A remark on a maximal function over a Cantor set of directions
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LetMe0 be the maximal operator over segments of length 1 with directions belonging to a Cantor set. It has been conjectured that this operator is bounded onL2. We consider a sequence of operators over finite sets of directions converging toMe0. We improve the previous estimate for the (L2,L2)-norm of these particular operators. We also prove thatMe0 is bounded from some subsets ofL2 toL2. These subsets are composed of positive functions whose Fourier transforms have a very weak decay or are supported in a vertical strip.
1980 Mathematics Subject Classification (1985 Revision)42B25
key words and phrasesMaximal Operators
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- [St1] Strömberg J. O.,Maximal Functions for Rectangles with Given Directions Thesis, Mittag-Leffler Institute, Djursholm, Suecia, 1976.Google Scholar