# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## The maximal theorem for weighted grand Lebesgue spaces

### Tom 188 / 2008

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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