Abstract
In this article we give some new necessary conditions for subsets of the unit circle to give collections of rectangles (by means of orientations) which differentiate Lp-functions or give Hardy-Littlewood type maximal functions which are bounded on Lp, p>1. This is done by proving that a well-known method, the construction of a Perron Tree, can be applied to a larger collection of subsets of the unit circle than was earlier known. As applications, we prove a partial converse of a well-known result of Nagel et al. [6] regarding boundedness of maximal functions with respect to rectangles of lacunary directions, and prove a result regarding the cardinality of subsets of arithmetic progressions in sets of the type described above.
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Communicated by Fernando Soria
Acknowledgements and Notes. This research was partially supported by NSERC.
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Hare, K.E., Rönning, JO. Applications of generalized perron trees to maximal functions and density bases. The Journal of Fourier Analysis and Applications 4, 215–227 (1998). https://doi.org/10.1007/BF02475990
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DOI: https://doi.org/10.1007/BF02475990