On two-weight norm estimates for multilinear fractional maximal function. (English) Zbl 1395.42052

In the paper [Stud. Math. 75, 1–11 (1982; Zbl 0508.42023)], E. T. Sawyer proved a general criterium for the boundedness of the fractional maximal function frow \(L^p(\sigma)\) to \(L^q(\omega)\), \( 1< p \leq q<\infty\). In the present work, it is proved a multilinear analogue by considering the multilinear fractional maximal functions associated to cubes in \(\mathcal Q\) with sides parallel to the coordinate axes, defined for \(x\in \mathbb R^n\) and \(\vec{f}=(f_1,\cdots, f_m)\) by \[ \mathcal M_\alpha \vec{f}(x):=\sup_{Q\in \mathcal Q}| Q|^{\alpha/n} \prod_{i=1}^m \frac{\chi_Q (x)}{|Q|} \int_Q |f_i(y)| dy, \] where the \(f_i\)’s are measurable functions and \(0\leq \alpha< mn\).
The main results consist in providing sufficient conditions for \({\mathcal M}_{\alpha}\) to be bounded from \(L^{p_1}(\sigma_1)\times \cdots \times L^{p_m}(\sigma_m) \) to \(L^q(\omega)\), the conditions obtained on the weights are close to the \(A_p\) characterization of Muckenhoupt. Once the problem is reduced to the corresponding dyadic maximal function, one of the main tools used in the proofs is the extension of Carlesons embedding theorem and its multilinear analogue.


42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B35 Function spaces arising in harmonic analysis


Zbl 0508.42023
Full Text: DOI arXiv


[1] S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, \doihref10.1090/S0002-9947-1993-1124164-0Trans. Amer. Math. Soc., 340 (1993), 253-272.
[2] W. Chen, and W. Damián, Weighted estimates for the multisublinear maximal function, \doihref10.1007/s12215-013-0131-9Rend. Circ. Mat. Palermo, 62 (2013), 379-391.
[3] W. Chen and P. D. Liu, Weighted norm inequalities for multisublinear maximal operator in martingale, \doihref10.2748/tmj/1432229196Tohoku Math. J., 66 (2014), 539-553. · Zbl 1321.42035
[4] D. Cruz-Uribe, New proofs of two-weight norm inequalities for the maximal operator, Georgian Math. J., 7 (2000), 33-42. · Zbl 0987.42019
[5] D. Cruz-Uribe and K. Moen, One and two weight norm inequalities for Riez potentials, Illinois J. Math., 57 (2013), 295-323. · Zbl 1297.42022
[6] W. Damián, A. K. Lerner and C. Pérez, Sharp weighted bounds for multilinear maximal functions and Calderón-Zygmund operators, \doihref10.1007/s00041-014-9364-zJ. Fourier Anal. Appl., 21 (2015), 161-181.
[7] P. L. Duren, Extension of a theorem of Carleson, \doihref10.1090/S0002-9904-1969-12181-6Bull. Amer. Math. Soc., 75 (1969), 143-146.
[8] N. Fujii, Weighted bounded mean oscillation and singular integrals, Math. Japon, 22 (1977/78), 529-534. · Zbl 0385.26010
[9] L. Grafakos, On multilinear fractional integrals, Studia Math., 102 (1992), 49-56. · Zbl 0808.42014
[10] L. Grafakos and N. Kalton, Some remarks on multilinear maps and interpolation, \doihref10.1007/PL00004426Math. Ann., 319 (2001), 151-180. · Zbl 0982.46018
[11] L. Grafakos and R. H. Torres, Multilinear Calderón-Zygmund theory, \doihref10.1006/aima.2001.2028Adv. Math., 165 (2002), 124-164.
[12] L. Grafakos and R. H. Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, \doihref10.1512/iumj.2002.51.2114Indiana Univ. Math. J., 51 (2002), 1261-1276. · Zbl 1033.42010
[13] S. V. Hurščev, A description of weights satisfying the \(A_∞\) condition of Muckenhoupt, \doihref10.1090/S0002-9939-1984-0727244-4Proc. Amer. Math. Soc., 90 (1984), 253-257.
[14] T. Hytönen and C. Pérez, Sharp weighted bounds involving \(A_∞\), \doihref10.2140/apde.2013.6.777Anal. and P.D.E., 6 (2013), 777-818.
[15] C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, \doihref10.4310/MRL.1999.v6.n1.a1Math. Res. Lett., 6 (1999), 1-15.
[16] A. K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, \doihref10.1016/j.aim.2010.11.009Adv. Math., 226 (2011), 3912-3926. · Zbl 1226.42010
[17] A. K. Lerner and S. Ombrosi, Personal communication.
[18] A. K. Lerner, S. Ombrosi, C. Pérez, R. H. Torres and R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, \doihref10.1016/j.aim.2008.10.014Adv. Math., 220 (2009), 1222-1264. · Zbl 1160.42009
[19] K. Li, K. Moen and W. Sun, Sharp weighted bounds for multilinear maximal functions and Calderón-Zygmund operators, \doihref10.1007/s00041-014-9326-5J. Fourier Anal. Appl., 20 (2014), 751-765.
[20] K. Li, K. Moen and W. Sun, Sharp weighted inequalities for multilinear fractional maximal operator and fractional integrals, \doihref10.1002/mana.201300287Math. Nach., 288 (2015), 619-632. · Zbl 1314.42021
[21] K. Li and W. Sun, Characterization of a two weight inequality for multilinear fractional maximal operators, Houston J. Math., 42 (2016), 977-990. · Zbl 1354.42035
[22] W. M. Li, L. M. Xue and X. F. Yan, Two-weight inequalities for multilinear maximal operators, \doihref10.1515/gmj-2011-0055Georgian Math. J., 19 (2012), 145-156.
[23] K. Moen, Sharp one-weight and two-weight bounds for maximal operators, Studia Math., 194 (2009), 163-180. · Zbl 1174.42020
[24] K. Moen, Weighted inequalities for multilinear fractional integral operators, \doihref10.1007/BF03191210Collect. Math., 60 (2009), 213-238. · Zbl 1172.26319
[25] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, \doihref10.1090/S0002-9947-1972-0293384-6 Trans. Amer. Math. Soc., 165 (1972), 207-226.
[26] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, \doihref10.1090/S0002-9947-1974-0340523-6Trans. Amer. Math. Soc., 192 (1974), 261-274. · Zbl 0289.26010
[27] C. Pérez, Two weight inequalities for potential and fractional type maximal operators, \doihref10.1512/iumj.1994.43.43028Indiana Univ. Math. J., 43 (1994), 663-683. · Zbl 0809.42007
[28] M. C. Pereyra, Lecture notes on dyadic harmonic analysis, \doihref10.1090/conm/289Contemp. Math., 289 (2001), 1-60. · Zbl 0991.42010
[29] S. Pott and M. C. Reguera, Sharp Békólle estimates for the Bergman Projection, \doihref10.1016/j.jfa.2013.08.018J. Funct. Anal., 265 (2013), 3233-3244. · Zbl 1295.46020
[30] E. Sawyer, A characterization of two-weight norm inequality for maximal operators, Studia Math., 75 (1982), 1-11. · Zbl 0508.42023
[31] H. Wang and W. Yi, Multilinear singular and fractional integral operators on weighted Morrey spaces, \doihref10.1155/2013/735795J. Funct. Spaces and Appl., 2013, Art. ID 735795, 11 pages.
[32] J. M. Wilson, Weighted inequalities for the dyadic square function without dyadic \(A_∞\), \doihref10.1215/S0012-7094-87-05502-5Duke Math. J., 55 (1987), 19-50.
[33] J. M. Wilson, Weighted norm inequalities for the continuous square function, \doihref10.1090/S0002-9947-1989-0972707-9Trans. Amer. Math. Soc., 314 (1989), 661-692.
[34] J. M.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.