Weighted estimates for the multilinear maximal function on the upper half-spaces. (English) Zbl 1414.42021
Summary: For a general dyadic grid, we give a Calderón-Zygmund type decomposition, which is the principle fact about the multilinear maximal function \(\mathfrak{M}\) on the upper half-spaces. Using the decomposition, we study the boundedness of \(\mathfrak{M}\). We obtain a natural extension to the multilinear setting of Muckenhoupt’s weak-type characterization. We also partially obtain characterizations of Muckenhoupt’s strong-type inequalities with one weight. Assuming the reverse Hölder’s condition, we get a multilinear analogue of Sawyer’s two weight theorem. Moreover, we also get Hytönen-Pérez type weighted estimates.
MSC:
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B35 | Function spaces arising in harmonic analysis |
Keywords:
Hytönen-Pérez type estimate; multilinear maximal function; reverse Hölder’s condition; upper half-space; weighted inequalityReferences:
| [1] | P.Byung‐Oh, A weak type inequality for generalized maximal operators on spaces of homogeneous type, Analysis Math.25 (1999), 179-186. |
| [2] | M. M.Cao and Q. Y.Xue, Characterization of two‐weighted inequalities for multilinear fractional maximal operator, Nonlinear Anal.130 (2016), 214-228. · Zbl 1331.42021 |
| [3] | M. M.Cao and Q. Y.Xue, Two weight entropy boundedness of multilinear fractional type operators, J. Math. Inequal.11 (2017), 441-453. · Zbl 1368.26015 |
| [4] | L.Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math. (2)76 (1962), 547-559. · Zbl 0112.29702 |
| [5] | W.Chen and W.Damián, Weighted estimates for the multisublinear maximal function, Rend. Circ. Mat. Palermo (2)62 (2013), 379-391. · Zbl 1281.42019 |
| [6] | W.Chen and P. D.Liu, Weighted inequalities for the generalized maximal operator in martingale spaces, Chin. Ann. Math. Ser. B.32 (2011), 781-792. · Zbl 1311.42047 |
| [7] | W.Chen and P. D.Liu, Weighted norm inequalities for multisublinear maximal operator in martingale spaces, Tohoku Math. J.66 (2014), 539-553. · Zbl 1321.42035 |
| [8] | D.Cruz‐Uribe, J. M.Martell, and C.Pérez, Weights, extrapolation and the theory of Rubio de Francia, Oper. Theory Adv. Appl., vol. 215, Birkhäuser, Basel, 2011. · Zbl 1234.46003 |
| [9] | D.Cruz‐Uribe and K.Moen, A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities, available at https://arxiv.org/pdf/1701.07800.pdf, Georgian Math. J. (to appear). · Zbl 1436.42017 |
| [10] | W.Damián, A. K.Lerner, and C.Peréz, Sharp weighted bounds for multilinear maximal functions and Calderón-Zygmund operators, J. Fourier Anal. Appl.21 (2015), 161-181. · Zbl 1321.42036 |
| [11] | C.Fefferman and E. M.Stein, Some maximal inequalities, Amer. J. Math.93 (1971), 107-115. · Zbl 0222.26019 |
| [12] | J.Garcia‐Cuerva and J. M.Martell, Two‐weight norm inequalities for maximal operators and fractional integrals on non‐homogeneous spaces, Indiana Univ. Math. J.50 (2001), 1241-1280. · Zbl 1023.42012 |
| [13] | J.García‐Cuerva and J. L. Rubiode Francia, Weighted norm inequalities and related topics, North‐Holland Math. Stud., vol. 116, North Holland Publishing Co., Amsterdam, 1985. · Zbl 0578.46046 |
| [14] | L.Grafakos et al., The multilinear strong maximal function, J. Geom. Anal.21 (2011), 118-149. · Zbl 1219.42008 |
| [15] | L.Grafakos, L. G.Liu, and D. C.Yang, Multiple‐weighted norm inequalities for maximal multi‐linear singular integrals with non‐smooth kernels, Proc. Roy. Soc. Edinburgh Sect. A141 (2011), 755-775. · Zbl 1225.42006 |
| [16] | L.Grafakos and R. H.Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, Indiana Univ. Math. J.51 (2002), 1261-1276. · Zbl 1033.42010 |
| [17] | T.Hytönen and C.Pérez, Sharp weighted bounds involving \(A_\infty \), Anal. PDE6 (2013), 777-818. · Zbl 1283.42032 |
| [18] | A. K.Lerner et al., New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math.220 (2009), 1222-1264. · Zbl 1160.42009 |
| [19] | K. W.Li, K.Moen, and W. C.Sun, The sharp weighted bound for multilinear maximal function and multilinear Calderón-Zygmund operators, J. Fourier Anal. Appl.20 (2014), 751-765. · Zbl 1318.42022 |
| [20] | W. M.Li, L. M.Xue, and X. F.Yan, Two‐weight inequalities for multilinear maximal operators, Georgian Math. J.19 (2012), 145-156. · Zbl 1239.42015 |
| [21] | B.Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc.165 (1972), 207-226. · Zbl 0236.26016 |
| [22] | C.Pérez and R. H.Torres, Sharp maximal function estimates for multilinear singular integrals, Contemp. Math.320 (2003), 323-331. · Zbl 1045.42011 |
| [23] | I. P.Rivera‐Ríos, A quantitative approach to weighted Carleson condition, Concr. Oper.4 (2017), 58-75. · Zbl 1374.42042 |
| [24] | F. J.Ruiz, A unified approach to Carleson measures and \(A_p\) weights, Pacific J. Math.117 (1985), 397-404. · Zbl 0587.42010 |
| [25] | F. J.Ruiz and J. L.Torrea, A unified approach to Carleson measures and \(A_p\) weights II, Pacific J. Math.120 (1985), 189-197. · Zbl 0644.42016 |
| [26] | E. T.Sawyer, A characterization of a two weight norm inequality for maximal operators, Studia Math.75 (1982), 1-11. · Zbl 0508.42023 |
| [27] | B. F.Sehba, On two‐weight norm estimates for multilinear fractional maximal function, J. Math. Soc. Japan70 (2018), 71-94. · Zbl 1395.42052 |
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