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Weighted norm inequalities for geometric fractional maximal operators

Authors:

David Cruz-Uribe C. J. Neugebauer V. Olesen

Affiliation:

(1) Department of Mathematics, Trinity College, Hartford, CT;(2) Department of Mathematics, Purdue University, West Lafayette, IN

Abstract:

For 0

let Tf denote one of the operators

$M_{alpha ,0} f(x) = mathop {sup }limits_{I mathrelbackepsilon x} left| I right|^alpha exp left( {frac{1}{{left| I right|}}int_I {log left| f right|} } right),M_{alpha ,0}^* f(x) = mathop {lim }limits_{r searrow 0} mathop {sup }limits_{I mathrelbackepsilon x} left| I right|^alpha left( {frac{1}{{left| I right|}}int_I {left| f right|^r } } right)^{{1 mathord{left/ {vphantom {1 r}} right. kern-nulldelimiterspace} r}} .$

We characterize the pairs of weights (u, v) for which T is a bounded operator from L^p(v) to L^q(u), 0 <p

q <

. This extends to agr

> 0 the norm inequalities for agr

=0 in [4, 16]. As an application we give lower bounds for convolutions phiv

f, where

is a radially decreasing function.

Keywords:

KeywordHeading" >Math subject classifications 42B25

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