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The minimal operator and the geometric maximal operator in ℝⁿ

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Cruz-Uribe D.

Studia Mathematica

2001

tom 144

nr 1

1-37

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 $M₀*f = lim_{r→0} M(|f|^{r})^{1/r}$.

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1 Studia Mathematica

1 Cruz-Uribe D.

1 2001

Wyniki wyszukiwania

The minimal operator and the geometric maximal operator in ℝⁿ