $, which forces $ s \in \mathbb{Z} $.

Experimental results

Research questions

• RQ1 Can the principal value of the signed Riesz transform exist for a non-integer dimension $ s \in (n, n+1) $ when the upper $ s $-density is bounded?
• RQ2 Does the existence of the principal value of the Riesz transform $ \mu $-a.e. imply that the dimension $ s $ must be an integer?
• RQ3 Can the classical tangent measure approach be bypassed to prove the integrality of $ s $ without assuming a positive lower density?
• RQ4 What is the relationship between the existence of the principal value of the Riesz transform and the rectifiability of the underlying set $ E $ with finite $ H^s $-measure?

Key findings

• The principal value of the signed Riesz transform of non-integer dimension $ s \in (n, n+1) $ cannot exist $ \mu $-a.e. if $ 0 < \theta^{s,*}_{\mu}(x) < \infty $ for $ \mu $-a.e. $ x $.
• For a Borel measure $ \mu $, if the principal value $ \lim_{\varepsilon \to 0} R^s_\varepsilon \mu(x) $ exists $ \mu $-a.e. and $ \theta^{s,*}_{\mu}(x) < \infty $, then $ s \in \mathbb{Z} $.
• For a set $ E \subset \mathbb{R}^m $ with $ 0 < H^s(E) < \infty $, the existence of the principal value of the Riesz transform $ H^s|_E $-a.e. on $ E $ implies $ s \in \mathbb{Z} $.
• The proof relies on a novel construction of a $ C^2 $ cutoff function $ \varphi $, which allows for precise control of the Riesz kernel and leads to a contradiction when $ s \notin \mathbb{Z} $.
• The contradiction arises from comparing a lower bound on the modified Riesz transform $ |U^\varepsilon(0)| \gtrsim \frac{\theta^s(\varepsilon)}{\varepsilon} $ with an upper bound of order $ \delta \cdot \theta^s(\varepsilon) $, where $ \delta \to 0 $, forcing $ s \in \mathbb{Z} $.