Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-12-01T01:30:30.672Z Has data issue: false hasContentIssue false
  • English
  • Français

SHARP GEOMETRIC POINCARÉ INEQUALITIES FOR VECTOR FIELDS AND NON-DOUBLING MEASURES

Published online by Cambridge University Press:  20 August 2001

BRUNO FRANCHI ,
CARLOS PÉREZ  and
RICHARD L. WHEEDEN
BRUNO FRANCHI
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta, S. Donato 5 I-40127, Bologna, Italy, [email protected]
CARLOS PÉREZ
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain, [email protected]
RICHARD L. WHEEDEN
Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ [email protected]
Get access

Abstract

We derive Sobolev--Poincaré inequalities that estimate the $L^q(d\mu)$ norm of a function on a metric ball when $\mu$ is an arbitrary Borel measure. The estimate is in terms of the $L^1(d\nu)$ norm on the ball of a vector field gradient of the function, where $d\nu/dx$ is a power of a fractional maximal function of $\mu$. We show that the estimates are sharp in several senses, and we derive isoperimetric inequalities as corollaries. 1991 Mathematics Subject Classification: 46E35, 42B25.

Keywords

Sobolev-Poincaré inequalitiesfractional maximal functionsisoperimetric estimatesvector field gradientsnon-doubling measuresweight functions
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 80 , Issue 3 , May 2000 , pp. 665 - 689
Copyright
1999 London Mathematical Society

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.)