Cartan type estimates for Riesz transforms
by
Vladimir Eiderman
Michigan State University
Coauthors: Fedor Nazarov and Alexander Volberg

For given complex numbers n₁, ..., n_N, and points y₁, ..., y_N, in R^d,  d ≥ 2, we define the set

Zs(P, n)={x ∈ Rd:| N å j=1  x-yj |x-yj|s+1  nj| > P},     s > 0,   P > 0.

Our goal is to give sharp upper bounds for the size of Z^s(P, n). This size will be measured by the Hausdorff content M_h with various gauge functions h. Among other things, we shall characterize all h for which the estimates do not blow up as N→∞. In this case a routine limiting arguments allow us to extend our bounds to s-Riesz transforms of all finite Borel measures.

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Date received: February 11, 2008