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Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems. (English) Zbl 0877.46023

The present paper deals with the following optimal embedding problem: given any \(N\)-function \(A\), which is the \(N\)-function \(B\) such that the Orlicz space \(L^B(G)\) is the smallest Orlicz space into which the Orlicz-Sobolev space \(W^{1,A}(G)\) is continuously embedded \[ W^{1,A}(G)\to L^B(G)? \] Here \(G\) is a sufficiently smooth open subset of \(\mathbb{R}^n\), \(n\geq 2\).
Reviewer: J.Wloka (Kiel)

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:

[1] R.A. Adams , Sobolev Spaces , Academic Press , New York , 1975 . MR 450957 | Zbl 0314.46030 · Zbl 0314.46030
[2] R.A. Adams , On the Orlicz-Sobolev imbedding theorem , J. Funct. Anal. 24 ( 1977 ), 241 - 257 . MR 430770 | Zbl 0344.46077 · Zbl 0344.46077 · doi:10.1016/0022-1236(77)90055-6
[3] A. Alvino - P.L. Lions - G. Trombetti , On optimization problems with prescribed rearrangements , Nonlinear Anal. 13 ( 1989 ), 185 - 220 . MR 979040 | Zbl 0678.49003 · Zbl 0678.49003 · doi:10.1016/0362-546X(89)90043-6
[4] T. Aubin , Problèms isopérimetriques et espaces de Sobolev , J. Differential Geom. 11 ( 1976 ), 573 - 598 . MR 448404 | Zbl 0371.46011 · Zbl 0371.46011
[5] A. Baernstein II , A unified approach to symmetrization, in: ”Partial Differential Equations of Elliptic Type” (A. Alvino, E. Fabes and G. Talenti, eds.), Symposia Mathematica 35 , Cambridge Univ. Press , Cambridge 1994 , 47 - 91 . MR 1297773 | Zbl 0830.35005 · Zbl 0830.35005
[6] C. Bennett - R. Sharpley , Interpolation of operators , Pure and Appl. Math. Vol. 129 , Academic Press , Boston , 1988 . MR 928802 | Zbl 0647.46057 · Zbl 0647.46057
[7] E.I. Berezhnoi , Weighted inequalities of Hardy type in general ideal spaces , Soviet. Math. Dokl. 43 ( 1991 ), 492 - 495 . MR 1121581 | Zbl 0770.46008 · Zbl 0770.46008
[8] D.W. Boyd , Indices for the Orlicz spaces , Pacific J. Math. 38 ( 1971 ), 315 - 323 . Article | MR 306887 | Zbl 0227.46039 · Zbl 0227.46039 · doi:10.2140/pjm.1971.38.315
[9] J.E. Brothers - W.P. Ziemer , Minimal rearrangements of Sobolev functions , J. Reine Angew Math. 384 ( 1988 ), 153 - 179 . Article | MR 929981 | Zbl 0633.46030 · Zbl 0633.46030
[10] A. Cianchi , One relative isoperimetric inequalities in the plane , Boll. Un. Mat. Ital . A ( 7 ) 3-B ( 1989 ), 289 - 325 . MR 997998 | Zbl 0674.49030 · Zbl 0674.49030
[11] A. Cianchi , A sharp form of Poincaré type inequalities on balls and spheres , Z. Angew. Math. Phys. 40 ( 1989 ), 558 - 569 . MR 1008923 | Zbl 0707.53034 · Zbl 0707.53034 · doi:10.1007/BF00944807
[12] A. Cianchi - D.E. Edmunds - P. Gurka , On weighted Poincaré inequalities , Math. Nachr. 180 ( 1996 ), 15 - 41 . MR 1397667 | Zbl 0858.26009 · Zbl 0858.26009
[13] T.K. Donaldson - N.S. Trudinger , Orlicz-Sobolev spaces and imbedding theorems , J. Funct. Anal. 8 ( 1971 ), 52 - 75 . MR 301500 | Zbl 0216.15702 · Zbl 0216.15702 · doi:10.1016/0022-1236(71)90018-8
[14] D.E. Edmunds - P. Gurka - B. Opic , Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces , Indiana Univ. Math. J. 45 ( 1995 ). MR 1336431 | Zbl 0826.47021 · Zbl 0826.47021 · doi:10.1512/iumj.1995.44.1977
[15] N. Fusco - P.L. Lions - C. Sbordone , Some remarks on Sobolev imbeddings in borderline cases , to appear in Proc. Amer. Math. Soc. MR 1301025
[16] S. Gallot , Inégalités isopérimétriques et analitiques sur les variétés riemanniennes , Asterisque n. 163 ( 1988 ), 31 - 91 . MR 999971 | Zbl 0674.53001 · Zbl 0674.53001
[17] J.A. Hempel - G.R. Morris - N.S. Trudinger , On the sharpness of a limiting case of the Sobolev imbedding theorem , Bull. Austral. Math. Soc. 3 ( 1970 ), 369 - 373 . MR 280998 | Zbl 0205.12801 · Zbl 0205.12801 · doi:10.1017/S0004972700046074
[18] J. Lindenstrauss - L. Tzafriri , Classical Banach spaces II , Springer-Verlag , Berlin , 1979 . MR 540367 | Zbl 0403.46022 · Zbl 0403.46022
[19] V.M. Maz’ja , Sobolev spaces , Springer-Verlag , Berlin , 1985 . MR 817985
[20] , O’neill , Fractional integration in Orlicz spaces , Trans. Amer. Math. Soc. 115 ( 1965 ), 300 - 328 . MR 194881 | Zbl 0132.09201 · Zbl 0132.09201 · doi:10.2307/1994271
[21] S.I. Pohozhaev , On the imbedding Sobolev theorem for pl = n , Doklady conference, Section Math. , Moscow Power Inst. ( 1965 ), 158 - 170 .
[22] J.M. Rakotoson - R. Teman , A co-area formula with applications to monotone rearrangement and to regularity , Arch. Rat. Mech. Anal. 109 ( 1990 ), 213 - 238 . MR 1025171 | Zbl 0735.49039 · Zbl 0735.49039 · doi:10.1007/BF00375089
[23] G. Talenti , Best constant in Sobolev inequality , Ann. Mat. Pura Appl. 110 ( 1976 ), 353 - 372 . MR 463908 | Zbl 0353.46018 · Zbl 0353.46018 · doi:10.1007/BF02418013
[24] G. Talenti , An embedding theorem, Essays of Math. Analysis in honour of E. De Giorgi , Birkhäuser Verlag , Boston , 1989 . MR 1034035 | Zbl 0702.46020 · Zbl 0702.46020
[25] G. Talenti , Boundedness ofminimizers , Hokkaido Math. J. 19 ( 1990 ), 259 - 279 . MR 1059170 | Zbl 0723.58015 · Zbl 0723.58015
[26] A. Torchinsky , Real variable methods in harmonic analysis , Academic Press , San Diego , 1986 . MR 869816 | Zbl 0621.42001 · Zbl 0621.42001
[27] N.S. Trudinger , On imbeddings into Orlicz spaces and some applications , J. Math. Mech. 17 ( 1967 ), 473 - 483 . MR 216286 | Zbl 0163.36402 · Zbl 0163.36402
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