BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES

doi:10.1007/s10473-022-0322-6

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (3): 1191-1208.doi: 10.1007/s10473-022-0322-6

BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES

何骞君¹, 吴新峰², 燕敦验³

1. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, China;
2. Department of Mathematics, China University Mining & Technology(Beijing), Beijing, 100083, China;
3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

收稿日期:2020-12-24 修回日期:2021-06-06 发布日期:2022-06-24
通讯作者: Xinfeng WU,E-mail:wuxf@cumtb.edu.cn E-mail:wuxf@cumtb.edu.cn
基金资助:
This work was supported by National Natural Science Foundation of China (11871452, 12071473) and the Beijing Information Science and Technology University Foundation (2025031).

BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES

Qianjun HE¹, Xinfeng WU², Dunyan YAN³

1. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, China;
2. Department of Mathematics, China University Mining & Technology(Beijing), Beijing, 100083, China;
3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

Received:2020-12-24 Revised:2021-06-06 Published:2022-06-24
Contact: Xinfeng WU,E-mail:wuxf@cumtb.edu.cn E-mail:wuxf@cumtb.edu.cn
Supported by:
This work was supported by National Natural Science Foundation of China (11871452, 12071473) and the Beijing Information Science and Technology University Foundation (2025031).

摘要/Abstract

摘要： In this paper, we study a boundedness property of the Adams type for multilinear fractional integral operators with the multilinear $L^{r^{\prime},\alpha}$-Hörmander condition and their commutators with vector valued BMO functions on a Morrey space and a predual Morrey space. Moreover, we give an endpoint estimate for multilinear fractional integral operators. As an application, we obtain the boundedness of multilinear Fourier multipliers with limited Sobolev regularity on a Morrey space.

关键词: Multilinear fractional integral, $L^{r^{\prime},\alpha}$-Hörmander condition, commutators, BMO spaces, Morrey spaces, multilinear Fourier multiplier

Abstract: In this paper, we study a boundedness property of the Adams type for multilinear fractional integral operators with the multilinear $L^{r^{\prime},\alpha}$-Hörmander condition and their commutators with vector valued BMO functions on a Morrey space and a predual Morrey space. Moreover, we give an endpoint estimate for multilinear fractional integral operators. As an application, we obtain the boundedness of multilinear Fourier multipliers with limited Sobolev regularity on a Morrey space.

Key words: Multilinear fractional integral, $L^{r^{\prime},\alpha}$-Hörmander condition, commutators, BMO spaces, Morrey spaces, multilinear Fourier multiplier

中图分类号:

42B20

何骞君, 吴新峰, 燕敦验. BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES[J]. 数学物理学报(英文版), 2022, 42(3): 1191-1208.

Qianjun HE, Xinfeng WU, Dunyan YAN. BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES[J]. Acta mathematica scientia,Series B, 2022, 42(3): 1191-1208.

参考文献

[1] Adams D R. A note on Riesz potentials. Duke Math J, 1975, 42:765-778
[2] Adams D R. A sharp inequality of J. Moser for higher order derivatives. Ann of Math, 1988, 128:385-398
[3] Adams D R, Xiao J. Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ Math J, 2004, 53:1629-1663
[4] Adams D R, Xiao J. Morrey potentials and harmonic maps. Commun Math Phys, 2011, 38:439-456
[5] Adams D R, Xiao J. Regularity of Morrey Commutators. Trans Amer Math Soc, 2012, 364:4801-4818
[6] Adams D R, Xiao J. Morrey spaces in harmonic analysis. Ark Math, 2012, 50:201-230
[7] Bui T A, Duong X T. Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers. Bull Sci Math, 2013, 137:63-75
[8] Caffarelli L. Elliptic second order equations. Rend Sem Mat Fis Milano, 1988, 58:253-284
[9] Chiarenza F, Frasca M. Morrey spaces and Hardy-Littlewood maximal functions. Rend Mat Applizioni, 1987, 7:273-279
[10] Chiarenza F, Frasca M, Longo P. Interior W^2,p-estimates for nondivergence elliptic equations with discontinuous coefficients. Ric Math, 1992, 40:149-168
[11] Chaffee L, Torres R H, Wu X. Multilinear weighted norm inequalities under integral type regularity conditions//Harmonic Analysis, Partial Differential Equations and Applications, Part of the series Applied and Numerical Harmonic Analysis. Birkhäuser, 2017:193-216
[12] Deng D, Duong X, Yan L. A characterization of the Morrey-Campanato spaces. Math Z, 2005, 250:641-655
[13] Di Fazio G, Palagachev G, Ragusa M. Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients. J Funct Anal, 1999, 166:179-196
[14] Di Fazio G, Ragusa M. Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with dicontinuous coefficients. J Funct Anal, 1993, 112:241-256
[15] Di Fazio G, Ragusa M. Commutators and Morrey spaces. Boll Univ Math Ital, VII Ser A, 1991, 5(3):321-332
[16] Ding Y. A characterization of BMO via commutators for some operators. Northeast Math J, 1997, 13(4):422-432
[17] Ding Y, Mei T. Boundedness and Compactness for the Commutators of Bilinear Operators on Morrey Spaces. Potential Anal, 2015, 42:717-748
[18] Duoandikoetxea J. Fourier Analysis. Graduate Studies in Math. Vol 29. Providence, Rhode Island:Amer Math Soc, 2001
[19] Duoandikoetxea J, Rosenthal M. Extension and boundedness of operators on Morrey spaces from extrapolation techniques and embeddings. J Geom Anal, 2018, 28:3081-3108
[20] Duong X, Xiao J, Yan L. Old and new Morrey spaces via heat kernel bounds. J Fourier Anal Appl, 2007, 13:87-111
[21] Fefferman C, Stein E. H^p spaces of several variables. Acta Math, 1972, 129:137-193
[22] Fujita J, Tomita N. A counter example to weighted estimates for multilinear Fourier multipliers with Sobolev regularity. J Math Anal Appl, 2014, 409:630-636
[23] Grafakos L. Modern Fourier analysis, Third edition. Graduate Texts in Mathematics, 250. New York:Springer, 2014
[24] Grafakos L, Miyachi A, Tomita N. On multilinear Fourier multipliers of limited smoothness. Canad J Math, 2013, 65:299-330
[25] Hart J, Torres R H, Wu X. Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces. Trans Amer Math Soc, 2018, 370:8581-8612
[26] Huang Q. Estimates on the generalized Morrey spaces $L_{\varphi}^{2,\lambda}$ and BMO$_{\psi}$ for linear elliptic systems. Indiana Univ Math J, 1996, 45:397-439
[27] Li P, Zhou J. Compactness for the commutator of the multilinear Fourier multiplier on the Morrey space. Acta Math Vietnam, 2016, 41:661-676
[28] Li K, Sun W. Weighted estimates for multilinear Fourier multipliers. Forum Math, 2015, 27:1101-1116
[29] Iida T, Sato E, Sawano Y, et al. Weighted norm inequalities for multilinear fractional operators on Morrey spaces. Studia Math, 2011, 205:139-170
[30] Iida T, Sato E, Sawano Y, et al. Multilinear fractional integrals on Morrey spaces. Acta Math Sin J, 2012, 28(7):1375-1384
[31] Iida T, Sato E, Sawano Y, et al. Sharp bounds for Multilinear fractional integral operators on Morrey type spaces. Positivity, 2012, 16:339-358
[32] Jiao Y L. Bilinear Fourier multiplier operators on Morrey spaces. J Math Res Appl, 2014, 34(6):703-712
[33] John F, Nirenberg L. On functions of bounded mean oscillation. Comm Pure Appl Math, 1961, 14:415-426
[34] Kato T. Strong solutions of the Navier-Stokes equation in morrey spaces. Bol Soc Brasil Math, 1992, 22:127-155
[35] Kenig C, Stein E, Multilinear estimates and fractional integration. Math Res Lett, 1999, 6:1-15
[36] Maharani D, Widjaja J, Budhi M. Boundedness of Mikhlin operator in Morrey space. J Phys:Vonf Ser, 2019, 1180:012002
[37] Mazzucato A. Besov-Morrey spaces:functons space theory and applications to non-linear PDE. Trans Amer Math Soc, 2003, 355:1297-1364
[38] Moser J. A sharp form of an inequality by N. Trudinger. Indiana Univ Math J, 1971, 21:1077-1092
[39] Morrey C. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43:126-166
[40] Nakai E. Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math Nachr, 1994, 166:95-103
[41] Palagachev D, Softova L. Singular integral operators, Morrey spaces and fine regularity of solutions to PDEs. Potential Anal, 2004, 20:237-263
[42] Pérez C. Two weighted norm inequalities for Riesz potentials and uniform L^p-weighted Sobolev inequalities. Indiana Univ Math J, 1990, 39:31-44
[43] Ruiz A, Vega L. Unique continuation for Schrödinger operators with potential in Morrey spaces. Publ Math, 1991, 35:291-298
[44] Shen Z. The periodic Schrödinger operators with potentials in the Morrey class. J Funct Anal, 2002, 193:314-345
[45] Shen Z. Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains. Amer J Math, 2003, 125:1079-1115
[46] Strichartz R S. A note on Trudinger's extension of Sobolev's inequalities. Indiana Univ Math J, 1972, 21:841-842
[47] Tang L. Endpoint estimates for multilinear fractional integrals. J Austral Math Soc, 2008, 84:419-429
[48] Taylor T. Analysis on Morrey spaces and applications to Navier and other evolution equations. Commun Partial Differ Equ, 1992, 17:1407-1456
[49] Trudinger N S. On imbeddings into Orlicz spaces and some applications. J Math Mech, 1967, 17:473-483
[50] Tomita N. A Hörmander type multiplier theorem for multilinear operators. J Funct Anal, 2010, 259:2028-2044
[51] Wang S, Jiang Y, Li P. Weighted Morrey estimates for multilinear Fourier multiplier operators. Abstr Appl Anal, 2014. Art ID 570450, 10pp
[52] Yan L. Classes of Hardy spaces associated with operators, duality theorem and applications. Trans Amer Math Soc, 2008, 360:4383-4408

BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES

BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES

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