The maximal theorem for weighted grand Lebesgue spaces

Volume 188 / 2008

Alberto Fiorenza, Babita Gupta, Pankaj Jain Studia Mathematica 188 (2008), 123-133 MSC: Primary 42B25; Secondary 46E30, 26D15. DOI: 10.4064/sm188-2-2


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.


  • Alberto FiorenzaDipartimento di Costruzioni
    e Metodi Matematici in Architettura
    Università di Napoli
    via Monteoliveto 3
    80134 Napoli, Italy
    Istituto per le Applicazioni
    del Calcolo “Mauro Picone”
    Sezione di Napoli
    Consiglio Nazionale delle Ricerche
    via Pietro Castellino 111
    80131 Napoli, Italy
  • Babita GuptaDepartment of Mathematics
    Shivaji College
    University of Delhi
    Raja Garden, Delhi 110027, India
  • Pankaj JainDepartment of Mathematics
    Deshbandhu College
    University of Delhi
    Kalkaji, New Delhi 110019, India

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