Milman, Mario; Pustylnik, Evgeniy On sharp higher order Sobolev embeddings. (English) Zbl 1108.46029 Commun. Contemp. Math. 6, No. 3, 495-511 (2004). It is well-known that the classical Sobolev embedding theorem \[ W_0^{k,p}(\Omega) \subset L^q(\Omega),\;{1\over q} = {1\over p} - {k \over n},\;1< p < {n \over k}, \tag{1} \]fails in the limit case \(p = {n \over k}\). The main result of the present paper is an extension of (1) to this case, by replacing \(L^\infty(\Omega)\) with a more exotic space denoted by \(L(\infty, p)(\Omega)\). If \(1 < p \leq \infty\), \(0 < q \leq \infty\), \(L(p,q)(\Omega)\) stands for the class of measurable functions such that \[ \int_0^\infty \{[(f^{**}(t) - f^*(t)) t^{1/p}]^q\, {dt/t}\}^{1/q} < \infty. \] Here, \(f^*\) is the usual decreasing rearrangement function, while, for \(t > 0\), \[ f^{**}(t) = {1\over t} \int_0^t f^*(u)\, du. \] The result is optimal in the sense that, if \(X(\Omega)\) is a rearrangement invariant space containing \(W_0^{{n \over k},p}(\Omega)\), then \(X(\Omega)\) is contained in \(L(\infty,p)(\Omega)\). Reviewer: Davide Guidetti (Bologna) Cited in 1 ReviewCited in 26 Documents 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 Keywords:Sobolev embedding theorem; rearrangement invariant space PDFBibTeX XMLCite \textit{M. Milman} and \textit{E. Pustylnik}, Commun. Contemp. Math. 6, No. 3, 495--511 (2004; Zbl 1108.46029) Full Text: DOI References: [1] DOI: 10.2307/1971445 · Zbl 0672.31008 · doi:10.2307/1971445 [2] DOI: 10.1016/0362-546X(89)90043-6 · Zbl 0678.49003 · doi:10.1016/0362-546X(89)90043-6 [3] DOI: 10.2307/2006999 · Zbl 0465.42015 · doi:10.2307/2006999 [4] Bennett C., Interpolation of Operators (1988) · Zbl 0647.46057 [5] Brézis H., Comm. Partial Differential Equations pp 773– [6] DOI: 10.1215/S0012-7094-00-10531-5 · Zbl 1017.46023 · doi:10.1215/S0012-7094-00-10531-5 [7] DOI: 10.1007/BF02384772 · Zbl 1035.46502 · doi:10.1007/BF02384772 [8] DOI: 10.1007/BF02498218 · Zbl 0930.46027 · doi:10.1007/BF02498218 [9] DOI: 10.1006/jfan.1999.3508 · Zbl 0955.46019 · doi:10.1006/jfan.1999.3508 [10] DOI: 10.1007/BF01774283 · Zbl 0639.46034 · doi:10.1007/BF01774283 [11] Hansson K., Math. Scand. 45 pp 77– · Zbl 0437.31009 · doi:10.7146/math.scand.a-11827 [12] DOI: 10.1070/SM1969v008n02ABEH001114 · Zbl 0193.09303 · doi:10.1070/SM1969v008n02ABEH001114 [13] DOI: 10.1070/SM1970v011n03ABEH001297 · Zbl 0216.15704 · doi:10.1070/SM1970v011n03ABEH001297 [14] DOI: 10.1070/RM1989v044n05ABEH002287 · Zbl 0715.41050 · doi:10.1070/RM1989v044n05ABEH002287 [15] DOI: 10.1090/S0002-9939-01-06060-9 · Zbl 0990.46022 · doi:10.1090/S0002-9939-01-06060-9 [16] Maz’ya V. G., Sobolev Spaces (1985) [17] Maz’ya V. G., Problems in Mathematical Analysis, Leningrad: LGU 3 pp 33– · Zbl 1187.42011 [18] DOI: 10.1215/S0012-7094-63-03015-1 · Zbl 0178.47701 · doi:10.1215/S0012-7094-63-03015-1 [19] DOI: 10.1007/BF01305216 · Zbl 0686.46029 · doi:10.1007/BF01305216 [20] DOI: 10.1006/jath.2002.3730 · Zbl 1032.46045 · doi:10.1006/jath.2002.3730 [21] Stein E. M., Singular Integrals and Differentiability of Functions (1970) · Zbl 0207.13501 [22] DOI: 10.1512/iumj.1972.21.21066 · Zbl 0241.46028 · doi:10.1512/iumj.1972.21.21066 [23] G. Talenti, Nonlinear Analysis, Function Spaces and Applications 5 (Prometheus, Prague, 1995) pp. 177–230. [24] Trudinger N. S., J. Math. Mech. 17 pp 473– [25] Yudovich V., Dokl. Akad. Nauk SSSR 138 pp 805– 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.