, Volume 44, Issue 2, pp 273-282

A remark on a maximal function over a Cantor set of directions

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Abstract

LetM e 0 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 e 0 . We improve the previous estimate for the (L 2,L 2)-norm of these particular operators. We also prove thatM e 0 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.

Partially supported by Spanish DGICYT grant no. PB90-0187.