Geodesic
The maximal theorem for weighted grand Lebesgue spaces
Alberto Fiorenza1 ; Babita Gupta2 ; Pankaj Jain3
1 Dipartimento di Costruzioni e Metodi Matematici in Architettura Università di Napoli via Monteoliveto 3 80134 Napoli, Italy and Istituto per le Applicazioni del Calcolo “Mauro Picone” Sezione di Napoli Consiglio Nazionale delle Ricerche via Pietro Castellino 111 80131 Napoli, Italy
2 Department of Mathematics Shivaji College University of Delhi Raja Garden, Delhi 110027, India
3 Department of Mathematics Deshbandhu College University of Delhi Kalkaji, New Delhi 110019, India
Studia Mathematica, Tome 188 (2008) no. 2, pp. 123-133

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval $(0,1)\subset\mathbb R$, and the maximal function is localized in $(0,1)$. Moreover, we prove that the inequality $\| Mf\|_{p),w}\le c\| f\|_{p),w}$ holds with some $c$ independent of $f$ iff $w$ belongs to the well known Muckenhoupt class $A_p$, and therefore iff $\| Mf\|_{p,w}\le c\| f\|_{p,w}$ for some $c$ independent of $f$. Some results of similar type are discussed for the case of small Lebesgue spaces.
DOI : 10.4064/sm188-2-2
Keywords: study hardy inequality derive maximal theorem hardy littlewood context grand lebesgue spaces considered underlying measure space interval subset mathbb maximal function localized moreover prove inequality holds independent belongs known muckenhoupt class therefore independent results similar type discussed small lebesgue spaces

Alberto Fiorenza 1 ; Babita Gupta 2 ; Pankaj Jain 3

1 Dipartimento di Costruzioni e Metodi Matematici in Architettura Università di Napoli via Monteoliveto 3 80134 Napoli, Italy and Istituto per le Applicazioni del Calcolo “Mauro Picone” Sezione di Napoli Consiglio Nazionale delle Ricerche via Pietro Castellino 111 80131 Napoli, Italy
2 Department of Mathematics Shivaji College University of Delhi Raja Garden, Delhi 110027, India
3 Department of Mathematics Deshbandhu College University of Delhi Kalkaji, New Delhi 110019, India 
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Alberto Fiorenza; Babita Gupta; Pankaj Jain. The maximal theorem for weighted grand Lebesgue
spaces. Studia Mathematica, Tome 188 (2008) no. 2, pp. 123-133. doi: 10.4064/sm188-2-2

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