Geodesic
Correction theorem for Sobolev spaces constructed by a~symmetric space
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 38-55.

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A sharp correction theorem is established for Sobolev spaces in which the norm (quasinorm) of generalized derivatives is calculated in an arbitrary symmetric space. The exact relation between the norm of a corrected function in the Lipschitz space and the measure of the set on which the corrected and original functions are different makes it possible to characterize the Sobolev spaces constructed on the basis of the Marcinkiewicz space in terms of correctability. This opens a way to constructing Sobolev–Marcinkiewicz spaces for functions with an arbitrary domain of definition. 
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     author = {E. I. Berezhnoi},
     title = {Correction theorem for {Sobolev} spaces constructed by a~symmetric space},
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E. I. Berezhnoi. Correction theorem for Sobolev spaces constructed by a~symmetric space. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 38-55. http://geodesic.mathdoc.fr/item/TM_2014_284_a2/

[1] Berezhnoi E.I., “Otsenki ravnomernogo modulya nepreryvnosti funktsii iz simmetrichnykh prostranstv”, Izv. RAN. Ser. mat., 60:2 (1996), 3–20 | DOI | MR | Zbl

[2] Bojarski B., Hajłasz P., Strzelecki P., “Improved $C^{k,\lambda}$ approximation of higher order Sobolev functions in norm and capacity”, Indiana Univ. Math. J., 51:3 (2002), 507–540 | DOI | MR | Zbl

[3] Burenkov V.I., Sobolev spaces on domains, Teubner-Texte Math., 137, Teubner, Stuttgart, 1998 | DOI | MR | Zbl

[4] Calderón A.P., Zygmund A., “Local properties of solutions of elliptic partial differential equations”, Stud. math., 20 (1961), 171–225 | MR | Zbl

[5] Evans L.K., Gariepi R.F., Teoriya mery i tonkie svoistva funktsii, Nauch. kn., Novosibirsk, 2002

[6] Federer G., Geometricheskaya teoriya mery, Nauka, M., 1987 | MR | Zbl

[7] Hajłasz P., “Sobolev spaces on an arbitrary metric space”, Potenial Anal., 5 (1996), 403–415 | MR | Zbl

[8] Hajłasz P., “Sobolev spaces on metric-measure spaces”, Heat kernels and analysis on manifolds, graphs, and metric spaces, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003, 173–218 | DOI | MR | Zbl

[9] Hedberg L.I., Netrusov Yu., An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation, Mem. AMS, 188, no. 882, Amer. Math. Soc., Providence, RI, 2007 | MR

[10] Krein S.G., Petunin Yu.I., Semenov E.M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR

[11] Lindenstrauss J., Tzafriri L., Classical Banach spaces. I, II, Springer, Berlin, 1977, 1979 | MR | Zbl

[12] Liu F.-C., “A Luzin type property of Sobolev functions”, Indiana Univ. Math. J., 26:4 (1977), 645–651 | DOI | MR | Zbl

[13] Liu F.-C., Tai W.-S., “Lusin properties and interpolation of Sobolev spaces”, Topol. Methods Nonlinear Anal., 9 (1997), 163–177 | MR | Zbl

[14] Mazya V.G., Prostranstva S.L. Soboleva, Izd-vo LGU, L., 1985 | MR | Zbl

[15] Michael J.H., Ziemer W.P., “A Lusin type approximation of Sobolev functions by smooth functions”, Classical real analysis, Contemp. Math., 42, Amer. Math. Soc., Providence, RI, 1985, 135–167 | DOI | MR

[16] Nguyen H.-M., “Some new characterizations of Sobolev spaces”, J. Funct. Anal., 237 (2006), 689–720 | DOI | MR | Zbl

[17] Stein I.M., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR

[18] Stocke B.M., “A Lusin type approximation of Bessel potentials and Besov functions by smooth functions”, Math. Scand., 77 (1995), 60–70 | MR | Zbl

[19] Swanson D., “Pointwise inequalities and approximation in fractional Sobolev spaces”, Stud. math., 149 (2002), 147–174 | DOI | MR | Zbl

[20] Tuominen H., “Pointwise behaviour of Orlicz–Sobolev functions”, Ann. Mat. Pura Appl. Ser. 4, 188:1 (2009), 35–59 | DOI | MR | Zbl