Abstract
We study maximal averages associated with singular measures on . Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension with for which the corresponding maximal operators are bounded on for . As a consequence, we are able to answer a question of Aversa and Preiss on density and differentiation theorems for singular measures in one dimension. Our proof combines probabilistic techniques with the methods developed in multidimensional Euclidean harmonic analysis; in particular, there are strong similarities to Bourgain's proof of the circular maximal theorem in two dimensions.
Citation
Malabika Pramanik. Izabella Łaba. "Maximal operators and differentiation theorems for sparse sets." Duke Math. J. 158 (3) 347 - 411, 15 June 2011. https://doi.org/10.1215/00127094-1345644
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