Hostname: page-component-5cf477f64f-xc2pj Total loading time: 0 Render date: 2025-03-29T20:35:03.572Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal operators associated to Fourier multipliers with an arbitrary set of parameters

Published online by Cambridge University Press:  14 November 2011

Javier Duoandikoetxea  and
Ana Vargas
Javier Duoandikoetxea
Affiliation:
Departamento de Matemáticas, Universidad del País Vasco, Apartado 644,48080, Bilbao, Spain E-mail: mtpduzuj@lg.ehu.es
Ana Vargas
Affiliation:
Department of Mathematics, Yale University, New Haven CT 06520, USA; Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain E-mail: ana.vargas@uam.es
Get access Rights & Permissions [Opens in a new window]

Abstract

We present here some general results of boundedness on LP for maximal operators of the form , where E is a subset of the positive real numbers and Tt is a dilation of a fixed multiplier operator. The range of values of p depends only on the decay at infinity of the multiplier and the Minkowski dimension of E. For the case being the maximal operator associated to a convex body, we prove that the norm of the operator is independent of the body.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 4 , 1998 , pp. 683 - 696
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bourgain, J.. Estimations de certaines fonctions maxiraales. C. R. Acad. Sci. Paris 301 (1985), 499–502.Google Scholar
2Bourgain, J.. On high dimensional maximal functions associated to convex bodies. Amer. J. Math. 108 (1986), 1467–76.CrossRefGoogle Scholar
3Bourgain, J.. On the LP-bounds for maximal functions associated to convex bodies in ℝn. Israel J. Math. 54 (1986), 257–65.CrossRefGoogle Scholar
4Carbery, A.. An almost-orthogonality principle with applications to maximal functions associated to convex bodies. Bull. Amer. Math. Soc. 14 (1986), 269–73.CrossRefGoogle Scholar
5Carbery, A.. Variants of the Calderón-Zygmund theory for LP-spaces. Rev. Mat. Iberoamericana 2 (1986), 381–96.CrossRefGoogle Scholar
6Duoandikoetxea, J. and Vargas, A.. Directional operators and radial functions on the plane. Ark. Mat. 33 (1995), 281–91.CrossRefGoogle Scholar
7Francia, J. L. Rubio de. Maximal functions and Fourier transforms. Duke Math. J. 53 (1986), 395–404.Google Scholar
8Seeger, A., Wainger, S. and Wright, J.. Pointwise convergence of spherical means. Math. Proc. Cambridge Philos. Soc. 118 (1995), 115–24.CrossRefGoogle Scholar
9Stein, E. M. and Weiss, G.. Introduction to Fourier Analysis in Euclidean Spaces (Princeton, NJ: Princeton University Press, 1971).Google Scholar
10Tricot, C.. Douze définitions de la densité logarithmique. C. R. Acad. Sci. Paris 293 (1981), 549–52.Google Scholar
11Wainger, S.. Special trigonometric series in k dimensions. Mem. Amer. Math. Soc. 59 (1965).Google Scholar
12Wainger, S.. Applications of Fourier transforms to averages over lower dimensional sets. In Proceedings of Symposia in Pure Mathematics, Weiss, G. and Wainger, S., eds, Vol. 35, pp. 85–94 (Providence, RI: American Mathematical Society, 1979).Google Scholar