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Calderón-Zygmund capacities and Wolff potentials on Cantor sets. (English) Zbl 1215.42024
The author shows that, for some Cantor sets in \(\mathbb R^d\), the capacity \(\gamma_s\) associated with the \(s\)-dimensional Riesz kernel \(x/|x|^{s+1}\) is comparable to the capacity \(\dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}\) 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 \(\mathbb R^d\). There are also discussed other open problems in connection with Riesz transforms and Wolff potentials.

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
31C45 Other generalizations (nonlinear potential theory, etc.)
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