Previous |  Up |  Next

Article

Keywords:
multi-Morrey space; multilinear maximal function; multilinear fractional integral operator; multilinear Calderón-Zygmund operator
Summary:
The spaces of multi-Morrey type for positive Radon measures satisfying a growth condition on $\mathbb {R}^{d}$ are introduced. After defining the spaces, we investigate the multilinear maximal function, the multilinear fractional integral operator and the multilinear Calderón-Zygmund operators, respectively, from multi-Morrey spaces to Morrey spaces.
References:
[1] Adams, D. R.: A note on Riesz potentials. Duke Math. J. 42 (1975), 765-778. DOI 10.1215/S0012-7094-75-04265-9 | MR 0458158 | Zbl 0336.46038
[2] García-Cuerva, J., Gatto, A. E.: Boundedness properties of fractional integral operators associated to non-doubling measures. Stud. Math. 162 (2004), 245-261. DOI 10.4064/sm162-3-5 | MR 2047654 | Zbl 1045.42006
[3] García-Cuerva, J., Martell, J. M.: Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces. Indiana Univ. Math. J. 50 (2001), 1241-1280. DOI 10.1512/iumj.2001.50.2100 | MR 1871355 | Zbl 1023.42012
[4] Iida, T., Sato, E., Sawano, Y., Tanaka, H.: Sharp bounds for multilinear fractional integral operators on Morrey type spaces. Positivity 16 (2012), 339-358. DOI 10.1007/s11117-011-0129-5 | MR 2929094 | Zbl 1256.42037
[5] Lerner, A. K., Ombrosi, S., Pérez, C., Torres, R. H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. Math. 220 (2009), 1222-1264. DOI 10.1016/j.aim.2008.10.014 | MR 2483720 | Zbl 1160.42009
[6] Lin, H.-B., Yang, D.-C.: Endpoint estimates for multilinear fractional integrals with non-doubling measures. Acta Math. Appl. Sin., Engl. Ser. 30 (2014), 755-764. DOI 10.1007/s10255-012-0194-y | MR 3285972 | Zbl 1304.42037
[7] C. B. Morrey, Jr.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43 (1938), 126-166. DOI 10.1090/S0002-9947-1938-1501936-8 | MR 1501936 | Zbl 0018.40501
[8] Nazarov, F., Treil, S., Volberg, A.: Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 1997 (1997), 703-726. DOI 10.1155/S1073792897000469 | MR 1470373 | Zbl 0889.42013
[9] Nazarov, F., Treil, S., Volberg, A.: Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 1998 (1998), 463-487. DOI 10.1155/S1073792898000312 | MR 1626935 | Zbl 0918.42009
[10] Nazarov, F., Treil, S., Volberg, A.: The {$Tb$}-theorem on non-homogeneous spaces. Acta Math. 190 (2003), 151-239. DOI 10.1007/BF02392690 | MR 1998349 | Zbl 1065.42014
[11] Sawano, Y.: Generalized Morrey spaces for non-doubling measures. NoDEA, Nonlinear Differ. Equ. Appl. 15 (2008), 413-425. DOI 10.1007/s00030-008-6032-5 | MR 2465971 | Zbl 1173.42317
[12] Sawano, Y., Tanaka, H.: Morrey spaces for non-doubling measures. Acta Math. Sin., Engl. Ser. 21 (2005), 1535-1544. DOI 10.1007/s10114-005-0660-z | MR 2190025 | Zbl 1129.42403
[13] Tolsa, X.: BMO, {$H^1$}, and Calderón-Zygmund operators for non doubling measures. Math. Ann. 319 (2001), 89-149. DOI 10.1007/PL00004432 | MR 1812821 | Zbl 0974.42014
[14] Tolsa, X.: Painlevé's problem and the semiadditivity of analytic capacity. Acta Math. 190 (2003), 105-149. DOI 10.1007/BF02393237 | MR 1982794 | Zbl 1060.30031
[15] Tolsa, X.: The space {$H^1$} for nondoubling measures in terms of a grand maximal operator. Trans. Am. Math. Soc. 355 (2003), 315-348. DOI 10.1090/S0002-9947-02-03131-8 | MR 1928090 | Zbl 1021.42010
[16] Tolsa, X.: The semiadditivity of continuous analytic capacity and the inner boundary conjecture. Am. J. Math. 126 (2004), 523-567. DOI 10.1353/ajm.2004.0021 | MR 2058383 | Zbl 1060.30032
[17] Zhou, J., Wang, D.: Lipschitz spaces and fractional integral operators associated with nonhomogeneous metric measure spaces. Abstr. Appl. Anal. (2014), Article ID 174010, 8 pages. DOI 10.1155/2014/174010 | MR 3198155 | Zbl 07021865
Partner of
EuDML logo