| Title:
|
Inequalities in rearrangement invariant function spaces (English) |
| Author:
|
Talenti, Giorgio |
| Language:
|
English |
| Journal:
|
Nonlinear Analysis, Function Spaces and Applications |
| Volume:
|
Vol. 5 |
| Issue:
|
1994 |
| Year:
|
|
| Pages:
|
177-230 |
| . |
| Category:
|
math |
| . |
| MSC:
|
26D20 |
| MSC:
|
46E30 |
| MSC:
|
46E35 |
| idZBL:
|
Zbl 0872.46020 |
| idMR:
|
MR1322313 |
| . |
| Date available:
|
2009-10-08T09:45:34Z |
| Last updated:
|
2012-08-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/702457 |
| . |
| Reference:
|
[Ada] Adams, R.A.: Sobolev spaces.Academic Press, 1975. Zbl 0314.46030, MR 0450957 |
| Reference:
|
[Al] Alvino, A.: Sulla disuguaglianza di Sobolev in spazi di Lorentz.Boll. Uni. Mat. Ital. 5 (1977), no. 14A, 148–156. MR 0438106 |
| Reference:
|
[ALT] Alvino, A., Lions, P.L., Trombetti, G.: On optimization problems with prescribed rearrangements.Nonlinear Anal. 13 (1989), 185–220. Zbl 0678.49003, MR 0979040 |
| Reference:
|
[Ar] Aronsson, G.: An integral inequality and plastic torsion.Arch. Rational Mech. Anal. 7 (1989), 23–39. MR 0540220 |
| Reference:
|
[AT] Aronsson, G., Talenti, G.: Estimating the integral of a function in terms of a distribution function of its gradient.Boll. Uni. Mat. Ital. 5 (1981), no. 18B, 885–894. Zbl 0476.49030, MR 0641744 |
| Reference:
|
[Au] Aubin, T.: Problèmes isopérimetriques et espaces de Sobolev.J. Differential Geom. 11 (1976), 573–598. Zbl 0371.46011, MR 0448404 |
| Reference:
|
[Bae] Baernstein, A.: A unified approach to symmetrization.To appear in a forthcoming volume of Sympos. Math. Zbl 0830.35005 |
| Reference:
|
[Bl] Bliss, G.A.: An integral inequality.J. London Math. Soc. 5 (1930), 40–46. MR 1574997 |
| Reference:
|
[BLL] Brascamp, H.J., Lieb, E.H., Luttinger, J.M.: A general rearrangement inequality for multiple integrals.J. Funct. Anal. 17 (1974), 227–237. Zbl 0286.26005, MR 0346109 |
| Reference:
|
[BZ] Brothers, J., Ziemer, W.: Minimal rearrangements of Sobolev functions.J. Reine Angew. Math. 384 (1988), 153–179. Zbl 0633.46030, MR 0929981 |
| Reference:
|
[BZa] Burago, Yu.D., Zalgaller, V.A.: Geometric inequalities.Grundlehren der Mathematischen Wissenschaften, Vol. 285, Springer-Verlag, Berlin – New York, 1988. Zbl 0633.53002, MR 0936419 |
| Reference:
|
[Bu] Burton, G.R.: Variational problems on classes of rearrangements and multiple configurations for steady vortices.Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 295–319. Zbl 0677.49005, MR 0998605 |
| Reference:
|
[Chi] Chiti, G.: Rearrangements of functions and convergence in Orlicz spaces.Appl. Anal. 9 (1979), 23–27. Zbl 0424.46023, MR 0536688 |
| Reference:
|
[CZ] Crowe, J.A., Zweibel, J.A.: Rearrangements of functions.J. Funct. Anal. 66 (1986), 432–438. Zbl 0612.46027, MR 0839110 |
| Reference:
|
[Ehr] Ehrhard, A.: Inégalités isopérimetriques et intégrales de Dirichlet gaussiennes.Ann. Sci. École Norm. Sup. 17 (1984), 317–332. Zbl 0546.49020, MR 0760680 |
| Reference:
|
[EST] Eydeland, A., Spruck, J., Turkington, B.: Multiconstrained variational problems of nonlinear eigenvalue type: new formulation and algorithms.Math. Comp. 55 (1990), 509–535. MR 1035931 |
| Reference:
|
[Fe] Federer, H.: Geometric measure theory.Grundlehren der Mathematischen Wissenschaften, Vol. 153, Springer-Verlag, New York, 1969. Zbl 0176.00801, MR 0257325 |
| Reference:
|
[FF] Federer, H., Fleming, W.H.: Normal and integral currents.Ann. of Math. 72 (1960), 458–520. Zbl 0187.31301, MR 0123260 |
| Reference:
|
[FP] Ferone, V., Posteraro, M.R.: Maximization on classes of functions with fixed rearrangement.Differential Integral Equations 4 (1991), 707–718. Zbl 0734.49002, MR 1108055 |
| Reference:
|
[GR] Garsia, A.M., Rodemich, E.: Monotonicity of certain functional under rearrangements.Ann. Inst. Fourier (Grenoble) 24 (1974), 67–116. MR 0414802 |
| Reference:
|
[GN] Giarrusso, E., Nunziante, D.: Symmetrization in a class of first-order Hamilton-Jacobi equations.Nonlinear Anal. 8 (1984), 289–299. Zbl 0543.35014, MR 0739660 |
| Reference:
|
[HLP] Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities.Cambridge Univ. Press, 1964. |
| Reference:
|
[He] Herz, C.: The Hardy-Littlewood maximal theorem.Symposium on harmonic analysis, Univ. of Warwick, 1968. |
| Reference:
|
[Hil] Hilden, K.: Symmetrization of functions in Sobolev spaces and the isoperimetric inequality.Manuscripta Math. 18 (1976), 215–235. Zbl 0365.46031, MR 0409773 |
| Reference:
|
[Hu] Hunt, R.: On $L(p,q)$ spaces.Enseign. Math. 12 (1966), 249–276. Zbl 0181.40301, MR 0223874 |
| Reference:
|
[Ka] Kawohl, B.: Rearrangements and convexity of level sets in PDE.Lecture Notes in Math., Vol. 1150, Springer-Verlag, Berlin – New York, 1985. Zbl 0593.35002, MR 0810619 |
| Reference:
|
[LS] Laurence, P., Stredulinsky, E.: A new approach to queer differential equations.Comm. Pure Appl. Math. 38 (1985), 333–355. Zbl 0818.35145, MR 0784478 |
| Reference:
|
[Le] Levine, H.A.: An estimate for the best constant in a Sobolev inequality involving three integral norms.Ann. Mat. Pura Appl. 124 (1980), 181–197. Zbl 0442.46028, MR 0591555 |
| Reference:
|
[Lb] Lieb, E.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities.Ann. of Math. 118 (1983), 349–374. Zbl 0527.42011, MR 0717827 |
| Reference:
|
[Lo1] Lorentz, G.G.: Some new functional spaces.Ann. of Math. 51 (1950), 37–55. Zbl 0035.35602, MR 0033449 |
| Reference:
|
[Lo2] Lorentz, G.G.: On the theory of spaces $\Lambda $.Pacific J. Math. 1 (1951), 411–429. Zbl 0043.11302, MR 0044740 |
| Reference:
|
[LZ] Luxemburg, W.A.J., Zaanen, A.C.: Notes on Banach function spaces.Indag. Math. 25, 26, 27 (1965, 1966, 1967). |
| Reference:
|
[Maz] Maz, V.G. ’j: Sobolev spaces.Springer-Verlag, Berlin – New York, 1985. MR 0817985 |
| Reference:
|
[McL] McLeod, J.B.: Rearrangements and extreme values of Dirichlet norms.Unpublished. |
| Reference:
|
[MPF] Mitrinović, D.S., Pecarić, J.E., Fink, A.M.: Classical and new inequalities in analysis.Kluwer Academic Publishers, 1993. MR 1220224 |
| Reference:
|
[Mrr] Morrey, Ch.B.: Multiple integrals in the calculus of variations.Grundlehren der Mathematischen Wissenschaften, Vol. 130, Springer-Verlag, New York, 1966. Zbl 0142.38701, MR 0202511 |
| Reference:
|
[Mos] Moser, J.: A sharp form of an inequality by N. Trudinger.Indiana Univ. Math. J. 20 (1971), 1077–1092. MR 0301504 |
| Reference:
|
[ON] O, R. ’Nei: Convolution operators and $L(p,q)$ spaces.Duke Math. J. 30 (1963), 129–142. MR 0146673 |
| Reference:
|
[ONW] O, R. ’Nei, Weiss, G.: The Hilbert transform and rearrangements of functions.Studia Math. 23 (1963), 189–198. MR 0160084 |
| Reference:
|
[Oss] Ossermann, R.: The isoperimetric inequality.Bull. Amer. Math. Soc. 84 (1978), 1182–1238. MR 0500557 |
| Reference:
|
[PT] Pelliccia, E., Talenti, G.: A proof of a logarithmic Sobolev inequality.Calculus of Variations 1 (1993), 237–242. Zbl 0796.49013, MR 1261545 |
| Reference:
|
[PS] Pólya, G., Szegö, G.: Isoperimetric inequalities in mathematical physics.Princeton Univ. Press, 1951. MR 0043486 |
| Reference:
|
[Po] Posteraro, M.R.: Un’osservazione sul riordinamento del gradiente di una funzione.Rend. Acad. Sci. Fis. Mat. Napoli 4 (1988), no. 55, 10 pp. |
| Reference:
|
[Rie] Riesz, F.: Sur une inégalité intégrale.J. London Math. Soc. 5 (1930), 162–168. MR 1574064 |
| Reference:
|
[So1] Sobolev, S.L.: On a theorem in functional analysis.Mat. Sb. 4 (1938), 471–497. |
| Reference:
|
[So2] Sobolev, S.L.: Applications of functional analysis in mathematical physics.Translations of Mathematical Monographs, Vol. 7, Amer. Math. Soc., 1963. Zbl 0123.09003, MR 0165337 |
| Reference:
|
[S1] Sperner, E.: Zur Symmetrisierung für Funktionen auf Sphären.Math. Z. 134 (1973), 317–327. MR 0340558 |
| Reference:
|
[S2] Sperner, E.: Symmetrisierung für Funktionen mehrerer reeller Variablen.Manuscripta Math. 11 (1974), 159–170. Zbl 0268.26011, MR 0328000 |
| Reference:
|
[Spi] Spiegel, W.: Über die Symmetrisierung stetiger Funktionen im Euklidischen Raum.Archiv. Math. (Basel) 24 (1973), 545–551. Zbl 0274.52011, MR 0412365 |
| Reference:
|
[SW] Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces.Princeton Mathematical Series, No. 32, Princeton Univ. Press, Princeton, N. J., 1971. Zbl 0232.42007, MR 0304972 |
| Reference:
|
[Ta1] Talenti, G.: Best constant in Sobolev inequality.Ann. Mat. Pura Appl. 110 (1976), 353–372. Zbl 0353.46018, MR 0463908 |
| Reference:
|
[Ta2] Talenti, G.: Elliptic equations and rearrangements.Ann. Scuola Norm. Sup. Pisa 3 (1976), 697–718. Zbl 0341.35031, MR 0601601 |
| Reference:
|
[Ta3] Talenti, G.: Rearrangements and partial differential equations.Inequalities (Birmingham, 1987), Lecture Notes in Pure and Appl. Math., Vol. 129, W.N. Everitt (ed.), Dekker, New York, 1991, pp. 211–230. MR 1112579 |
| Reference:
|
[Ta4] Talenti, G.: An inequality between $u^*$ and $|\text{grad $u|^*$}$.General Inequalities, 6 (Oberwolfach, 1990), Internat. Ser. Numer. Math., Vol. 103, W. Walter (ed.), Birkhäuser, Basel, 1992, pp. 175–182. Zbl 0783.26015, MR 1213004 |
| Reference:
|
[Ta5] Talenti, G.: The standard isoperimetric theorem.Handbook of Convex Geometry, P.M. Gruber & J.M. Wills (eds.), Elsevier, 1993, pp. 75–123. Zbl 0799.51015, MR 1242977 |
| Reference:
|
[Ta6] Talenti, G.: On functions whose gradients have a prescribed rearrangement.Inequalities and Applications, World Scientific Publishing Co., to appear. Zbl 0886.49009, MR 1299585 |
| Reference:
|
[Zie] Ziemer, W.P.: Weakly differentiable functions.Graduate Texts in Math., Vol. 120, Springer-Verlag, New York – Berline – Heidelberg – London – Paris – Tokyo – Hong Kong, 1989. Zbl 0692.46022, MR 1014685 |
| . |
