Variable Hajłasz-Sobolev spaces on compact metric spaces
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Michał Gaczkowski
Abstract
We study variable exponent Sobolev spaces on compact metric spaces. Without the assumption of log-Hölder continuity of the exponent, the compact Sobolev-type embeddings theorems for these spaces are shown.
Acknowledgement
The authors thank to the referees for their careful reading of the paper, which helped to improve the text. Moreover, we wish to thank Marcin Dudziński for reading our manuscript.
References
[1] Almeida, A.—Samko, S.: Pointwise inequalities in variable Sobolev spaces and applications, Z. Anal. Anwend. 26 (2007), 179–193.10.4171/ZAA/1317Search in Google Scholar
[2] Almeida, A.—Samko, S.: Embeddings of variable Hajłasz-Sobolev spaces into Hölder spaces of variable order, J. Math. Anal. Appl. 353 (2009), no. 2, 489–496.10.1016/j.jmaa.2008.12.034Search in Google Scholar
[3] Cruz-Uribe, D. V.—Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Heidelberg, 2013.10.1007/978-3-0348-0548-3Search in Google Scholar
[4] Diening, L.: Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces Lp(·) and Wk,p(·), Math. Narch. 263 (2004), 31–43.Search in Google Scholar
[5] Diening, L.—Hästö, P.—Nekvinda, A.: Open problems in variable exponent Lebesgue and Sobolev spaces, in: Proc. Int. Conference Function Spaces, Differential Operators and Nonlinear Analysis, Milovy, Czech Rep., 2004, Math. Inst. Acad. Sci. Czech Rep., Prague, 2005, pp. 38–58.Search in Google Scholar
[6] Diening, L.—Harjulehto, P.—Hästö, P.—Růžička, M.: Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, Springer, Heidelberg, 2011.10.1007/978-3-642-18363-8Search in Google Scholar
[7] Fan, X.—Zhao, D.: On the Spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446.10.1006/jmaa.2000.7617Search in Google Scholar
[8] 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.Search in Google Scholar
[9] Gaczkowski, M.—Górka, P.: Sobolev spaces with variable exponents on Riemannian manifolds, Nonlinear Anal. 92 (2013), 47–59.10.1016/j.na.2013.06.012Search in Google Scholar
[10] Górka, P.—Macios, A.: Almost everything you need to know about relatively compact sets in variable Lebesgue spaces, J. Funct. Anal. 269 (2015), 1925–1949.10.1016/j.jfa.2015.06.024Search in Google Scholar
[11] Hajłasz, P.: Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403–415.Search in Google Scholar
[12] Hajłasz, P.: Sobolev spaces on metric-measure spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), Contemp. Math. 338 (2003), 173–218.10.1090/conm/338/06074Search in Google Scholar
[13] Hajłasz, P.—Koskela, P.: Sobolev met Poincaré, Memoirs Amer. Math. Soc. 688 (2000), 1–101.10.1090/memo/0688Search in Google Scholar
[14] Harjulehto, P.—Hästö, P.: Sobolev inequalities for variable exponents attaining the values 1 and n, Publ. Mat. 52 (2008), 347–367.10.5565/PUBLMAT_52208_05Search in Google Scholar
[15] Harjulehto, P.—Hästö, P.—Latvala, V.: Sobolev embeddings in metric measure spaces with variable dimension, Math. Z. 254 (2006), 591–609.10.1007/s00209-006-0960-8Search in Google Scholar
[16] Harjulehto, P.—Hästö, P.—Pere, M.: Variable exponent Lebesgue spaces on metric spaces: The Hardy-Littlewood maximal operator, Real Anal. Exchange 30 (2004), 87–104.10.14321/realanalexch.30.1.0087Search in Google Scholar
[17] P. Harjulehto, P. Hästö-Pere, M.: Variable exponent Sobolev spaces on metric measure spaces, Funct. Approx. Comment. Math. 36 (2006), 79–94.10.7169/facm/1229616443Search in Google Scholar
[18] Kałamajska, A.: On compactness of embedding for Sobolev spaces defined on metric spaces, Ann. Acad. Sci. Fenn. 24 (1999), 123–132.Search in Google Scholar
[19] Kováčik, O.—Rákosník, J.: On spaces Lp(x)(Ω) and Wk,p(x)(Ω), Czechoslovak Math. J. 41 (1991), 592–618.10.21136/CMJ.1991.102493Search in Google Scholar
[20] Li, F.—Li, Z.—Pi, L.: Variable exponent functionals in image restoration, Appl. Math. Comput. 216 (2010), 870–882.10.1016/j.amc.2010.01.094Search in Google Scholar
[21] Mizuta, Y.—Shimomura, T.: Continuity of Sobolev functions of variable exponent on metric spaces, Proc. Japan Acad. Ser. A Math. Sci., 80, (2004), 96–99.10.3792/pjaa.80.96Search in Google Scholar
[22] Ross, B.—Samko, S.: Fractional integration operator of variable order in the spaces Hλ, Int. J. Math. Sci. 18 (1995), 777–788.10.1155/S0161171295001001Search in Google Scholar
[23] Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.10.1007/BFb0104029Search in Google Scholar
[24] Samko, S.: On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms Spec. Funct. 16 (2005), 461–482.10.1080/10652460412331320322Search in Google Scholar
[25] Zhikov, V. V.: Averaging of functionals in the calculus of variations and elasticity, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 675–710.10.1070/IM1987v029n01ABEH000958Search in Google Scholar
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Articles in the same Issue
- State hoops
- On derivations of partially ordered sets
- Interior and closure operators on commutative basic algebras
- When is the cayley graph of a semigroup isomorphic to the cayley graph of a group
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- δ-Fibonacci and δ-lucas numbers, δ-fibonacci and δ-lucas polynomials
- Law of inertia for the factorization of cubic polynomials – the case of primes 2 and 3
- Closed hereditary coreflective subcategories in epireflective subcategories of Top
- Method of upper and lower solutions for coupled system of nonlinear fractional integro-differential equations with advanced arguments
- Difference of two strong Światkowski lower semicontinuous functions
- Fejér-type inequalities (II)
- Representation of maxitive measures: An overview
- On a conjecture of Y. H. Cao and X. B. Zhang
- On the generalized orthogonal stability of the pexiderized quadratic functional equations in modular spaces
- Almost everywhere convergence of some subsequences of Fejér means for integrable functions on some unbounded Vilenkin groups
- Homological properties of banach modules over abstract segal algebras
- Variable Hajłasz-Sobolev spaces on compact metric spaces
- Commuting pairs of self-adjoint elements in C*-algebras
- Additivity of maps preserving products AP ± PA* on C*-algebras
- A note on derived connections from semi-symmetric metric connections
- Lindelöf P-spaces need not be Sokolov
- Strong convergence properties for arrays of rowwise negatively orthant dependent random variables
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