1Dipartimento 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 2Department of Mathematics Shivaji College University of Delhi Raja Garden, Delhi 110027, India 3Department of Mathematics Deshbandhu College University of Delhi Kalkaji, New Delhi 110019, India
Studia Mathematica, Tome 188 (2008) no. 2, pp. 123-133
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.
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
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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author = {Alberto Fiorenza and Babita Gupta and Pankaj Jain},
title = {The maximal theorem for weighted grand {Lebesgue
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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
