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Keywords:
grand Morrey space; variable exponent; non-doubling measure; metric measure space; Riesz potential; maximal operator; Sobolev's inequality; Trudinger's exponential inequality; continuity
Summary:
Our aim in this paper is to deal with the boundedness of the Hardy-Littlewood maximal operator on grand Morrey spaces of variable exponents over non-doubling measure spaces. As an application of the boundedness of the maximal operator, we establish Sobolev's inequality for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces. We are also concerned with Trudinger's inequality and the continuity for Riesz potentials.
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] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory. Fundamental Principles of Mathematical Sciences 314 Springer, Berlin (1995). MR 1411441 | Zbl 0834.46021
[3] Almeida, A., Hasanov, J., Samko, S. G.: Maximal and potential operators in variable exponent Morrey spaces. Georgian Math. J. 15 (2008), 195-208. MR 2428465 | Zbl 1263.42002
[4] Bojarski, B., Hajłasz, P.: Pointwise inequalities for Sobolev functions and some applications. Stud. Math. 106 (1993), 77-92. MR 1226425 | Zbl 0810.46030
[5] Chiarenza, F., Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. Appl., VII. Ser. 7 (1987), 273-279. MR 0985999 | Zbl 0717.42023
[6] Cruz-Uribe, D., Fiorenza, A., Neugebauer, C. J.: The maximal function on variable $L^{p}$ spaces. Ann. Acad. Sci. Fenn., Math. 28 (2003), 223-238; Corrections to ``The maximal function on variable $L^{p}$ spaces'' Ann. Acad. Sci. Fenn., Math. 29 (2004), 247-249. MR 2041952 | Zbl 1064.42500
[7] Diening, L.: Maximal function in generalized Lebesgue spaces $L^{p(\cdot)}$. Math. Inequal. Appl. 7 (2004), 245-253. MR 2057643
[8] Diening, L.: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces $L^{p(\cdot)}$ and $W^{k,p(\cdot)}$. Math. Nachr. 268 (2004), 31-43. DOI 10.1002/mana.200310157 | MR 2054530 | Zbl 1065.46024
[9] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017 Springer, Berlin (2011). MR 2790542 | Zbl 1222.46002
[10] Edmunds, D. E., Gurka, P., Opic, B.: Double exponential integrability, Bessel potentials and embedding theorems. Stud. Math. 115 (1995), 151-181. MR 1347439 | Zbl 0829.47024
[11] Edmunds, D. E., Gurka, P., Opic, B.: Sharpness of embeddings in logarithmic Besselpotential spaces. Proc. R. Soc. Edinb., Sect. A 126 (1996), 995-1009. MR 1415818
[12] Edmunds, D. E., Hurri-Syrjänen, R.: Sobolev inequalities of exponential type. Isr. J. Math. 123 (2001), 61-92. DOI 10.1007/BF02784120 | MR 1835289 | Zbl 0991.46019
[13] Edmunds, D. E., Krbec, M.: Two limiting cases of Sobolev imbeddings. Houston J. Math. 21 (1995), 119-128. MR 1331250 | Zbl 0835.46027
[14] Fiorenza, A., Gupta, B., Jain, P.: The maximal theorem for weighted grand Lebesgue spaces. Stud. Math. 188 (2008), 123-133. DOI 10.4064/sm188-2-2 | MR 2430998 | Zbl 1161.42011
[15] Fiorenza, A., Krbec, M.: On the domain and range of the maximal operator. Nagoya Math. J. 158 (2000), 43-61. DOI 10.1017/S0027763000007285 | MR 1766576 | Zbl 1039.42015
[16] Fiorenza, A., Sbordone, C.: Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$. Stud. Math. 127 (1998), 223-231. MR 1489454 | Zbl 0891.35039
[17] Futamura, T., Mizuta, Y.: Continuity properties of Riesz potentials for function in $L^{p(\cdot)}$ of variable exponent. Math. Inequal. Appl. 8 (2005), 619-631. MR 2174890
[18] Futamura, T., Mizuta, Y., Shimomura, T.: Integrability of maximal functions and Riesz potentials in Orlicz spaces of variable exponent. J. Math. Anal. Appl. 366 (2010), 391-417. DOI 10.1016/j.jmaa.2010.01.053 | MR 2600488 | Zbl 1193.46016
[19] Futamura, T., Mizuta, Y., Shimomura, T.: Sobolev embeddings for variable exponent Riesz potentials on metric spaces. Ann. Acad. Sci. Fenn., Math. 31 (2006), 495-522. MR 2248828 | Zbl 1100.31002
[20] Greco, L., Iwaniec, T., Sbordone, C.: Inverting the $p$-harmonic operator. Manuscr. Math. 92 (1997), 249-258. DOI 10.1007/BF02678192 | MR 1428651 | Zbl 0869.35037
[21] Guliyev, V. S., Hasanov, J. J., Samko, S. G.: Boundedness of maximal, potential type, and singular integral operators in the generalized variable exponent Morrey type spaces. Problems in mathematical analysis 50. J. Math. Sci. (N.Y.) 170 (2010), 423-443. DOI 10.1007/s10958-010-0095-7 | MR 2839874
[22] Guliyev, V. S., Hasanov, J. J., Samko, S. G.: Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces. Math. Scand. 107 (2010), 285-304. DOI 10.7146/math.scand.a-15156 | MR 2735097 | Zbl 1213.42077
[23] Gunawan, H., Sawano, Y., Sihwaningrum, I.: Fractional integral operators in nonhomogeneous spaces. Bull. Aust. Math. Soc. 80 (2009), 324-334. DOI 10.1017/S0004972709000343 | MR 2540365 | Zbl 1176.42012
[24] Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 688 (2000), 101 pages. MR 1683160 | Zbl 0954.46022
[25] Harjulehto, P., Hästö, P., Pere, M.: Variable exponent Lebesgue spaces on metric spaces: the Hardy-Littlewood maximal operator. Real Anal. Exch. 30 (2004/2005), 87-104. MR 2126796
[26] Hedberg, L. I.: On certain convolution inequalities. Proc. Am. Math. Soc. 36 (1972), 505-510. DOI 10.1090/S0002-9939-1972-0312232-4 | MR 0312232
[27] Iwaniec, T., Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses. Arch. Ration. Mech. Anal. 119 (1992), 129-143. DOI 10.1007/BF00375119 | MR 1176362 | Zbl 0766.46016
[28] Iwaniec, T., Sbordone, C.: Riesz transforms and elliptic PDEs with VMO coefficients. J. Anal. Math. 74 (1998), 183-212. DOI 10.1007/BF02819450 | MR 1631658 | Zbl 0909.35039
[29] Kinnunen, J.: The Hardy-Littlewood maximal function of a Sobolev function. Isr. J. Math. 100 (1997), 117-124. DOI 10.1007/BF02773636 | MR 1469106 | Zbl 0882.43003
[30] Kokilashvili, V., Meskhi, A.: Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure. Complex Var. Elliptic Equ. 55 (2010), 923-936. MR 2674873 | Zbl 1205.26014
[31] Kokilashvili, V., Samko, S. G.: Boundedness of weighted singular integral operators in grand Lebesgue spaces. Georgian Math. J. 18 (2011), 259-269. MR 2805980 | Zbl 1239.42014
[32] Meskhi, A.: Maximal functions, potentials and singular integrals in grand Morrey spaces. Complex Var. Elliptic Equ. 56 (2011), 1003-1019. MR 2838234 | Zbl 1261.42022
[33] Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Boundedness of fractional integral operators on Morrey spaces and Sobolev embeddings for generalized Riesz potentials. J. Math. Soc. Japan 62 (2010), 707-744. DOI 10.2969/jmsj/06230707 | MR 2648060 | Zbl 1200.26007
[34] Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponents. Complex Var. Elliptic Equ. 56 (2011), 671-695. MR 2832209 | Zbl 1228.31004
[35] Mizuta, Y., Shimomura, T.: Continuity properties of Riesz potentials of Orlicz functions. Tohoku Math. J. 61 (2009), 225-240. DOI 10.2748/tmj/1245849445 | MR 2541407 | Zbl 1181.46026
[36] Mizuta, Y., Shimomura, T.: Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent. J. Math. Soc. Japan 60 (2008), 583-602. DOI 10.2969/jmsj/06020583 | MR 2421989 | Zbl 1161.46305
[37] Jr., C. B. Morrey: 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
[38] Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166 (1994), 95-103. DOI 10.1002/mana.19941660108 | MR 1273325 | Zbl 0837.42008
[39] Peetre, J.: On the theory of $L_{p,\lambda}$ spaces. J. Funct. Anal. 4 (1969), 71-87. DOI 10.1016/0022-1236(69)90022-6 | MR 0241965
[40] 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
[41] 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
[42] Sbordone, C.: Grand Sobolev spaces and their application to variational problems. Matematiche 51 (1996), 335-347. MR 1488076 | Zbl 0915.46030
[43] Serrin, J.: A remark on Morrey potential. Control Methods in PDE-Dynamical Systems. AMS-IMS-SIAM joint summer research conference, 2005 F. Ancona et al. Contemporary Mathematics 426 American Mathematical Society, Providence (2007), 307-315. MR 2311532
[44] Stein, E. M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30 Princeton University Press, Princeton (1970). MR 0290095 | Zbl 0207.13501
[45] Trudinger, N. S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473-483. MR 0216286 | Zbl 0163.36402
[46] Ziemer, W. P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics 120 Springer, Berlin (1989). DOI 10.1007/978-1-4612-1015-3 | MR 1014685 | Zbl 0692.46022
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