Arkiv för Matematik

  • Ark. Mat.
  • Volume 45, Number 1 (2007), 123-139.

Characterization of Orlicz–Sobolev space

Heli Tuominen

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We give a new characterization of the Orlicz–Sobolev space W1,Ψ(Rn) in terms of a pointwise inequality connected to the Young function Ψ. We also study different Poincaré inequalities in the metric measure space.

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Ark. Mat., Volume 45, Number 1 (2007), 123-139.

Received: 19 September 2005
Revised: 21 December 2005
First available in Project Euclid: 31 January 2017

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2007 © Institut Mittag-Leffler


Tuominen, Heli. Characterization of Orlicz–Sobolev space. Ark. Mat. 45 (2007), no. 1, 123--139. doi:10.1007/s11512-006-0023-8.

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  • Adams, R. A., Sobolev spaces, Pure and Applied Mathematics, 65, Academic Press, New York–London, 1975.
  • Adams, D. R. and Hurri-Syrjänen, R., Capacity estimates, Proc. Amer. Math. Soc. 131 (2003), 1159–1167.
  • Adams, D. R. and Hurri-Syrjänen, R., Vanishing exponential integrability for functions whose gradients belong to Ln(log(e+L))α, J. Funct. Anal. 197 (2003), 162–178.
  • Bhattacharya, T. and Leonetti, F., A new Poincaré inequality and its application to the regularity of minimizers of integral functionals with nonstandard growth, Nonlinear Anal. 17 (1991), 833–839.
  • Bojarski, B. and Hajłasz, P., Pointwise inequalities for Sobolev functions and some applications, Studia Math. 106 (1993), 77–92.
  • Edmunds, D. E., Gurka, P. and Opic, B., Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces, Indiana Univ. Math. J. 44 (1995), 19–43.
  • Franchi, B., Hajłasz, P. and Koskela, P., Definitions of Sobolev classes on metric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), 1903–1924.
  • Fusco, N., Lions, P.-L. and Sbordone, C., Sobolev imbedding theorems in borderline cases, Proc. Amer. Math. Soc. 124 (1996), 561–565.
  • Hajłasz, P., Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403–415.
  • Hajłasz, P., A new characterization of the Sobolev space, Studia Math. 159 (2003), 263–275.
  • Hajłasz, P. and Koskela, P., Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), 1–101.
  • Heinonen, J., Lectures on Analysis on Metric Spaces, Universitext, Springer, New York, 2001.
  • Iwaniec, T., Koskela, P. and Onninen, J., Mappings of finite distortion: monotonicity and continuity, Invent. Math. 144 (2001), 507–531.
  • Kauhanen, J., Koskela, P., Malý, J., Onninen, J. and Zhong, X., Mappings of finite distortion: sharp Orlicz-conditions, Rev. Mat. Iberoamericana 19 (2003), 857–872.
  • Krasnosel’skiı̆, M. A. and Rutickiı̆, J. B., Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
  • Kufner, A., John, O. and Fučík, S., Function Spaces, Noordhoff, Leyden; Academia, Prague, 1977.
  • Rao, M. M. and Ren, Z. D., Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, New York, 1991.
  • Tuominen, H., Orlicz–Sobolev spaces on metric measure spaces, Ann. Acad. Sci. Fenn. Math. Diss. 135 (2004).