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Morrey spaces and fractional integral operators. (English) Zbl 1181.26014
The aim of this paper is to establish the boundedness of fractional integral operators in Morrey spaces defined on quasimetric measure spaces. The author also derives Sobolev-, trace- and two-weight-inequalities for fractional integrals. In the case that the measure satisfies the doubling condition, the derived conditions are necessary and sufficient for appropriate inequalities.
##### MSC:
 26A33 Fractional derivatives and integrals (real functions) 42B35 Function spaces arising in harmonic analysis 47B38 Operators on function spaces (general)
##### Keywords:
fractional integrals; Morrey spaces; trace inequality
##### References:
 [1] Adams, D. R.: A trace inequality for generalized potentials, Studia math. 48, 99-105 (1973) · Zbl 0237.46037 [2] Adams, D. R.: A note on Riesz potentials, Duke math. J. 42, No. 4, 765-778 (1975) · Zbl 0336.46038 · doi:10.1215/S0012-7094-75-04265-9 [3] Burenkov, V.; Guliyev, H. V.: Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces, Studia math. 163, No. 2, 157-176 (2004) · Zbl 1044.42015 · doi:10.4064/sm163-2-4 · doi:http://journals.impan.gov.pl/sm/Inf/163-2-4.html [4] Burenkov, I. V.; Guliyev, H. V.; Guliyev, V. S.: Necessary and sufficient conditions for the boundedness of the fractional maximal operator in local Morrey-type spaces, Dokl. akad. Nauk 409, No. 4, 443-447 (2006) · Zbl 1135.42011 · doi:10.1134/S1064562406040181 [5] Edmunds, D.; Kokilashvili, V.; Meskhi, A.: Bounded and compact integral operators, mathematics and its applications, Bounded and compact integral operators, mathematics and its applications 543 (2002) · Zbl 1023.42001 [6] Eridani, V. Kokilashvili, A. Meskhi, Morrey spaces and fractional integral operators, Preprint No. 65, School of Mathematical Sciences, GC University, Lahore, 2007. [7] García-Cuerva, J.; Gatto, A. E.: Boundedness properties of fractional integral operators associated to non-doubling measures, Studia math. 162, No. 3, 245-261 (2004) · Zbl 1045.42006 · doi:10.4064/sm162-3-5 · doi:http://journals.impan.gov.pl/sm/Inf/162-3-5.html [8] 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, No. 3, 1241-1280 (2001) · Zbl 1023.42012 · doi:10.1512/iumj.2001.50.2100 [9] Genebashvili, I.; Gogatishvili, A.; Kokilashvili, V.: Solution of two-weight problems for integral transforms with positive kernels, Georgian math. J. 3, No. 1, 319-342 (1996) · Zbl 1056.42507 · doi:10.1007/BF02256723 · doi:emis:journals/GMJ/vol3/contents.htm [10] Genebashvili, I.; Gogatishvili, A.; Kokilashvili, V.; Krbec, M.: Weight theory for integral transforms on spaces of homogeneous type, Pitman monographs and surveys in pure and applied mathematics, Weight theory for integral transforms on spaces of homogeneous type, Pitman monographs and surveys in pure and applied mathematics 92 (1998) · Zbl 0955.42001 [11] V. Kokilashvili, Weighted estimates for classical integral operators, in: Nonlinear Analysis, Function Spaces and Applications IV, Roudnice nad Labem, vol. 119, Teubner-Texte Math, Leipzig, 1990, pp. 86 – 103. · Zbl 0746.47027 [12] Kokilashvili, V.; Meskhi, A.: Fractional integrals on measure spaces, Frac. calc. Appl. anal. 4, No. 4, 1-24 (2001) · Zbl 1065.47503 [13] Kokilashvili, V.; Meskhi, A.: On some weighted inequalities for fractional integrals on non-homogeneous spaces, Z. anal. Anwend. 24, No. 4, 871-885 (2005) · Zbl 1094.42010 [14] Komori, Y.; Mizuhara, T.: Notes on commutators and Morrey spaces, Hokkaido math. J. 32, 345-353 (2003) · Zbl 1044.42011 [15] Peetre, J.: On the theory of lp,$\lambda$ spaces, J. funct. Anal. 4, 71-87 (1969) · Zbl 0175.42602 · doi:10.1016/0022-1236(69)90022-6 [16] Samko, S.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives. Theory and applications, (1993) · Zbl 0818.26003 [17] Sawano, Y.; Tanaka, H.: Morrey spaces for nondoubling measures, Acta math. Sinica 21, No. 6, 1535-1544 (2005) · Zbl 1129.42403 · doi:10.1007/s10114-005-0660-z [18] Stein, E. M.; Weiss, G.: Fractional integrals on n-dimensional Euclidean spaces, J. math. Mech. 7, No. 4, 503-514 (1958) · Zbl 0082.27201 [19] Strömberg, J. O.; Torchinsky, A.: Weighted Hardy spaces, lecture notes in mathematics, Weighted Hardy spaces, lecture notes in mathematics 1381 (1989) · Zbl 0676.42021