# zbMATH — the first resource for mathematics

Characterization of Orlicz–Sobolev space. (English) Zbl 1166.46308
Summary: We give a new characterization of the Orlicz–Sobolev space $$W ^{1,\Psi}(\mathbb{R}^n)$$ in terms of a pointwise inequality connected to the Young function $$\Psi$$. We also study different Poincaré inequalities in the metric measure space.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26D10 Inequalities involving derivatives and differential and integral operators
Full Text:
##### References:
 [1] Adams, R. A., Sobolev spaces, Pure and Applied Mathematics, 65, Academic Press, New York–London, 1975. · Zbl 0314.46030 [2] Adams, D. R. and Hurri-Syrjänen, R., Capacity estimates, Proc. Amer. Math. Soc. 131 (2003), 1159–1167. · Zbl 1032.46046 [3] Adams, D. R. and Hurri-Syrjänen, R., Vanishing exponential integrability for functions whose gradients belong to L n (log(e+L)){$$\alpha$$}, J. Funct. Anal. 197 (2003), 162–178. · Zbl 1029.46021 [4] 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. · Zbl 0779.49046 [5] Bojarski, B. and Hajłasz, P., Pointwise inequalities for Sobolev functions and some applications, Studia Math. 106 (1993), 77–92. · Zbl 0810.46030 [6] 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. · Zbl 0826.47021 [7] Franchi, B., Hajłasz, P. and Koskela, P., Definitions of Sobolev classes on metric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), 1903–1924. · Zbl 0938.46037 [8] Fusco, N., Lions, P.-L. and Sbordone, C., Sobolev imbedding theorems in borderline cases, Proc. Amer. Math. Soc. 124 (1996), 561–565. · Zbl 0841.46023 [9] Hajłasz, P., Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403–415. · Zbl 0859.46022 [10] Hajłasz, P., A new characterization of the Sobolev space, Studia Math. 159 (2003), 263–275. · Zbl 1059.46021 [11] Hajłasz, P. and Koskela, P., Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), 1–101. · Zbl 0954.46022 [12] Heinonen, J., Lectures on Analysis on Metric Spaces, Universitext, Springer, New York, 2001. · Zbl 0985.46008 [13] Iwaniec, T., Koskela, P. and Onninen, J., Mappings of finite distortion: monotonicity and continuity, Invent. Math. 144 (2001), 507–531. · Zbl 1006.30016 [14] 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. · Zbl 1059.30017 [15] Krasnosel’skiı, M. A. and Rutickiı, J. B., Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961. [16] Kufner, A., John, O. and Fučík, S., Function Spaces, Noordhoff, Leyden; Academia, Prague, 1977. [17] 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. [18] Tuominen, H., Orlicz–Sobolev spaces on metric measure spaces, Ann. Acad. Sci. Fenn. Math. Diss. 135 (2004). · Zbl 1068.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.