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We study differentiability properties of Riesz potentials of finite Borelmeasures in dimension d larger than 2. The Riesz kernel has homogeneity 2-d. Indimension 2 we consider logarithmic potentials. We introduce a notion ofdifferentiability in the capacity sense, capacity being Newtonian capacity indimension larger than 2 and Wiener capacity in the plane. It turns out thatdifferentiability in the capacity sense at a point is related to the existenceof principal values of the measure with respect to the vector valued Rieszpotential x/|x|^d of homogeneity 1-d. This leads to Calder\'on-Zygmund theoryfor non-doubling measures. We prove that the Riesz potential of a finite Borelmeasure is differentiable in the capacity sense except for a set of zeroC^1-harmonic capacity. This result is sharp. Surprisingly in the plane thereare two distinct notions of differentiability in the capacity sense. For eachof them we obtain the best possible result on the size of the exceptional setin terms of Hausdorff measures. We obtain for dimension larger than 2 resultson Peano second order differentiability in the capacity sense with exceptionalsets of zero Lebesgue measure. Finally, we apply our results to find a newproof of the fact that equilibrium measure is singular with respect to Lebesguemeasure.

Differentiability properties of Riesz potentials of finite measures and non-doubling Calder\'on-Zygmund theory