The maximal theorem for weighted grand Lebesgue spaces

Tom 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

Streszczenie

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.

Autorzy

  • Alberto FiorenzaDipartimento 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
    e-mail
  • Babita GuptaDepartment of Mathematics
    Shivaji College
    University of Delhi
    Raja Garden, Delhi 110027, India
    e-mail
  • Pankaj JainDepartment of Mathematics
    Deshbandhu College
    University of Delhi
    Kalkaji, New Delhi 110019, India
    e-mail

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