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Sharp generalized Trudinger inequalities via truncation. (English) Zbl 1115.46026

Imbedding problems between Sobolev spaces are particularly delicate in the case of so-called “critical exponents”. For instance, N.S.Trudinger showed in the pioneering paper [J. Math.Mech.17, 473–484 (1967; Zbl 0163.36402)] that \(W_0^{1,n}(\Omega)\) is imbedded into the Orlicz space \(\exp L^{n/(n-1)}(\Omega)\) whose generating Young function does not satisfy a \(\Delta_2\) condition. Since then, a large variety of results of this type have been published which aim at choosing “sharper” spaces among other classes of rearrangement-invariant function spaces. In the paper under review, the author discusses this problem by passing from Orlicz to Orlicz–Lorentz spaces, which yields, under some natural hypotheses on truncations, essentially stronger imbedding results.

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.)

Citations:

Zbl 0163.36402
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References:

[1] Brézis, H.; Wainger, S., A note on limiting case of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5, 773-789 (1980) · Zbl 0437.35071
[2] Cianchi, A., A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J., 45, 39-65 (1996) · Zbl 0860.46022
[3] Cianchi, A., Optimal Orlicz-Sobolev embeddings, Rev. Mat. Iberoamericana, 20, 427-474 (2004) · Zbl 1061.46031
[4] Edmunds, D. E.; Gurka, P.; Opic, B., Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces, Indiana Univ. Math. J., 44, 19-43 (1995) · Zbl 0826.47021
[5] Edmunds, D. E.; Gurka, P.; Opic, B., Double exponential integrability, Bessel potentials and embedding theorems, Studia Math., 115, 151-181 (1995) · Zbl 0829.47024
[6] Edmunds, D. E.; Gurka, P.; Opic, B., Sharpness of embeddings in logarithmic Bessel-potential spaces, Proc. Roy. Soc. Edinburgh Sect. A, 126, 995-1009 (1996) · Zbl 0860.46024
[7] Edmunds, D. E.; Hurri-Syrjänen, R., Sobolev inequalities of exponential type, Israel J. Math., 123, 61-92 (2001) · Zbl 0991.46019
[8] Edmunds, D. E.; Kerman, R.; Pick, L., Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms, J. Funct. Anal., 170, 307-355 (2000) · Zbl 0955.46019
[9] Fusco, N.; Lions, P. L.; Sbordone, C., Sobolev imbedding theorems in borderline cases, Proc. Amer. Math. Soc., 124, 561-565 (1996) · Zbl 0841.46023
[10] Hajlasz, P.; Koskela, P., Sobolev met Poincaré, Mem. Amer. Math. Soc., 145, 101 (2000) · Zbl 0954.46022
[11] Hansson, K., Imbeddings theorems of Sobolev type in potential theory, Math. Scand., 49, 77-102 (1979) · Zbl 0437.31009
[12] Hempel, J. A.; Morris, G. R.; Trudinger, N. S., On the sharpness of a limiting case of the Sobolev imbedding theorem, Bull. Austral. Math. Soc., 3, 369-373 (1970) · Zbl 0205.12801
[13] Hencl, S., A sharp form of an embedding into exponential and double exponential spaces, J. Funct. Anal., 204, 196-227 (2003) · Zbl 1034.46031
[14] Koskela, P.; Onninen, J., Sharp inequalities via truncation, J. Math. Anal. Appl., 278, 324-334 (2003) · Zbl 1019.26003
[15] Maz’ya, V., Sobolev Spaces (1975), Springer: Springer Berlin · Zbl 0407.46028
[16] Maz’ya, V., A theorem on multidimensional Schrödinger operator, Izv. Akad. Nauk, 28, 1145-1172 (1964), (in Russian) · Zbl 0148.35602
[17] Malý, J.; Pick, L., An elementary proof of sharp Sobolev embeddings, Proc. Amer. Math. Soc., 130, 555-563 (2002) · Zbl 0990.46022
[18] Martio, O.; Sarvas, J., Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math., 4, 383-401 (1979) · Zbl 0406.30013
[19] O’Neil, R., Convolution operators and \(L_{(p, q)}\) spaces, Duke Math. J., 30, 129-142 (1963) · Zbl 0178.47701
[20] Peetre, J., Espaces d’interpolation et théorème de Soboleff, Ann. Inst. Fourier, 16, 279-317 (1966) · Zbl 0151.17903
[21] Rao, M. M.; Ren, Z. D., Theory of Orlicz Spaces, Pure Appl. Math. (1991) · Zbl 0724.46032
[22] Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (1971), Princeton Univ. Press · Zbl 0232.42007
[23] Trudinger, N. S., On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17, 473-484 (1967) · Zbl 0163.36402
[24] Yudovič, V. I., Some estimates connected with integral operators and with solutions of elliptic equations, Soviet Math. Dokl., 2, 746-749 (1961) · Zbl 0144.14501
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