Affine Moser-trudinger and Morrey-sobolev inequalities

Calculus of Variations and Partial Differential Equations

Volumn 36, Issue 3, 2009, Pages 419-436

  Cianchi, Andrea ^a   Lutwak, Erwin ^b   Yang, Deane ^b   Zhang, Gaoyong ^b  

^a UNIVERSITY OF FLORENCE   (Italy)

^b POLYTECHNIC UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70350389542     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-009-0235-4     Document Type: Article

References (56)

1. Adams D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128, 385-398 (1988).
2. Adams D.R., Hedberg L.I.: Function Spaces and Potential Theory. Springer, Berlin (1996).
3. Alberico A.: Moser type inequalities for higher order derivatives in Lorentz spaces. Potential Anal. 28, 389-400 (2008).
4. Alvino A., Ferone V., Trombetti G.: Moser-type inequalities in Lorentz spaces. Potential Anal. 5, 273-299 (1996).
5. Aubin T.: Problèmes isopérimetriques et espaces de Sobolev. J. Differ. Geom. 11, 573-598 (1976).
6. Balogh Z.M., Manfredi J.J., Tyson J.T.: Fundamental solution for the Q-Laplacian and sharp Moser-Trudinger inequality in Carnot groups. J. Funct. Anal. 204, 35-49 (2003).
7. Beckner W.: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. Math. 138, 213-242 (1993).
8. Brothers J.E., Ziemer W.P.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384, 153-179 (1988).
9. p-Busemann-Petty centroid inequality. Adv. Math. 167, 128-141 (2002).
10. Carleson L., Chang S.Y.A.: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. 110, 113-127 (1986).
11. 2. J. Differ. Geom. 27, 259-296 (1988).
12. p Minkowski problem with not necessarily positive data. Adv. Math. 201, 77-89 (2006).
13. Chiti G.: Rearrangements of functions and convergence in Orlicz spaces. Appl. Anal. 9, 23-27 (1979).
14. p-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv. Math. 205, 33-83 (2006).
15. Cianchi A.: Moser-Trudinger inequalities without boundary conditions and isoperimetric problems. Indiana Univ. Math. J. 54, 669-705 (2005).
16. Cianchi A.: Moser-Trudinger trace inequalities. Adv. Math. 217, 2005-2044 (2008).
17. Cianchi A., Fusco N.: Functions of bounded variation and rearrangements. Arch. Ration. Mech. Anal. 165, 1-40 (2002).
18. Cianchi A., Fusco N.: Steiner symmetric extremals in Pólya-Szegö type inequalities. Adv. Math. 203, 673-728 (2006).
19. Cohn W.S., Lu G.: Best constants for Moser-Trudinger inequalities on the Heisenberg group. Indiana Univ. Math. J. 50, 1567-1591 (2001).
20. de Figuereido D.G., do Ó, Marcos J., Ruf B.: On an inequality by N.Trudinger and J.Moser and related elliptic equations. Comm. Pure Appl. Math. 55, 135-152 (2002).
21. Federer H., Fleming W.: Normal and integral currents. Ann. Math. 72, 458-520 (1960).
22. Ferone A., Volpicelli R.: Convex rearrangement: equality cases in the Pólya-Szegö inequality. Calc. Var. Partial Differ. Equ. 21, 259-272 (2004).
23. Flucher M.: Extremal functions for Trudinger-Moser inequality in 2 dimensions. Comment. Math. Helvetici 67, 471-497 (1992).
24. Fontana L.: Sharp borderline Sobolev inequalities on compact Riemannian manifolds. Comment. Math. Helvetici 68, 415-454 (1993).
25. Gardner R.J.: The Brunn-Minkowski inequality. Bull. Am. Math. Soc. 39, 355-405 (2002).
26. Hang F.: On the higher order conformal covariant operators on the sphere. Commun. Contemp. Math. 9, 279-299 (2007).
27. Hu C., Ma X.-N., Shen C.: On the Christoffel-Minkowski problem of Firey's p-sum. Calc. Var. Partial Differ. Equ. 21, 137-155 (2004).
28. Kawohl, B.: Rearrangements and convexity of level sets in PDE. Lecture Notes in Math., vol. 1150. Springer, Berlin (1985).
29. Kawohl B.: On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems. Arch. Ration. Mech. Anal. 94, 227-243 (1986).
30. Lin K.C.: Extremal functions for Moser's inequality. Trans. Am. Math. Soc. 348, 2663-2671 (1996).
31. Ludwig M.: Ellipsoids and matrix-valued valuations. Duke Math. J. 119, 159-188 (2003).
32. Ludwig M.: Minkowski valuations. Trans. Am. Math. Soc. 357, 4191-4213 (2005).
33. Lutwak E.: The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131-150 (1993).
34. Lutwak E.: The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118, 244-294 (1996).
35. Lutwak E., Oliker V.: On the regularity of solutions to a generalization of the Minkowski problem. J. Differ. Geom. 41, 227-246 (1995).
36. Lutwak E., Yang D., Zhang G.: A new ellipsoid associated with convex bodies. Duke Math. J. 104, 375-390 (2000).
37. p affine isoperimetric inequalities. J. Differ. Geom. 56, 111-132 (2000).
38. p Sobolev inequalities. J. Differ. Geom. 62, 17-38 (2002).
39. Lutwak E., Yang D., Zhang G.: The Cramer-Rao inequality for star bodies. Duke Math. J. 112, 59-81 (2002).
40. p Minkowski problem. Trans. Am. Math. Soc. 356, 4359-4370 (2004).
41. p Minkowski Problem. Int. Math. Res. Not. (2006) Art. ID 62987, 21 pp.
42. Maz'ya, V.M.: Classes of domains and imbedding theorems for function spaces, Dokl. Akad. Nauk. SSSR 133 (1960), 527-530 (Russian); English translation: Soviet Math. Dokl. 1 (1960), 882-885.
43. Marshall D.E.: A new proof of a sharp inequality concerning the Dirichlet integral. Ark. Math. 27, 131-137 (1989).
44. Meyer M., Werner E.: On the p-affine surface area. Adv. Math. 152, 288-313 (2000).
45. Moser J.: A sharp form of an inequality by Trudinger. Indiana Univ. Math. J. 20, 1077-1092 (1970/71).
46. Pohozhaev, S.I.: On the imbedding Sobolev theorem for pl = n. Doklady Conference, Section Math. Moscow Power Inst, pp. 158-170 (1965) (Russian).
47. 2. J. Funct. Anal. 219, 340-367 (2005).
48. Ryabogin D., Zvavitch A.: The Fourier transform and Firey projections of convex bodies. Indiana Univ. Math. J. 53, 667-682 (2004).
49. Schütt C., Werner E.: Surface bodies and p-affine surface area. Adv. Math. 187, 98-145 (2004).
50. 0-Minkowski problem. Adv. Math. 167, 160-174 (2002).
51. Schneider R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993).
52. Talenti G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353-372 (1976).
53. Talenti, G.: Inequalities in rearrangement invariant function spaces. In: Krbec, M., Kufner, A., Opic, B., Rákosnik, J. (eds.) Nonlinear Analysis, Function Spaces and Applications, vol. 5, pp. 177-230. Prometheus, Prague (1994).
54. Trudinger N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473-483 (1967).
55. Yudovic, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Dokl. Akad.Nauk SSSR, vol. 138, pp. 805-808 (1961), English translation in Soviet Math. Doklady 2 (1961), 746-749.
56. Zhang G.: The affine Sobolev inequality. J. Differ. Geom. 53, 183-202 (1999).