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AN OPTIMAL REPRESENTATION FORMULA FOR CARNOT–CARATHÉODORY VECTOR FIELDS

Published online by Cambridge University Press:  01 November 1998

GUOZHEN LU  and
RICHARD L. WHEEDEN
GUOZHEN LU
Affiliation:
Department of Mathematics, Wright State University, Dayton, OH 45435, USA
RICHARD L. WHEEDEN
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA
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Abstract

The simplest example of the sort of representation formula that we shall study is the following familiar inequality for a smooth, real-valued function f(x) defined on a ball B in N-dimensional Euclidean space IRN:

formula here

where ∇f denotes the gradient of f, fB is the average [mid ]B[mid ]−1∫ Bf(y)dy, [mid ]B[mid ] is the Lebesgue measure of B, and C is a constant which is independent of f, x and B. This formula can be found, for example, in [4] and [12]; see also the closely related estimates in [20, pp. 228{231]. Indeed, such a formula holds in any bounded convex domain.

Type
Notes and Papers
Information
Bulletin of the London Mathematical Society , Volume 30 , Issue 6 , November 1998 , pp. 578 - 584
Copyright
© The London Mathematical Society 1998

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