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

  • BRUNO FRANCHI (a1), CARLOS PÉREZ (a2) and RICHARD L. WHEEDEN (a3)

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.

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

  • BRUNO FRANCHI (a1), CARLOS PÉREZ (a2) and RICHARD L. WHEEDEN (a3)

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