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Applications of generalized Perron trees to maximal functions and density bases. (English) Zbl 0911.42010
The authors establish some new necessary conditions for subsets of the unit circle to give collections of rectangles (by means of orientations) which differentiate $$L^p$$-functions or give Hardy-Littlewood type maximal functions which are bounded on $$L^p$$, $$p>1$$. This is done by proving that a well-known method, the construction of a Perron tree, can be applied to a large collection of subsets of the unit circle than was earlier known. As applications, the authors prove a partial converse of a well-known result in [A. Nagel, E. M. Stein and S. Wainger, Proc. Natl. Acad. Sci. USA 75, 1060-1062 (1978; Zbl 0391.42015)] 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.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42C99 Nontrigonometric harmonic analysis
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