The minimal operator and the geometric maximal operator in ℝⁿ

• Autorzy: David Cruz-Uribe, SFO
• Języki publikacji: EN

Abstrakty

EN

We prove two-weight norm inequalities in ℝⁿ for the minimal operator
$𝗆 f(x) = inf_{Q∋x} 1/|Q| ∫_{Q} |f|dy$,
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
$M₀f(x) = {sup}_{Q∋x} exp(1/|Q| ∫_{Q} log|f|dx)$,
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 $M₀*f = lim_{r→0} M(|f|^{r})^{1/r}$.

Kategorie tematyczne

• 42B25: Maximal functions, Littlewood-Paley theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2001
• Tom: 144
• Numer: 1
• Strony: 1-37

Daty

wydano

2001

Twórcy

autor

David Cruz-Uribe, SFO

• Department of Mathematics, Trinity College, Hartford, CT 06106-3100, U.S.A.

Identyfikatory

DOI

10.4064/sm144-1-1