Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces

Ohno, Takao   ; Shimomura, Tetsu

Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 209-228

Abstract (VO)

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.

Détail

DOI : 10.1007/s10587-014-0095-8

Classification : 31B15, 46E35
Keywords: grand Morrey space; variable exponent; non-doubling measure; metric measure space; Riesz potential; maximal operator; Sobolev's inequality; Trudinger's exponential inequality; continuity

Bibliographie

[1] Adams, D. R.: A note on Riesz potentials. Duke Math. J. 42 (1975), 765-778. | DOI | MR | Zbl

[2] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory. Fundamental Principles of Mathematical Sciences 314 Springer, Berlin (1995). | MR | Zbl

[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 | Zbl

[4] Bojarski, B., Hajłasz, P.: Pointwise inequalities for Sobolev functions and some applications. Stud. Math. 106 (1993), 77-92. | MR | Zbl

[5] Chiarenza, F., Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. Appl., VII. Ser. 7 (1987), 273-279. | MR | Zbl

[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 | Zbl

[7] Diening, L.: Maximal function in generalized Lebesgue spaces $L^{p(\cdot)}$. Math. Inequal. Appl. 7 (2004), 245-253. | MR

[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 | MR | Zbl

[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 | Zbl

[10] Edmunds, D. E., Gurka, P., Opic, B.: Double exponential integrability, Bessel potentials and embedding theorems. Stud. Math. 115 (1995), 151-181. | MR | Zbl

[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

[12] Edmunds, D. E., Hurri-Syrjänen, R.: Sobolev inequalities of exponential type. Isr. J. Math. 123 (2001), 61-92. | DOI | MR | Zbl

[13] Edmunds, D. E., Krbec, M.: Two limiting cases of Sobolev imbeddings. Houston J. Math. 21 (1995), 119-128. | MR | Zbl

[14] Fiorenza, A., Gupta, B., Jain, P.: The maximal theorem for weighted grand Lebesgue spaces. Stud. Math. 188 (2008), 123-133. | DOI | MR | Zbl

[15] Fiorenza, A., Krbec, M.: On the domain and range of the maximal operator. Nagoya Math. J. 158 (2000), 43-61. | DOI | MR | Zbl

[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 | Zbl

[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

[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 | MR | Zbl

[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 | Zbl

[20] Greco, L., Iwaniec, T., Sbordone, C.: Inverting the $p$-harmonic operator. Manuscr. Math. 92 (1997), 249-258. | DOI | MR | Zbl

[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 | MR

[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 | MR | Zbl

[23] Gunawan, H., Sawano, Y., Sihwaningrum, I.: Fractional integral operators in nonhomogeneous spaces. Bull. Aust. Math. Soc. 80 (2009), 324-334. | DOI | MR | Zbl

[24] Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 688 (2000), 101 pages. | MR | Zbl

[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

[26] Hedberg, L. I.: On certain convolution inequalities. Proc. Am. Math. Soc. 36 (1972), 505-510. | DOI | MR

[27] Iwaniec, T., Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses. Arch. Ration. Mech. Anal. 119 (1992), 129-143. | DOI | MR | Zbl

[28] Iwaniec, T., Sbordone, C.: Riesz transforms and elliptic PDEs with VMO coefficients. J. Anal. Math. 74 (1998), 183-212. | DOI | MR | Zbl

[29] Kinnunen, J.: The Hardy-Littlewood maximal function of a Sobolev function. Isr. J. Math. 100 (1997), 117-124. | DOI | MR | Zbl

[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 | Zbl

[31] Kokilashvili, V., Samko, S. G.: Boundedness of weighted singular integral operators in grand Lebesgue spaces. Georgian Math. J. 18 (2011), 259-269. | MR | Zbl

[32] Meskhi, A.: Maximal functions, potentials and singular integrals in grand Morrey spaces. Complex Var. Elliptic Equ. 56 (2011), 1003-1019. | MR | Zbl

[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 | MR | Zbl

[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 | Zbl

[35] Mizuta, Y., Shimomura, T.: Continuity properties of Riesz potentials of Orlicz functions. Tohoku Math. J. 61 (2009), 225-240. | DOI | MR | Zbl

[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 | MR | Zbl

[37] Jr., C. B. Morrey: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43 (1938), 126-166. | DOI | MR | Zbl

[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 | MR | Zbl

[39] Peetre, J.: On the theory of $L_{p,\lambda}$ spaces. J. Funct. Anal. 4 (1969), 71-87. | DOI | MR

[40] Sawano, Y.: Generalized Morrey spaces for non-doubling measures. NoDEA, Nonlinear Differ. Equ. Appl. 15 (2008), 413-425. | DOI | MR | Zbl

[41] Sawano, Y., Tanaka, H.: Morrey spaces for non-doubling measures. Acta Math. Sin., Engl. Ser. 21 (2005), 1535-1544. | DOI | MR | Zbl

[42] Sbordone, C.: Grand Sobolev spaces and their application to variational problems. Matematiche 51 (1996), 335-347. | MR | Zbl

[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

[44] Stein, E. M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30 Princeton University Press, Princeton (1970). | MR | Zbl

[45] Trudinger, N. S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473-483. | MR | Zbl

[46] Ziemer, W. P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics 120 Springer, Berlin (1989). | DOI | MR | Zbl