Premium
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.