| Title:
|
Sharp generalized Trudinger inequalities via truncation for embedding into multiple exponential spaces (English) |
| Author:
|
Černý, Robert |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
51 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
577-593 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We prove that the generalized Trudinger inequality for Orlicz-Sobolev spaces embedded into multiple exponential spaces implies a version of an inequality due to Brézis and Wainger. (English) |
| Keyword:
|
Orlicz spaces |
| Keyword:
|
Sobolev inequalities |
| MSC:
|
46E30 |
| MSC:
|
46E35 |
| idZBL:
|
Zbl 1224.46063 |
| idMR:
|
MR2858262 |
| . |
| Date available:
|
2010-11-30T16:19:27Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140839 |
| . |
| Reference:
|
[1] Brézis H., Wainger S.: A note on limiting case of Sobolev embeddings and convolution inequalities.Comm. Partial Differential Equations 5 (1980), no. 7, 773–789. MR 0579997, 10.1080/03605308008820154 |
| Reference:
|
[2] Černý R., Mašková S.: A sharp form of an embedding into multiple exponential spaces.Czechoslovak Math. J. 60 (2010), no. 3, 751–782. MR 2672414, 10.1007/s10587-010-0048-9 |
| Reference:
|
[3] Cianchi A.: A sharp embedding theorem for Orlicz-Sobolev spaces.Indiana Univ. Math. J. 45 (1996), 39–65. Zbl 0860.46022, MR 1406683, 10.1512/iumj.1996.45.1958 |
| Reference:
|
[4] Cianchi A.: Optimal Orlicz-Sobolev embeddings.Rev. Mat. Iberoamericana 20 (2004), 427–474. Zbl 1061.46031, MR 2073127, 10.4171/RMI/396 |
| Reference:
|
[5] Edmunds D.E., Gurka P., Opic B.: Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces.Indiana Univ. Math. J. 44 (1995), 19–43. Zbl 0826.47021, MR 1336431, 10.1512/iumj.1995.44.1977 |
| Reference:
|
[6] Edmunds D.E., Gurka P., Opic B.: Double exponential integrability, Bessel potentials and embedding theorems.Studia Math. 115 (1995), 151–181. Zbl 0829.47024, MR 1347439 |
| Reference:
|
[7] Edmunds D.E., Gurka P., Opic B.: Sharpness of embeddings in logarithmic Bessel-potential spaces.Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 995–1009. Zbl 0860.46024, MR 1415818 |
| Reference:
|
[8] Edmunds D.E., Kerman R., Pick L.: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms.J. Funct. Anal. 170 (2000), no. 2, 307–355. Zbl 0955.46019, MR 1740655, 10.1006/jfan.1999.3508 |
| Reference:
|
[9] Fusco N., Lions P.L., Sbordone C.: Sobolev imbedding theorems in borderline cases.Proc. Amer. Math. Soc. 124 (1996), 561–565. Zbl 0841.46023, MR 1301025, 10.1090/S0002-9939-96-03136-X |
| Reference:
|
[10] Hajlasz P., Koskela P.: Sobolev met Poincaré.Memoirs of the Amer. Math. Soc 145 (2000), 101pp. Zbl 0954.46022, MR 1683160 |
| Reference:
|
[11] Hansson K.: Imbeddings theorems of Sobolev type in potential theory.Math. Scand. 49 (1979), 77–102. MR 0567435 |
| Reference:
|
[12] Hedberg L.I.: On certain convolution inequalities.Proc. Amer. Math. Soc. 36 (1972), 505–512. Zbl 0283.26003, MR 0312232, 10.1090/S0002-9939-1972-0312232-4 |
| Reference:
|
[13] 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 (1970), 369–373. Zbl 0205.12801, MR 0280998, 10.1017/S0004972700046074 |
| Reference:
|
[14] Hencl S.: A sharp form of an embedding into exponential and double exponential spaces.J. Funct. Anal. 204 (2003), no. 1, 196–227. Zbl 1034.46031, MR 2004749, 10.1016/S0022-1236(02)00172-6 |
| Reference:
|
[15] Hencl S.: Sharp generalized Trudinger inequalities via truncation.J. Math. Anal. Appl. 326 (2006), no. 1, 336–348. Zbl 1115.46026, MR 2239242, 10.1016/j.jmaa.2005.07.041 |
| Reference:
|
[16] Koskela P., Onninen J.: Sharp inequalities via truncation.J. Math. Anal. Appl. 278 (2003), 324–334. Zbl 1019.26003, MR 1974010, 10.1016/S0022-247X(02)00465-1 |
| Reference:
|
[17] Maz'ya V.: Sobolev Spaces.Springer, Berlin, 1975. Zbl 1152.46002, MR 0817985 |
| Reference:
|
[18] Maz'ya V.: A theorem on multidimensional Schrödinger operator.(Russian), Izv. Akad. Nauk 28 (1964), 1145–1172. |
| Reference:
|
[19] Malý J., Pick L.: An elementary proof of sharp Sobolev embeddings.Proc. Amer. Math. Soc. 130 (2002), no. 2, 555–563. MR 1862137, 10.1090/S0002-9939-01-06060-9 |
| Reference:
|
[20] O'Neil R.: Convolution operators and $L_{(p,q)}$ spaces.Duke Math. J. 30 (1963), 129–142. Zbl 0178.47701, MR 0146673, 10.1215/S0012-7094-63-03015-1 |
| Reference:
|
[21] Peetre J.: Espaces d'interpolation et théorème de Soboleff.Ann. Inst. Fourier 16 (1966), 279–317. Zbl 0151.17903, MR 0221282, 10.5802/aif.232 |
| Reference:
|
[22] Pohozhaev S.I.: On the imbedding Sobolev theorem for $pl=n$.Doklady Conference, Section Math. Moscow Power Inst. (1965), 158–170. |
| Reference:
|
[23] Rao M.M., Ren Z.D.: Theory of Orlicz Spaces.Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, New York, 1991. Zbl 0724.46032, MR 1113700 |
| Reference:
|
[24] Strichartz R.S.: A note on Trudinger's extension of Sobolev's inequality.Indiana Univ. Math. J. 21 (1972), 841–842. MR 0293389, 10.1512/iumj.1972.21.21066 |
| Reference:
|
[25] Trudinger N.S.: On imbeddings into Orlicz spaces and some applications.J. Math. Mech. 17 (1967), 473–484. Zbl 0163.36402, MR 0216286 |
| Reference:
|
[26] Yudovič V.I.: Some estimates connected with integral operators and with solutions of elliptic equations.Soviet Math. Doklady 2 (1961), 746–749. |
| . |
