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Differentiation of Besov spaces and the Nikodym maximal operator


Author: Jason Murcko
Journal: Proc. Amer. Math. Soc. 145 (2017), 2139-2153
MSC (2010): Primary 42B25, 42B35
DOI: https://doi.org/10.1090/proc/13396
Published electronically: December 30, 2016
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Abstract: We study several questions related to differentiation of integrals for Besov spaces relative to the basis $ \mathcal {R}$ of arbitrarily oriented rectangular parallelepipeds in $ \mathbb{R}^{d}$, $ d \geq 2$. We improve on positive and negative differentiation results of Aimar, Forzani, and Naibo and on capacitary and dimensional bounds for exceptional sets of Naibo. Our main tool in obtaining these improvements involves showing that bounds for the Nikodym maximal operator can be used to deduce boundedness properties of the local maximal operator associated to $ \mathcal {R}$.


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Additional Information

Jason Murcko
Affiliation: Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: jmurcko@gmail.com

DOI: https://doi.org/10.1090/proc/13396
Received by editor(s): July 11, 2016
Published electronically: December 30, 2016
Additional Notes: The author would like to thank his advisor, Andreas Seeger, for his guidance and support
The author was supported in part by the National Science Foundation.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society