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SELF-IMPROVING PROPERTIES OF GENERALIZED POINCARÉ TYPE INEQUALITIES THROUGH REARRANGEMENTS

ANDREI K. LERNER and CARLOS PÉREZ
Mathematica Scandinavica
Vol. 97, No. 2 (2005), pp. 217-234
Published by: Mathematica Scandinavica
Stable URL: http://www.jstor.org/stable/24493533
Page Count: 18
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SELF-IMPROVING PROPERTIES OF GENERALIZED POINCARÉ TYPE INEQUALITIES THROUGH REARRANGEMENTS
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Abstract

We prove, within the context of spaces of homogeneous type, Lp and exponential type self-improving properties for measurable functions satisfying the following Poincaré type inequality: $\underset{\mathrm{\alpha }}{\mathrm{inf}}\left(\right(\mathrm{f}-\mathrm{\alpha }\left){\mathrm{\chi }}_{\mathrm{B}}{)}_{\mathrm{\mu }}^{*}\right(\mathrm{\lambda }\mathrm{\mu }\left(\mathrm{B}\right))\le {\mathrm{c}}_{\mathrm{\lambda }}\mathrm{a}(\mathrm{B})$. Here, ${\mathrm{f}}_{\mathrm{\mu }}^{*}$ denotes the non-increasing rearrangement of f, and a is a functional acting on balls B, satisfying appropriate geometric conditions. Our main result improves the work in [11], [12] as well as [2], [3] and [14]. Our method avoids completely the "good-λ" inequality technique and any kind of representation formula.

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