We consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on $L^2$ if the Cantor set has positive Hausdorff dimension.

Keywords:

maximal functions, Cantor set, lacunary set maximal functions, Cantor set, lacunary set

MSC Classifications:

42B25, 43A46 show english descriptions Maximal functions, Littlewood-Paley theory
Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) 42B25 - Maximal functions, Littlewood-Paley theory
43A46 - Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)

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