Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces

• Published in: Bulletin des sciences mathématiques 2005-09, Vol.129 (8), p.657-700
• Main Author: Diening, Lars
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fourier analysis; Functional analysis; Lebesgue spaces with variable exponent; Mathematical analysis; Mathematics; Maximal functions; Musielak–Orlicz spaces; Sciences and techniques of general use
• ISSN: 0007-4497; 1952-4773
• DOI: 10.1016/j.bulsci.2003.10.003

We consider the Hardy–Littlewood maximal operator M on Musielak–Orlicz Spaces L φ ( R d ) . We give a necessary condition for the continuity of M on L φ ( R d ) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces L p ( ⋅ ) ( R d ) we show that this condition is also sufficient. Moreover, we show that the condition is “left-open” in the sense that not only M but also M q is continuous for some q > 1 , where M q f = ( M ( | f | q ) ) 1 q . On considère l'opérateur maximal Hardy–Littlewood sur les espaces Musielak–Orlicz L φ ( R d ) . Une condition nécessaire est donnée pour la continuité de M sur L φ ( R d ) , qui généralise la conception des classes Muckenhaupt. Pour le cas spécial des espaces Lebesgues généralisés L p ( ⋅ ) ( R d ) on justifie que cette condition est aussi suffisante. En plus, on prouve que la condition est “left-open” au sens que non seulement M mais aussi M q est continu pour certes q > 1 , où M q f = ( M ( | f | q ) ) 1 q .