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

doi:10.1007/s10473-022-0322-6

Abstract

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

CLC Number:

42B20

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.

Trendmd

References

[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