The maximal theorem for weighted grand Lebesgue spaces

Alberto Fiorenza; Babita Gupta; Pankaj Jain

Alberto Fiorenza ; Babita Gupta ; Pankaj Jain

Studia Mathematica, Tome 187 (2008), p. 123-133 / Harvested from The Polish Digital Mathematics Library

Résumé

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) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality ${| | M f | |}_{p), w} {\leq c | | f | |}_{p), w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_{p}$ , and therefore iff ${| | M f | |}_{p, w} {\leq c | | f | |}_{p, w}$ for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces.

Publié le : 2008-01-01

Zbl 1161.42011

EUDML-ID : urn:eudml:doc:285100

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     title = {The maximal theorem for weighted grand Lebesgue spaces},
     journal = {Studia Mathematica},
     volume = {187},
     year = {2008},
     pages = {123-133},
     zbl = {1161.42011},
     language = {en},
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Alberto Fiorenza; Babita Gupta; Pankaj Jain. The maximal theorem for weighted grand Lebesgue spaces. Studia Mathematica, Tome 187 (2008) pp. 123-133. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-2-2/