The Multilinear Strong Maximal Function

Grafakos, Loukas; Liu, Liguang; Pérez, Carlos; Torres, Rodolfo H.

doi:10.1007/s12220-010-9174-8

Cited by 45 publications

(58 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is here that we use condition A on the weight u (see inequality (3.1)). We begin with a lemma from [9].…”

Section: Two Weight Bump Conditionsmentioning

confidence: 99%

“…We will now use a selection process from [9] (see also [13]). In [9] the selection process was calculated for the basis R but the same process works for a general basis.…”

Section: It Easily Follows Thatmentioning

confidence: 99%

“…In [9] the selection process was calculated for the basis R but the same process works for a general basis.…”

Section: It Easily Follows Thatmentioning

confidence: 99%

See 2 more Smart Citations

On the harmonic and geometric maximal operators

Duffee¹

2018

Mathematical Inequalities &Amp; Applications

View full text Add to dashboard Cite

In this dissertation, we attempt to characterize the boundedness of two operators over certain spaces of functions. The operators in question, the harmonic maximal operator and the geometric maximal operator, arise naturally from consideration of the paradigmatic maximal operator, the Hardy-Littlewood maximal function, and from consideration of well-known analogues of the simple arithmetic mean. In particular, we seek to improve upon earlier work by removing certain unwieldy assumptions, thus moving closer to a complete characterization of the L p -boundedness of both our operators for a pair of weights.This work primarily concerns weighted norm inequalities for two operators, the harmonic maximal operator and geometric maximal operator, given either a dyadic basis or a general basis. For any basis of cubes Q, we define those operators as follows. The harmonic maximal operator,and the geometric maximal operatorWe begin by considering our two operators restricted to the dyadic basis. The geometry of this basis allows us to decompose and recompose integrals of our operators taken over R n , a kind of discretization that simplifies the methods necessary for characterization. Within this framework we are able to remove the technical obstacles that ii lead to the unwieldy doubling conditions found in previous work.Having achieved a complete characterization in our well-behaved dyadic basis, we turn to a more difficult situation: that of a general basis. The principles necessary to treat our maximal operators in the standard basis of cubes or the dyadic basis can be abstracted further, leading to more general results that apply to a large extent to any measure regardless of its geometry.We conclude by investigating the necessary geometric theory to recover the standard basis results from our dyadic results.iii

show abstract

“…It is here that we use condition A on the weight u (see inequality (3.1)). We begin with a lemma from [9].…”

Section: Two Weight Bump Conditionsmentioning

confidence: 99%

“…We will now use a selection process from [9] (see also [13]). In [9] the selection process was calculated for the basis R but the same process works for a general basis.…”

Section: It Easily Follows Thatmentioning

confidence: 99%

See 1 more Smart Citation

On the harmonic and geometric maximal operators

Duffee¹

2018

Mathematical Inequalities &Amp; Applications

View full text Add to dashboard Cite

show abstract

“…The following definition concerns the concept of α-scattered families, which was considered in the works [17] and [16] and implicitly in [2] and [3]. The proof of the following lemma is in [16, p. 370, Lemma 1.5]; see also [14].…”

Section: Proof Of the Strong Type Estimate In The (M + 1)-weight Casementioning

confidence: 99%

A $B_p$ condition for the strong maximal function

Liu

Luque

2014

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. A strong version of the Orlicz maximal operator is introduced and a natural Bp condition for the rectangle case is defined to characterize its boundedness. This fact let us to describe a sufficient condition for the two weight inequalities of the strong maximal function in terms of power and logarithmic bumps. Results for the multilinear version of this operator and for others multi(sub)linear maximal functions associated with bases of open sets are also studied.

show abstract

“…It turns out that this operator can be used to characterize the class of multiple A p weights introduced in [32] suitably modified for rectangles, see [21]. Here, we will be concerned with endpoint boundedness properties of M R .…”

Section: The Multilinear Strong Maximal Functionmentioning

confidence: 99%