Let $ 0<\beta <1$ and $\Omega$ be a proper open and non-empty subset of $\mathbf{R}^n$. In this paper, the object of our investigation is the multilinear local maximal operator $\mathcal{M}_{\beta}$, defined by $$\mathcal{M}_{\beta}(\vec{f})(x)= \sup_{\substack{Q \ni x \\ Q\in{\mathcal{F}_{\beta}}}} \prod_{i=1}^m \frac{1}{|Q|} \int_Q |f_i(y_i)|{\rm d}y_i,$$ where $\mathcal{F}_{\beta}=\{Q(x,l):x \in \Omega, l< \beta {\rm d}(x, \Omega^c)\}$, $Q=Q(x,l)$ is denoted as a cube with sides parallel to the axes, and $x$ and $l$ denote its center and half its side length. Two-weight characterizations for the multilinear local maximal operator $\mathcal{M}_{\beta}$ are obtained. A formulation of the Carleson embedding theorem in the multilinear setting is proved.
Yali PAN
,
Qingying XUE
. TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS[J]. Acta mathematica scientia, Series B, 2021
, 41(2)
: 596
-608
.
DOI: 10.1007/s10473-021-0219-9
[1] Cao M, Tanaka H, Xue Q, Yabuta K. A note on Carleson embedding theorems. Publ Math Debrecen, 2018, 93(3/4):517-523
[2] Cao M, Xue Q. Characterization of two-weighted inequalities for multilinear fractional maximal operator. Nonlinear Analysis, 2016, 130:214-228
[3] Chen W, Damián W. Weighted estimates for the multilinear maximal function. Rend Circ Mat Palermo, 2013, 62:379-391
[4] Chen X, Xue Q. Weighted estimates for a class of multilinear fractional type operators. J Math Anal Appl, 2010, 362:355-373
[5] Damián W, Lerner A K, Pérez C. Sharp weighted bounds for multilinear maximal functions and CalderónZygmund operators. J Fourier Anal Appl, 2015, 21:161-181
[6] Harboure E, Salinas O, Viviani B. Local maximal function and weights in a general setting. Math Ann, 2014, 358:609-628
[7] Hart J, Liu F, Xue Q. Regularity and continuity of local multilinear maximal type operators. J Geom Anal, 2020, DIO:10.1007/s12220-020-00400-7
[8] Hu G. Weighted estimates with general weights for multilinear Calderón-Zygmund operators. Acta Mathematica Scientia, 2012, 32B(4):1529-1544
[9] Hytönen T, Pérez C. Sharp weighted bounds involving A∞. Anal PDE, 2013, 6(4):777-818
[10] Lerner A K, Ombrosi S, Pérez C, Torres R H, Trujillo-González R. New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv Math, 2009, 220:1222-1264
[11] Li K, Sun W. Characterization of a two weight inequality for multilinear fractional maximal operators. Houston J Math, 2016, 42(3):977-990
[12] Lin C, Stempak K, Wang Y. Local maximal operators on measure metric spaces. Publ Math, 2013, 57:239-264
[13] Moen K. Weighted inequalities for multilinear fractional integral operators. Collect Math, 2009, 60:213-238
[14] Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function. Trans Amer Math Soc, 1972, 165:207-226
[15] Nowak A, Stempak K. Weighted estimates for the Hankel transform transplatation operator. Tohoku Math J, 2006, 58:277-301
[16] Ramseyer M, Salinas O, Viviani B. Two-weight norminequalities for the local maximal function. J Geom Anal, 2017, 27:120-141
[17] Sawyer E. A characterization of a two weight norm inequality for maximal operators. Studia Math, 1982, 75:1-11
[18] Uribe D C, Moen K. A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities. Georgian Mathematical Journal, 2018, 27(1):37-42
[19] Xue Q, Yan J. Multilinear version of reversed Hölder inequality and its applications to multilinear CalderónZygmund operators. J Math Soc Japan, 2012, 64(4):1-17
[20] Yin X, Muckenhoupt B. Weighted inequalities for the maximal geometric mean operator. Proc Am Math Soc, 1996, 124:75-81
