Riesz transforms of non-integer homogeneity on uniformly disconnected sets (Q2796085)

\textit{J. Mateu} et al. [J. Reine Angew. Math. 578, 201--223 (2005; Zbl 1086.31005)] proved precise estimates for the \(L^p\) norm of the \(s\)-dimensional Riesz transforms of measures \(\mu\) supported on any arbitrary compact set in \({\mathbb R}^d\), where \(0<s<1\). \textit{J. Mateu} and \textit{X. Tolsa} [Proc. Lond. Math. Soc., III. Ser. 89, No. 3, 676--696 (2004; Zbl 1089.42009)] and \textit{X. Tolsa} [J. Geom. Anal. 21, No. 1, 195--223 (2011; Zbl 1215.42024)] studied the case \(0<s<d\), where \(\mu\) is a probability measure supported on sets which are some kind of high-dimensional variants of the \(1/4\) planar Cantor set. The authors consider very general Cantor sets. As a corollary, the capacity \(\gamma_s\) associated with the \(s\)-dimensional Riesz kernel is comparable to the capacity \(\dot{C}_{\frac{2}{3}(d-s), \frac{3}{2}}\) from non-linear potential theory.NEWLINENEWLINEFor \(0<s<d\) and \(x \in {\mathbb R}^d\), let NEWLINE\[NEWLINE K^s (x) = \frac{x}{ | x |^{s+1}} \quad \quad \text{and} \quad \quad R^s \mu (x) = \int K^s (x-y) \, d\mu (y). NEWLINE\]NEWLINE They define very general Cantor sets \(E\) by some algorithm. They denote by \(\mathcal{D}\) the family of all the cubes \(Q_i^k\) in the construction of \(E\). Given a measure \(\mu\) supported on \(E\) and a cube \(Q \in \mathcal{D}\), the \(s\)-dimensional density of \(\mu\) on \(Q\) is defined by NEWLINE\[NEWLINE \Theta_{\mu}^s (Q) = \frac{ \mu (Q)}{ \ell (Q)^s}. NEWLINE\]NEWLINE They prove the following: let \( d-1 < s < d\). If \, \( \sup_{Q \in \mathcal{D}} \Theta_{\mu}^s (Q) \leq C\), then NEWLINE\[NEWLINE \| R^s \mu \|_{L^2(\mu)}^2 \approx \sum_{Q \in \mathcal{D}} \Theta_{\mu}^s (Q)^2 \mu (Q). NEWLINE\]NEWLINENEWLINENEWLINEAs a corollary they prove the following result. Given a compact set \(F \subset {\mathbb R}^d\), the \(s\)-dimensional Calderón-Zygmund capacity is defined by NEWLINE\[NEWLINE \gamma_s(F) = \sup | \langle T,1 \rangle |, NEWLINE\]NEWLINE where the supremum runs over all distributions \(T\) supported on \(F\) such that NEWLINE\[NEWLINE \| R^s(T) \|_{L^{\infty}({\mathbb R}^d)} \leq 1. NEWLINE\]NEWLINE The capacity \(\dot{C}_{\alpha, p}\) of \(F\) is defined by NEWLINE\[NEWLINE \dot{C}_{\alpha, p}(F) = \sup \mu (F)^p, NEWLINE\]NEWLINE where the supremum is taken over all positive measures \(\mu\) supported on \(F\) such that NEWLINE\[NEWLINE I_{\alpha}(\mu)(x) = \int\frac{1}{ | x-y |^{d - \alpha}} d\mu(y) NEWLINE\]NEWLINE satisfies \(\| I_{\alpha} (\mu)\|_{p'} \leq 1\). If \(d-1 < s <d\), then NEWLINE\[NEWLINE \gamma_s(F) \approx \dot{C}_{\frac{2}{3}(d-s), \frac{3}{2}}(F) NEWLINE\]NEWLINE for all compact uniformly disconnected sets.NEWLINENEWLINENote that the class of Cantor sets considered above coincides with the class of compact uniformly disconnected sets, cf. [\textit{G. David} and \textit{S. Semmes}, Fractured fractals and broken dreams. Self-similar geometry through metric and measure. Oxford Lecture Series in Mathematics and its Applications 7. Oxford: Clarendon Press (1997; Zbl 0887.54001)].NEWLINENEWLINEMateu, Prat and Verdera proved this estimate when \(0<s<1\). \textit{V. Eiderman} et al. [Proc. Lond. Math. Soc. (3) 101, No. 3, 727--758 (2010; Zbl 1209.42008)] showed that NEWLINE\[NEWLINE \gamma_s(F) \gtrsim \dot{C}_{\frac{2}{3}(d-s), \frac{3}{2}}(F) NEWLINE\]NEWLINE for all compact sets, where \(0<s<d\).