Abstract. Our aim is to give sharp upper bounds for the size of the set of points where the Riesz transform of a linear combination of N point masses is large. This size will be measured by the Hausdorff content with various gauge functions. Among other things, we shall characterize all gauge functions for which the estimates do not blow up as N tends to infinity (in this case a routine limiting argument will allow us to extend our bounds to all finite Borel measures). We also show how our techniques can be applied to estimates for certain capacities.

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In this paper, we prove that for s ∈ (1, 2) there exists no totally lower irregular finite positive Borel measure µ in R 2 with H s (supp µ) < +∞ such that Rµ L ∞ (m2) < +∞, where Rµ = µ * x |x| s+1 and m 2 is the Lebesgue measure in R 2 . Combined with known results of Prat and Vihtilä, this shows that for any noninteger s ∈ (0, 2) and any finite positive Borel measure in R 2 with H s (supp µ) < +∞, we have Rµ L ∞ (m2) = ∞.

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The aim of this paper is to estimate the L 2 -norms of vector-valued Riesz transforms R s ν and the norms of Riesz operators on Cantor sets in R d , as well as to study the distribution of values of R s ν . Namely, we show that this distribution is "uniform" in the following sense. The values of |R s ν | 2 which are comparable with its average value are attended on a "big" portion of a Cantor set. We apply these results to give examples demonstrating the sharpness of our previous estimates for the set of points where Riesz transform is large, and for the corresponding Riesz capacities. The Cantor sets under consideration are different from the usual corner Cantor sets. They are constructed by means a certain process of regularization introduced in the paper.