Abstract
LetM 0e 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 onL 2. We consider a sequence of operators over finite sets of directions converging toM 0e . We improve the previous estimate for the (L 2,L 2)-norm of these particular operators. We also prove thatM 0e is bounded from some subsets ofL 2 toL 2. These subsets are composed of positive functions whose Fourier transforms have a very weak decay or are supported in a vertical strip.
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Partially supported by Spanish DGICYT grant no. PB90-0187.
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Vargas, A.M. A remark on a maximal function over a Cantor set of directions. Rend. Circ. Mat. Palermo 44, 273–282 (1995). https://doi.org/10.1007/BF02850835
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DOI: https://doi.org/10.1007/BF02850835