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15 June 2011 Maximal operators and differentiation theorems for sparse sets
Malabika Pramanik, Izabella Łaba
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Duke Math. J. 158(3): 347-411 (15 June 2011). DOI: 10.1215/00127094-1345644

Abstract

We study maximal averages associated with singular measures on R. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension 1ε with 0ε<1/3 for which the corresponding maximal operators are bounded on Lp(R) for p>(1+ε)/(1ε). 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.

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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

Information

Published: 15 June 2011
First available in Project Euclid: 1 June 2011

zbMATH: 1242.42011
MathSciNet: MR2805064
Digital Object Identifier: 10.1215/00127094-1345644

Subjects:
Primary: 42B25
Secondary: 26A24 , 26A99 , 28A78

Rights: Copyright © 2011 Duke University Press

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Vol.158 • No. 3 • 15 June 2011
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