The minimal operator and the geometric maximal operator in Rⁿ

Abstract

Abstract. We prove two-weight norm inequalities in R n for the minimal operatorextending to higher dimensions results obtained by Cruz-Uribe, Neugebauer and Olesen [8] on the real line. As an application we extend to R n weighted norm inequalities for the geometric maximal operatorproved 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 = lim r→0 M (|f | r )

Weighted integral inequalities for the maximal geometric mean operator in martingales

“…Recently, Refs. [2][3][4] and [1] also considered the minimal operator m and the maximal geometric mean operator G.…”

On the harmonic and geometric maximal operators

“…The smallest such C will be called the doubling constant of µ and denoted d(µ). We can now state the higher dimensional results for M −1 and M 0 both due to Cruz-Uribe [3]. …”

“…In Chapter 2, we follow CruzUribe's suggested method in [3] by refining earlier results by restricting our basis to that of dyadic cubes, and we achieve the first results for the harmonic maximal operator in higher dimensions without doubling. The dyadic basis allows a clean decomposition that removes the necessity of the doubling condition on our weight.…”

“…However, as Cruz-Uribe notes in [3], the minimal operator is simply the harmonic maximal operator in disguise:…”

“…The results in this chapter proceed from the method suggested by Cruz-Uribe in [3], with an alteration of proof in the strong-type inequality's equivalence with our testing condition.…”