A Note on a Maximal Function Over Arbitrary Sets of Directions | Zendy

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A Note on a Maximal Function Over Arbitrary Sets of Directions

Author(s) -

Pereyra María C.,

Vargas Ana

Publication year - 2000

Publication title -

bulletin of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 2.396

H-Index - 48

eISSN - 1469-2120

pISSN - 0024-6093

DOI - 10.1112/s0024609399006347

Subject(s) - equidistributed sequence , mathematics , bounded function , corollary , operator (biology) , sequence (biology) , logarithm , unit interval , function (biology) , uniform boundedness , discrete mathematics , pure mathematics , combinatorics , mathematical analysis , biochemistry , chemistry , genetics , repressor , evolutionary biology , biology , transcription factor , gene

Let M S be the universal maximal operator over unit vectors of arbitrary directions. This operator is not bounded in L 2 ( R 2 ). We consider a sequence of operators over sets of finite equidistributed directions converging to M S . We provide a new proof of N. Katz's bound for such operators. As a corollary, we deduce that M S is bounded from some subsets of L 2 to L 2 . These subsets are composed of positive functions whose Fourier transforms have a logarithmic decay or which are supported on a disc. 1991 Mathematics Subject Classification 42B25.

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