[Paper Review] Calderon-Zygmund capacities and Wolff potentials on Cantor sets

Xavier Tolsa|arXiv (Cornell University)|Jan 18, 2010

Analytic and geometric function theory2 references4 citations

TL;DR

This paper establishes that for certain Cantor sets in $\mathbb{R}^d$, the $s$-dimensional Calderón-Zygmund capacity $\gamma_s$ is comparable to the Wolff capacity $\dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}$, providing a key link between potential theory and singular integrals. The result supports a broader open conjecture on the comparability of these capacities for all compact sets when $s \in (0,d)$ is non-integer.

ABSTRACT

We show that, for some Cantor sets in R^d, the capacity g_s associated to the s-dimensional Riesz kernel x/|x|^{s+1} is comparable to the capacity C_{2(d-s)/3,3/2} from non linear potential theory. It is an open problem to show that, when s is positive and non integer, they are comparable for all compact sets in R^d. We also discuss other open questions in the area.

Motivation & Objective

• To investigate the relationship between $s$-dimensional Calderón-Zygmund capacities $\gamma_s$ and Wolff capacities $\dot{C}_{\alpha,p}$ for compact sets in $\mathbb{R}^d$.
• To determine whether $\gamma_s(E) \lesssim \dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}(E)$ holds for all compact sets $E \subset \mathbb{R}^d$ when $s \in (0,d)$ is non-integer.
• To explore open problems in the theory of Riesz transforms, capacity comparability, and invariance under bilipschitz and affine maps.
• To examine the role of square functions $Q^s_\mu$ in characterizing $L^2(\mu)$ boundedness of Riesz transforms and its connection to rectifiability.

Proposed method

• Uses the definition of $\gamma_s(E)$ as the supremum of $|\langle T,1\rangle|$ over distributions $T$ supported on $E$ with $\|R^s(T)\|_{L^\infty} \leq 1$.
• Applies the Wolff potential characterization $\dot{C}_{\alpha,p}(E) \approx \sup_\mu \mu(E)$, where $\mu$ satisfies $\dot{W}^\mu_{\alpha,p}(x) \leq 1$ for all $x \in E$.
• Analyzes the $s$-Riesz transform $R^s_\mu(f) = \int K^s(y-x) f(y)\,d\mu(y)$ with $K^s(x) = x/|x|^{s+1}$, and its truncated version $R^s_{\mu,\varepsilon}$.
• Introduces the square function $Q^s_\mu(f)(x) = \left(\sum_{j\in\mathbb{Z}} |R^s_j\mu(f)(x)|^2\right)^{1/2}$ to study $L^2(\mu)$ boundedness of $R^s_\mu$.
• Employs techniques from harmonic analysis, including the $T(1)$ theorem and symmetrization of kernels, to compare $\gamma_s$ and $\dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}$.
• Considers the case of Cantor sets to construct explicit examples where $\gamma_s \approx \dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}$, leveraging self-similarity and energy estimates.

Experimental results

Research questions

• RQ1 Is $\gamma_s(E) \lesssim \dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}(E)$ true for all compact sets $E \subset \mathbb{R}^d$ when $s \in (0,d)$ is non-integer?
• RQ2 Does the $L^2(\mu)$ boundedness of the Riesz transform $R^s_\mu$ imply the boundedness of the associated square function $Q^s_\mu$ for non-integer $s$?
• RQ3 Is the $L^2(\mu)$ boundedness of $R^s_\mu$ preserved under bilipschitz or affine maps when $d > 2$?
• RQ4 For $s > 1$, is the capacity $\widetilde{\gamma}_s(E)$, defined via component-wise $L^\infty$ bounds on $R^s_{(j)}\nu$, comparable to $\gamma_s(E)$?
• RQ5 Does the $L^2(\mu)$ boundedness of $d-1$ components of the $s$-Riesz transform imply the boundedness of the full vector-valued transform?

Key findings

• For certain Cantor sets in $\mathbb{R}^d$, the $s$-dimensional Calderón-Zygmund capacity $\gamma_s$ is comparable to the Wolff capacity $\dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}$.
• The inequality $\gamma_s(E) \gtrsim \dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}(E)$ holds for all compact sets $E \subset \mathbb{R}^d$ and all $0 < s < d$.
• The reverse inequality $\gamma_s(E) \lesssim \dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}(E)$ fails when $s$ is an integer, as shown by examples in $s$-planes with positive $s$-measure.
• When $0 < s < 1$, the comparability $\gamma_s \approx \dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}$ holds for all compact sets, as established in [MPV05].
• For $s = 1$ and $d = 2$, the result follows from curvature methods and the $T(1)$ theorem, yielding $\gamma_1(E) \gtrsim \dot{C}_{\frac{2}{3},\frac{3}{2}}(E)$.
• The open problem remains whether $\gamma_s(E) \lesssim \dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}(E)$ holds for all compact sets when $s \in (0,d)$ is non-integer.