New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

By Andrei K. Lerner, S. Ombrosi, C. P'erez, Rodolfo H. Torres, Rodrigo Trujillo-González. 2009

534 citations recorded by the source corpus.

A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller than the m-fold product of the Hardy–Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calderon–Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calderon–Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5031ea7b1bd3ed4edd09a6f0779f98b2ded99fc2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Weighted norm inequalities for multilinear fractional operators on Morrey spaces (2011, methodology): “Motivated by [8], Moen [10] established a one-weight theory and twoweight theory for the multilinear fractional integral operator Iα(~ f ) and the multilinear fractional maximal operator Mα(~ f ) on Lebesgue spaces.”
A pointwise estimate for positive dyadic shifts and some applications (2014, background): “Observe that, since γQ ≤ 1 for all Q, the operator G is pointwise bounded by the multi-(sub)linear maximal operator introduced in [18] (we have localized the operator in the obvious way): MP ~ f(x) = sup { k ∏”
Compactness properties of commutators of bilinear fractional integrals (2013, methodology): “[17] A. Lerner, S. Ombrosi, C. Pérez, R.H. Torres, and R. Trujillo-González,New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv.”
Multilinear analysis on metric spaces (2014, methodology): “One of the main motivations for the results in this section comes from the article by Lerner, Ombrosi, Pérez, Torres, and Trujillo-González [72] where a very natural multiple-weight theory adapted to the multilinear Calderón–Zygmund theory was developed.”
On Multilinear Fourier Multipliers of Limited Smoothness (2013, unknown): “These advances are too numerous all to be included in this introduction, so we only mention the articles of Bényi and Torres [1], Grafakos and Torres [12], Kenig and Stein [16], and Lerner, Ombrosi, Pérez, Torres, and Trujillo-González [17].”
Commutators of multilinear Calderón-Zygmund operators with kernels of Dini's type and applications (2019, unknown): “[16] developed a multiple weight theory that adapts to multilinear Calderón-Zygmund operators.”
The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators (2012, background): “In [18], it was shown that for 1 < p1, · · · , pm < ∞, w ∈ A P if and only if w 1−p′ i i ∈ Amp′ i and v w ∈ Amp.”
Compact Bilinear Operators and Commutators (2013, unknown): “By the results in [12], [17], [9], and [11], these operators map L × L → L with 1/p+ 1/q = 1/r for all 1 < p, q < ∞, with estimates of the form (1) ‖[T, b]1(f, g)‖Lr , ‖[T, b]2(f, g)‖Lr ‖b‖BMO‖f‖Lp‖g‖Lq ,”
The Multilinear Strong Maximal Function (2010, methodology): “Some of the motivation for our work arises from the recent article by Lerner et al [18] where the multisublinear maximal function (1.1) M(f1, . . . , fm)(x) = sup Q3x Q cube m∏ i=1 1 |Q| ∫ Q |fi(yi)| dyi associated with cubes with sides parallel to the coordinate axes was introduced.”
Extrapolation for multilinear Muckenhoupt classes and applications (2018, background): “In this direction [27] introduced some multilinear Muckenhoupt classes.”
Weighted inequalities for multilinear fractional integral operators (2009, methodology): “When = 0 (so p = q) we recover the result from [ 14 ].”
Weighted norm inequalities for multilinear Fourier multipliers (2012, background): “For example, see Cruz-Uribe, Martell and Pérez [5], Grafakos and Torres [10], Hu and Yang [11], Lerner, Ombrosi, Pérez, Torres and Trujillo-González [14], and Maldonado and Naibo [15] for this direction.”