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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]
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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

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