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A NOTE ON A MAXIMAL FUNCTION OVER ARBITRARY SETS OF DIRECTIONS

Published online by Cambridge University Press:  01 January 2000

MARÍA C. PEREYRA  and
ANA VARGAS
MARÍA C. PEREYRA
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA
ANA VARGAS
Affiliation:
Departmento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
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Abstract

Let [Mscr ]S be the universal maximal operator over unit vectors of arbitrary directions. This operator is not bounded in L2(R2). We consider a sequence of operators over sets of finite equidistributed directions converging to [Mscr ]S. We provide a new proof of N. Katz's bound for such operators. As a corollary, we deduce that [Mscr ]S is bounded from some subsets of L2 to L2. These subsets are composed of positive functions whose Fourier transforms have a logarithmic decay or which are supported on a disc.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 32 , Issue 1 , January 2000 , pp. 71 - 74
Copyright
© The London Mathematical Society 2000

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