Movatterモバイル変換

[0]ホーム
URL:

×

zbMATH Open — the first resource for mathematics

from until
Reset all

Examples

GeometrySearch for the termGeometry inany field. Queries arecase-independent.
Funct*Wildcard queries are specified by* (e .g.functions,functorial, etc.). Otherwise the search isexact.''Topological group'':Phrases (multi - words) should be set in''straight quotation marks''.
au: Bourbaki & ti: AlgebraSearch forauthorBourbaki andtitleAlgebra. Theand-operator & is default and can be omitted.
Chebyshev | TschebyscheffTheor-operator| allows to search forChebyshev orTschebyscheff.
Quasi* map* py: 1989The resulting documents havepublicationyear1989.
so:Eur* J* Mat* Soc* cc:14Search for publications in a particularsource with aMathematics SubjectClassificationcode in14.
cc:*35 ! any:ellipticSearch for documents about PDEs (prefix with * to search only primary MSC); the not-operator ! eliminates all results containing the wordelliptic.
dt: b & au: HilbertThedocumenttype is set tobooks; alternatively:j forjournal articles,a forbookarticles.
py: 2000 - 2015 cc:(94A | 11T)Numberranges when searching forpublicationyear are accepted . Terms can be grouped within( parentheses).
la: chineseFind documents in a givenlanguage .ISO 639 - 1 (opens in new tab) language codes can also be used.
st: c r sFind documents that arecited, havereferences and are from asingle author.

Fields

ab Text from the summary or review (for phrases use “. ..”)
an zbMATH ID, i.e.: preliminary ID, Zbl number, JFM number, ERAM number
any Includes ab, au, cc, en, rv, so, ti, ut
arxiv arXiv preprint number
au Name(s) of the contributor(s)
br Name of a person with biographic references (to find documents about the life or work)
cc Code from the Mathematics Subject Classification (prefix with* to search only primary MSC)
ci zbMATH ID of a document cited in summary or review
db Database: documents in Zentralblatt für Mathematik/zbMATH Open (db:Zbl), Jahrbuch über die Fortschritte der Mathematik (db:JFM), Crelle's Journal (db:eram), arXiv (db:arxiv)
dt Type of the document: journal article (dt:j), collection article (dt:a), book (dt:b)
doi Digital Object Identifier (DOI)
ed Name of the editor of a book or special issue
en External document ID: DOI, arXiv ID, ISBN, and others
in zbMATH ID of the corresponding issue
la Language (use name, e.g.,la:French, orISO 639-1, e.g.,la:FR)
li External link (URL)
na Number of authors of the document in question. Interval search with “-”
pt Reviewing state: Reviewed (pt:r), Title Only (pt:t), Pending (pt:p), Scanned Review (pt:s)
pu Name of the publisher
py Year of publication. Interval search with “-”
rft Text from the references of a document (for phrases use “...”)
rn Reviewer ID
rv Name or ID of the reviewer
se Serial ID
si swMATH ID of software referred to in a document
so Bibliographical source, e.g., serial title, volume/issue number, page range, year of publication, ISBN, etc.
st State: is cited (st:c), has references (st:r), has single author (st:s)
sw Name of software referred to in a document
ti Title of the document
ut Keywords

Operators

a & bLogical and (default)
a | bLogical or
!abLogical not
abc*Right wildcard
ab cPhrase
(ab c)Term grouping

See also ourGeneral Help.

An asymmetric affine Pólya-Szegő principle.(English)Zbl 1241.26014

The classical Pólya-Szegő principle states that the symmetric rearrangement \(f^{\ast}\) of a function \(f\in W^{1,p}(\mathbb R^n)\), \(p\geq1\), remains in \(W^{1,p}(\mathbb R^n)\) and, moreover, \(\|\nabla f^{\ast}\|_p\leq\|\nabla f\|_p\). In the affine Pólya-Szegő inequality the \(L^p\) norm of the gradient \(\|\nabla f\|_p\) is replaced by the \(L^p\) affine energy \[\mathcal E_p(f)=c_{n,p}\bigl(\int_{S^{n-1}}\| D_uf\|_p^{-n}du\bigr)^{-1/n},\] where \(D_u f\) is the directional derivative of \(f\) in the direction \(u\) and the constant \(c_{n,p}\) is such that \[\mathcal E_p(f^{\ast})=\|\nabla f^{\ast}\|_p\] for \(f\in W^{1,p}(\mathbb R^n)\). Note that, unlike \(\|\nabla f\|_p\), \(\mathcal E_p(f)\) is invariant under volume preserving affine transformations on \(\mathbb R^n\).A. Cianchi,E. Lutwak,D. Yang andG. Zhang [Calc. Var. Partial Differ. Equ. 36, No. 3, 419–436 (2009;Zbl 1202.26029)] proved the affine Pólya-Szegő principle \(\mathcal E_p(f^{\ast})\leq\mathcal E_p(f)\).E. Lutwak,D. Yang andG. Zhang [J. Differ. Geom. 62, No. 1, 17–38 (2002;Zbl 1073.46027)] proved that \[\mathcal E_p(f)\leq\|\nabla f\|_p\] thus showing that the affine inequality is stronger than the original Euclidean one.
The authors introduce the asymmetric \(L^p\) affine energy \[\mathcal E_p^+(f)=2^{1/p}c_{n,p}\bigl(\int_{S^{n-1}}\| D_u^+f\|_p^{-n}du\bigr)^{-1/n},\] where \(D_u^+ f=\max\{D_uf,0\}\). The asymmetric affine energy \(\mathcal E_p^+(f)\) is again invariant under volume preserving affine transformations on \(\mathbb R^n\), but it differs from the (symmetric) affine energy \(\mathcal E_p(f)\) by the fact that the odd parts of the directional derivative do not vanish. The authors prove that for \(p\geq1\) and \(f\in W^{1,p}(\mathbb R^n)\) also \(f^\ast\) belongs to \(W^{1,p}(\mathbb R^n)\) and the estimate \(\mathcal E_p^+(f^\ast)\leq\mathcal E_p^+(f)\) holds. Using this result they derive new sharp asymmetric affine versions of various inequalities like the classical Sobolev and logarithmic Sobolev inequalities, the Nash inequality, the Moser-Trudinger inequality, the Morrey-Sobolev inequality, and the Gagliardo-Nirenberg inequality.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Cite

References:

[1]Adams R.A.: General logarithmic Sobolev inequalities and Orlicz embedding. J. Funct. Anal. 34, 292–303 (1979) ·Zbl 0425.46020 ·doi:10.1016/0022-1236(79)90036-3
[2]Aubin T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976) ·Zbl 0371.46011
[3]Bakry D., Coulhon T., Ledoux M., Saloff–Coste L.: Sobolev inequalities in disguise. Indiana Univ. Math. J. 44, 1033–1074 (1995) ·Zbl 0857.26006 ·doi:10.1512/iumj.1995.44.2019
[4]Beckner W.: Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality. Ann. Math. 138, 213–242 (1993) ·Zbl 0826.58042 ·doi:10.2307/2946638
[5]Beckner, W.: Geometric proof of Nash’s inequality. Int. Math. Res. Not. 67–71 (1998) ·Zbl 0895.35015
[6]Beckner W.: Geometric asymptotics and the logarithmic Sobolev inequality. Forum Math. 11, 105–137 (1999) ·Zbl 0917.58049 ·doi:10.1515/form.11.1.105
[7]Bendikov A.D., Maheux P.: Nash type inequalities for fractional powers of non-negative self-adjoint operators. Trans. Am. Math. Soc. 359, 3085–3097 (2007) ·Zbl 1122.47014 ·doi:10.1090/S0002-9947-07-04020-2
[8]Brothers J.E., Ziemer W.P.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384, 153–179 (1988) ·Zbl 0633.46030
[9]Burchard A.: Steiner symmetrization is continuous in W 1,p . Geom. Funct. Anal. 7, 823–860 (1997) ·Zbl 0912.46034 ·doi:10.1007/s000390050027
[10]Campi S., Gronchi P.: The L p -Busemann–Petty centroid inequality. Adv. Math. 167, 128–141 (2002) ·Zbl 1002.52005 ·doi:10.1006/aima.2001.2036
[11]Carlen E.A.: Superadditivity of Fisher’s information and logarithmic Sobolev inequalities. J. Funct. Anal. 101, 194–211 (1991) ·Zbl 0732.60020 ·doi:10.1016/0022-1236(91)90155-X
[12]Carlen, E.A., Loss, M.: Sharp constant in Nash’s inequality. Int. Math. Res. Not. 213–215 (1993) ·Zbl 0822.35018
[13]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) ·Zbl 0619.58013
[14]Chiti G.: Rearrangements of functions and convergence in Orlicz spaces. Appl. Anal. 9, 23–27 (1979) ·Zbl 0424.46023 ·doi:10.1080/00036817908839248
[15]Chou K.-S., Wang X.-J.: The L p -Minkowski problem and the Minkowski problem in centroaffine geometry. Adv. Math. 205, 33–83 (2006) ·Zbl 1245.52001 ·doi:10.1016/j.aim.2005.07.004
[16]Cianchi A.: Second-order derivatives and rearrangements. Duke Math. J. 105, 355–385 (2000) ·Zbl 1017.46023 ·doi:10.1215/S0012-7094-00-10531-5
[17]Cianchi A.: Moser–Trudinger inequalities without boundary conditions and isoperimetric problems. Indiana Univ. Math. J. 54, 669–705 (2005) ·Zbl 1097.46016 ·doi:10.1512/iumj.2005.54.2589
[18]Cianchi A.: Moser–Trudinger trace inequalities. Adv. Math. 217, 2005–2044 (2008) ·Zbl 1138.46020 ·doi:10.1016/j.aim.2007.09.007
[19]Cianchi A., Esposito L., Fusco N., Trombetti C.: A quantitative Pólya–Szegö principle. J. Reine Angew. Math. 614, 153–189 (2008) ·Zbl 1175.46021
[20]Cianchi A., Fusco N.: Functions of bounded variation and rearrangements. Arch. Ration. Mech. Anal. 165, 1–40 (2002) ·Zbl 1028.49035 ·doi:10.1007/s00205-002-0214-9
[21]Cianchi A., Fusco N.: Steiner symmetric extremals in Pólya–Szegö type inequalities. Adv. Math. 203, 673–728 (2006) ·Zbl 1110.46021 ·doi:10.1016/j.aim.2005.05.007
[22]Cianchi A., Fusco N.: Minimal rearrangements, strict convexity and minimal points. Appl. Anal. 85, 67–85 (2006) ·Zbl 1104.46016 ·doi:10.1080/00036810500277777
[23]Cianchi A., Lutwak E., Yang D., Zhang G.: Affine Moser–Trudinger and Morrey–Sobolev inequalities. Calc. Var. Partial Differ. Equ. 36, 419–436 (2009) ·Zbl 1202.26029 ·doi:10.1007/s00526-009-0235-4
[24]Cohn W.S., Lu G.: Best constants for Moser–Trudinger inequalities on the Heisenberg group. Indiana Univ. Math. J. 50, 1567–1591 (2001) ·Zbl 1019.43009 ·doi:10.1512/iumj.2001.50.2138
[25]Cordero-Erausquin D., Nazaret B., Villani C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 307–332 (2004) ·Zbl 1048.26010 ·doi:10.1016/S0001-8708(03)00080-X
[26]Del Pino M., Dolbeault J.: Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81, 847–875 (2002) ·Zbl 1112.35310
[27]Del Pino M., Dolbeault J.: The optimal Euclidean L p -Sobolev logarithmic inequality. J. Funct. Anal. 197, 151–161 (2003) ·Zbl 1091.35029 ·doi:10.1016/S0022-1236(02)00070-8
[28]Esposito L., Trombetti C.: Convex symmetrization and Pólya–Szegö inequality. Nonlinear Anal. 56, 43–62 (2004) ·Zbl 1038.26014 ·doi:10.1016/j.na.2003.07.010
[29]Federer H.: Geometric Measure Theory. Springer, Berlin (1969) ·Zbl 0176.00801
[30]Federer H., Fleming W.: Normal and integral currents. Ann. Math. 72, 458–520 (1960) ·Zbl 0187.31301 ·doi:10.2307/1970227
[31]Ferone A., Volpicelli R.: Convex symmetrization: the equality case in the Pólya–Szegö inequality. Calc. Var. Part. Differ. Equ. 21, 259–272 (2004) ·Zbl 1116.49022
[32]Flucher M.: Extremal functions for Trudinger–Moser inequality in 2 dimensions. Comment. Math. Helvetici 67, 471–497 (1992) ·Zbl 0763.58008 ·doi:10.1007/BF02566514
[33]Gardner R.J.: The Brunn–Minkowksi inequality. Bull. Am. Math. Soc. (N.S.) 39, 355–405 (2002) ·Zbl 1019.26008 ·doi:10.1090/S0273-0979-02-00941-2
[34]Gross L.: Logarithmic Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1975) ·Zbl 0318.46049 ·doi:10.2307/2373688
[35]Haberl C., Schuster F.E.: General L p affine isoperimetric inequalities. J. Differ. Geom. 83, 1–26 (2009) ·Zbl 1185.52005
[36]Haberl C., Schuster F.E.: Asymmetric affine L p Sobolev inequalities. J. Funct. Anal. 257, 641–658 (2009) ·Zbl 1180.46023 ·doi:10.1016/j.jfa.2009.04.009
[37]Hardy G., Littlewood J.E., Pólya G.: Inequalities. Cambridge University Press, Cambridge (1952)
[38]Hug D., Lutwak E., Yang D., Zhang G.: On the L p Minkowski problem for polytopes. Discrete Comput. Geom. 33, 699–715 (2005) ·Zbl 1078.52008 ·doi:10.1007/s00454-004-1149-8
[39]Humbert E.: Extremal functions for the sharp L 2-Nash inequality. Calc. Var. Partial Differ. Equ. 22, 21–44 (2005) ·Zbl 1065.58025 ·doi:10.1007/s00526-003-0265-2
[40]Kawohl, B.: Rearrangements and convexity of level sets in PDE. In: Lecture Notes in Mathematics, vol. 1150. Springer, Berlin (1985) ·Zbl 0593.35002
[41]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) ·Zbl 0603.49030 ·doi:10.1007/BF00279864
[42]Kesavan S.: Symmetrization and Applications. Series in Analysis 3. World Scientific, Hackensack (2006) ·Zbl 1110.35002
[43]Ledoux, M.: Isoperimetry and Gaussian analysis. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1994). Lecture Notes in Mathematics, vol. 1648, pp. 165–294. Springer, Berlin (1996) ·Zbl 0874.60005
[44]Lieb E., Loss M.: Analysis. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (2001) ·Zbl 0966.26002
[45]Lin K.C.: Extremal functions for Moser’s inequality. Trans. Am. Math. Soc. 348, 2663–2671 (1996) ·Zbl 0861.49001 ·doi:10.1090/S0002-9947-96-01541-3
[46]Ludwig M.: Ellipsoids and matrix-valued valuations. Duke Math. J. 119, 159–188 (2003) ·Zbl 1033.52012 ·doi:10.1215/S0012-7094-03-11915-8
[47]Ludwig M.: Minkowski valuations. Trans. Am. Math. Soc. 357, 4191–4213 (2005) ·Zbl 1077.52005 ·doi:10.1090/S0002-9947-04-03666-9
[48]Ludwig M., Reitzner M.: A classification of SL(n) invariant valuations. Ann. Math. 172, 1219–1267 (2010) ·Zbl 1223.52007 ·doi:10.4007/annals.2010.172.1223
[49]Ludwig, M., Xiao, J., Zhang, G.: Sharp convex Lorentz-Sobolev inequalities. Math. Ann., 29. doi: 10.1007/s00208-010-055-x ·Zbl 1220.26020
[50]Lutwak E.: On some affine isoperimetric inequalities. J. Differ. Geom. 23, 1–13 (1986) ·Zbl 0592.52005
[51]Lutwak E.: The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993) ·Zbl 0788.52007
[52]Lutwak E.: The Brunn–Minkowski–Firey theory. II: Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996) ·Zbl 0853.52005 ·doi:10.1006/aima.1996.0022
[53]Lutwak E., Oliker V.: On the regularity of solutions to a generalization of the Minkowski problem. J. Differ. Geom. 41, 227–246 (1995) ·Zbl 0867.52003
[54]Lutwak E., Yang D., Zhang G.: L p affine isoperimetric inequalities. J. Differ. Geom. 56, 111–132 (2000) ·Zbl 1034.52009
[55]Lutwak E., Yang D., Zhang G.: A new ellipsoid associated with convex bodies. Duke Math. J. 104, 375–390 (2000) ·Zbl 0974.52008 ·doi:10.1215/S0012-7094-00-10432-2
[56]Lutwak E., Yang D., Zhang G.: Sharp affine L p Sobolev inequalities. J. Differ. Geom. 62, 17–38 (2002) ·Zbl 1073.46027
[57]Lutwak E., Yang D., Zhang G.: On the L p Minkowski problem. Trans. Am. Math. Soc 356, 4359–4370 (2004) ·Zbl 1069.52010 ·doi:10.1090/S0002-9947-03-03403-2
[58]Lutwak, E., Yang, D., Zhang, G.: Optimal Sobolev norms and the L p Minkowski problem. Int. Math. Res. Not. 1–21 (2006) ·Zbl 1110.46023
[59]Maz’ya V.G.: Classes of domains and imbedding theorems for function spaces. Dokl. Akad. Nauk. SSSR 133, 527–530 (1960)
[60]Moser J.: A sharp form of an inequality by Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970) ·Zbl 0213.13001 ·doi:10.1512/iumj.1971.20.20101
[61]Petty, C.M.: Isoperimetric problems. In: Proc. Conf. Convexity and Combinatorial Geometry (Univ. Oklahoma 1971), pp. 26–41. University of Oklahoma (1972)
[62]Pólya, G., Szegö, G.: Isoperimetric inequalities in mathematical physics. Ann. Math. Stud. 27. Princeton University Press (1951) ·Zbl 0044.38301
[63]Ruf B.: A sharp Trudinger–Moser type inequality for unbounded domains in $${\(\backslash\)mathbb{R}\^2}$$ . J. Funct. Anal. 219, 340–367 (2005) ·Zbl 1119.46033 ·doi:10.1016/j.jfa.2004.06.013
[64]Schneider R.: Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press, Cambridge (1993) ·Zbl 0798.52001
[65]Stam A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Control 2, 255–269 (1959) ·Zbl 0085.34701 ·doi:10.1016/S0019-9958(59)90348-1
[66]Talenti G.: Best constant in Sobolev inequality. Ann. Math. Pure Appl. 110, 353–372 (1976) ·Zbl 0353.46018 ·doi:10.1007/BF02418013
[67]Talenti G.: On isoperimetric theorems in mathematical physics. In: Gruber, P.M., Wills, J.M. (eds) Handbook of Convex Geometry, North-Holland, Amsterdam (1993) ·Zbl 0804.35005
[68]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)
[69]Trudinger N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967) ·Zbl 0163.36402
[70]Weissler F.B.: Logarithmic Sobolev inequalities for the heat-diffusion semigroup. Trans. Am. Math. Soc. 237, 255–269 (1978) ·Zbl 0376.47019 ·doi:10.1090/S0002-9947-1978-0479373-2
[71]Xiao J.: The sharp Sobolev and isoperimetric inequalities split twice. Adv. Math. 211, 417–435 (2007) ·Zbl 1125.26026 ·doi:10.1016/j.aim.2006.08.006
[72]Zhang G.: The affine Sobolev inequality. J. Differ. Geom. 53, 183–202 (1999) ·Zbl 1040.53089
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.
© 2025FIZ Karlsruhe GmbHPrivacy PolicyLegal NoticesTerms & Conditions
  • Mastodon logo
 (opens in new tab)
[8]ページ先頭
©2009-2025 Movatter.jp