The minimal operator and the geometric maximal operator in ℝⁿ

David Cruz-Uribe, SFO

Studia Mathematica, Tome 147 (2001), p. 1-37 / Harvested from The Polish Digital Mathematics Library

Résumé

We prove two-weight norm inequalities in ℝⁿ for the minimal operator $f (x) = i n f_{Q ∋ x} 1 / | Q | \int_{Q} | f | d y$ , extending to higher dimensions results obtained by Cruz-Uribe, Neugebauer and Olesen [8] on the real line. As an application we extend to ℝⁿ weighted norm inequalities for the geometric maximal operator $M ₀ f (x) = {s u p}_{Q ∋ x} e x p (1 / | Q | \int_{Q} l o g | f | d x)$ , proved by Yin and Muckenhoupt [27]. We also give norm inequalities for the centered minimal operator, study powers of doubling weights and give sufficient conditions for the geometric maximal operator to be equal to the closely related limiting operator $M ₀ * f = l i m_{r \to 0} {M (| f |}^{r})^{1 / r}$ .

Publié le : 2001-01-01

Zbl 0964.42006

EUDML-ID : urn:eudml:doc:284790

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     title = {The minimal operator and the geometric maximal operator in Rn},
     journal = {Studia Mathematica},
     volume = {147},
     year = {2001},
     pages = {1-37},
     zbl = {0964.42006},
     language = {en},
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David Cruz-Uribe, SFO. The minimal operator and the geometric maximal operator in ℝⁿ. Studia Mathematica, Tome 147 (2001) pp. 1-37. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm144-1-1/