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Lipschitz estimates for rough fractional multilinear integral operators on local generalized Morrey spaces. (English) Zbl 1439.42016
Summary: We obtain the Lipschitz boundedness for a class of fractional multilinear operators \(I_{\Omega,\alpha}^{A,m}\) with rough kernels \(\Omega\in L_s(\mathbb{S}^{n-1})\), \(s>n/(n-\alpha)\) on the local generalized Morrey spaces \(LM_{p,\varphi}^{\{x_0\}}\), generalized Morrey spaces \(M_{p,\varphi}\) and vanishing generalized Morrey spaces \(VM_{p,\varphi}\), where the functions \(A\) belong to homogeneous Lipschitz space \(\dot{\Lambda}_{\beta}, 0<\beta<1\). We find the sufficient conditions on the pair \((\varphi_1,\varphi_2)\) which ensures the boundedness of the operators \(I_{\Omega,\alpha}^{A,m}\) from \(LM_{p,\varphi_1}^{\{x_0\}}\) to \(LM_{q,\varphi_2}^{\{x_0\}}\), from \(M_{p,\varphi_1}\) to \(M_{q,\varphi_2}\) and from \(VM_{p,\varphi_1}\) to \(VM_{q,\varphi_2}\) for \(1<p<q <\infty\) and \(1/p-1/q=(\alpha+\beta)/n\). In all cases the conditions for the boundedness of the operator \(I_{\Omega,\alpha}^{A,m}\) is given in terms of Zygmund-type integral inequalities on \((\varphi_1,\varphi_2)\), which do not assume any assumption on monotonicity of \(\varphi_1(x,r)\), \(\varphi_2(x,r)\) in \(r\).

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B35 Function spaces arising in harmonic analysis
47B38 Linear operators on function spaces (general)
47G10 Integral operators
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[1] J. Alvarez, J. Lakey, M. Guzman-Partida, Spaces of bounded \(\lambda \)-central mean oscillation, Morrey spaces, and \(\lambda \)-central Carleson measures, Collect. Math. 51 (1) (2000), 1-47. · Zbl 0948.42013
[2] A. Akbulut, V.S. Guliyev, R. Mustafayev, Boundedness of the maximal operator and singular integral operator in generalized Morrey spaces, Math. Bohem. 137 (1) (2012), 27-43. · Zbl 1250.42038
[3] V.I. Burenkov, V.S. Guliyev, Necessary and sufficient conditions for the boundedness of the Riesz potential in Local Morrey-type spaces, Pot. Anal. 30 (3) 2009, 211-249. · Zbl 1171.42003
[4] J. Cohen, J. Gosselin, A \(BMO\) estimate for multilinear singular integrals, Illinois J. Math. 30 (1986), 445-464. · Zbl 0619.42012
[5] Y. Ding, A note on multilinear fractional integrals with rough kernel, Adv. Math. (China) 30 (2001), 238-246. · Zbl 0989.42003
[6] Y. Ding, D.S. Yang, Z. Zhou, Boundedness of sublinear operators and commutators on \(L_p(\Rn)\), Yokohama Math. J. 46 (1998), 15-27. · Zbl 0969.42009
[7] A. Eroglu, V.S. Guliyev, C.V. Azizov, Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups, Math. Notes 102 (5) (2017), 127-139. · Zbl 1432.42010
[8] G. Di Fazio and M.A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal. 112 (1993), 241-256. · Zbl 0822.35036
[9] D. Fan, S. Lu and D. Yang, Boundedness of operators in Morrey spaces on homogeneous spaces and its applications, Acta Math. Sinica (N. S.) 14 (1998), suppl., 625-634. · Zbl 0916.43006
[10] R.A. DeVore, R.C. Sharpley, Maximal functions measuring smoothness, Mem. Am. Math. Soc. 47 (293) (1984), 115 pp. · Zbl 0529.42005
[11] V.S. Guliyev, Integral operators on function spaces on the homogeneous groups and on domains in \(\Rn \), Doctoral dissertation, Moscow, Mat. Inst. Steklov, 1994, 329 pp. (in Russian)
[12] V.S. Guliyev, Function spaces, integral operators and two weighted inequalities on homogeneous groups. Some applications, Baku, Elm. 1999, 332 pp. (Russian)
[13] V.S. Guliyev, Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl. 2009, Art. ID 503948, 20 pp. · Zbl 1193.42082
[14] V.S. Guliyev, S.S. Aliyev, T. Karaman, P. S. Shukurov, Boundedness of sublinear operators and commutators on generalized Morrey space, Integral Equations Operator Theory 71 (3) (2011), 327-355. · Zbl 1247.42014
[15] V.S. Guliyev, Local generalized Morrey spaces and singular integrals with rough kernel, Azerb. J. Math. 3 (2) (2013), 79-94. · Zbl 1290.42036
[16] V.S. Guliyev, Generalized local Morrey spaces and fractional integral operators with rough kernel, J. Math. Sci. (N. Y.) 193 (2) (2013), 211-227. · Zbl 1277.42021
[17] V.S. Guliyev, F. Deringoz, J.J. Hasanov, \( \Phi \)-admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces, J. Inequal. Appl. 2014, 2014:143, 18 pp. · Zbl 1375.42014
[18] V.S. Guliyev, M.N. Omarova, M.A. Ragusa, A. Scapellato, Commutators and generalized local Morrey spaces, J. Math. Anal. Appl. 457 (2) (2018), 1388-1402. · Zbl 1376.42021
[19] Sh. Lu, Pu Zhang, Lipschitz estimates for generalized commutators of fractional integrals with rough kernel, Math. Nachr. 252 (2003), 70-85. · Zbl 1049.42009
[20] T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (S. Igari, Editor), ICM 90 Satellite Proceedings, Springer - Verlag, Tokyo (1991), 183-189. · Zbl 0771.42007
[21] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166. · Zbl 0018.40501
[22] E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994), 95-103. · Zbl 0837.42008
[23] D.K. Palagachev and L.G. Softova, Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s, Potential Anal. 20 (2004), 237-263. · Zbl 1036.35045
[24] M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J. 44 (1995), 1-17. · Zbl 0838.42006
[25] T. Qian, On the estimates of the multilinear singular integrals, Science in China (Ser. A) 7 (1984), 613-623.
[26] M.A. Ragusa, Commutators of fractional integral operators on vanishing-Morrey spaces, J. Global Optim. 40 (1-3) (2008), 361-368. · Zbl 1143.42020
[27] M.A. Ragusa, Embeddings for Morrey-Lorentz Spaces, J. Optim. Theory Appl. 154 (2) (2012), 491-499. · Zbl 1270.46025
[28] M.A. Ragusa, Necessary and sufficient condition for a VMO function, Abstr. Appl. Anal. 218 (24) (2012), 11952-11958. · Zbl 1280.42019
[29] N. Samko, Maximal, potential and singular operators in vanishing generalized Morrey spaces, J. Global Optim. 57 (4) (2013), 1385-1399. · Zbl 1287.46024
[30] Y. Sawano, H. Gunawan, V. Guliyev, H. Tanaka, Morrey spaces and related function spaces [Editorial]. J. Funct. Spaces 2014, Art. ID 867192, 2 pp. · Zbl 1298.00180
[31] Y. Sawano, A thought on generalized Morrey spaces, arXiv:1812.08394v1, 2018, 78 pp.
[32] L. Softova, Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 3, 757-766. · Zbl 1129.42372
[33] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton (1970). · Zbl 0207.13501
[34] C Vitanza, Functions with vanishing Morrey norm and elliptic partial differential equations, In: Proceedings of methods of real analysis and partial differential equations, Capri, Springer (1990), 147-150.
[35] H.L. Wu, J.C. Lan, Lipschitz estimates for fractional multilinear singular integral on variable exponent Lebesgue spaces, Abstr. Appl. Anal. Volume 2013, Article ID 632384, 6 pages. · Zbl 1432.42018
[36] Q. Wu, D. Yang, On fractional multilinear singular integrals, Math. Nachr. 239/240 (2002),215-235. · Zbl 1004.42019
[37] R. Wheeden, A. Zygmund, Measure and integral, An introduction to real analysis, Pure and Applied Mathematics, 43. Marcel Dekker, Inc., New York- Basel, 1977. · Zbl 0362.26004
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