scholarly journals Modular inequalities for the maximal operator in variable Lebesgue spaces

2018 ◽  
Vol 177 ◽  
pp. 299-311 ◽  
Author(s):  
David Cruz-Uribe ◽  
Giovanni Di Fratta ◽  
Alberto Fiorenza
Keyword(s):  
Maximal Operator ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Modular Inequalities

scholarly journals Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Joaquín Motos ◽  
María Jesús Planells ◽  
César F. Talavera
Keyword(s):  
Analytic Functions ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Previous Article ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Complemented Subspace ◽  
Open Set ◽  
Hörmander Spaces ◽  
Littlewood Maximal Operator

We show that the dual Bp·locΩ′ of the variable exponent Hörmander space Bp(·)loc(Ω) is isomorphic to the Hörmander space B∞c(Ω) (when the exponent p(·) satisfies the conditions 0<p-≤p+≤1, the Hardy-Littlewood maximal operator M is bounded on Lp(·)/p0 for some 0<p0<p- and Ω is an open set in Rn) and that the Fréchet envelope of Bp(·)loc(Ω) is the space B1loc(Ω). Our proofs rely heavily on the properties of the Banach envelopes of the p0-Banach local spaces of Bp(·)loc(Ω) and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for p(·)≡p, 0<p<1, are also given (e.g., all quasi-Banach subspace of Bploc(Ω) is isomorphic to a subspace of lp, or l∞ is not isomorphic to a complemented subspace of the Shapiro space hp-). Finally, some questions are proposed.

The maximal operator on weighted variable Lebesgue spaces

2011 ◽  
Vol 14 (3) ◽  
Author(s):  
David Cruz-Uribe ◽  
Lars Diening ◽  
Peter Hästö
Keyword(s):  
Maximal Operator ◽  
Variable Exponent ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lebesgue Spaces ◽  
Hölder Continuous ◽  
Variable Lebesgue Spaces ◽  
Muckenhoupt Condition ◽  
Necessary And Sufficient ◽  
Weighted Variable

AbstractWe study the boundedness of the maximal operator on the weighted variable exponent Lebesgue spaces L ωp(·) (Ω). For a given log-Hölder continuous exponent p with 1 < inf p ⩽ supp < ∞ we present a necessary and sufficient condition on the weight ω for the boundedness of M. This condition is a generalization of the classical Muckenhoupt condition.

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