Abstract
New sharp Lorentz–Sobolev inequalities are obtained by convexifying level sets in Lorentz integrals via the L p Minkowski problem. New L p isocapacitary and isoperimetric inequalities are proved for Lipschitz star bodies. It is shown that the sharp convex Lorentz–Sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities.
Similar content being viewed by others
References
Adams D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128, 385–398 (1988)
Adams D.R.: Choquet integrals in potential theory. Publ. Mat. 42, 3–66 (1998)
Alesker S.: Continuous rotation invariant valuations on convex sets. Ann. Math. 149(2), 977–1005 (1999)
Alvino A.: Sulla diseguaglianza di Sobolev in spazi di Lorentz. Boll. Un. Mat. Ital. A 14(5), 148–156 (1977)
Aubin T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)
Bastero J., Milman M., Ruiz Blasco F.J.: A note on L ∞,q spaces and Sobolev embeddings. Indiana Univ. Math. J. 52, 1215–1230 (2003)
Bobkov S., Ledoux M.: From Brunn-Minkowski to sharp Sobolev inequalities. Ann. Mat. Pura Appl. 187, 369–384 (2008)
Böröczky K., Bárány I., Makai I. Jr, Pach J.: Maximal volume enclosed by plates and proof of the chessboard conjecture. Discrete Math. 60, 101–120 (1986)
Brothers J., Morgan F.: The isoperimetric theorem for general integrands. Michigan Math. J. 41, 419–431 (1994)
Campi S., Gronchi P.: The L p-Busemann-Petty centroid inequality. Adv. Math. 167, 128–141 (2002)
Cassani D.: Lorentz–Sobolev spaces and systems of Schrödinger equations in \({\mathbb{R}^n}\). Nonlinear Anal. 70, 2846–2854 (2009)
Chavel, I.: Isoperimetric inequalities. In: Cambridge Tracts in Mathematics, vol. 145. Cambridge University Press, Cambridge (2001)
Chen W.: L p Minkowski problem with not necessarily positive data. Adv. Math. 201, 77–89 (2006)
Cianchi A., Lutwak E., Yang D., Zhang G.: Affine Moser-Trudinger and Morrey-Sobolev inequalities. Calc. Var. Partial Differ. Equ. 36, 419–436 (2009)
Chou K., Wang X.-J.: The L p -Minkowski problem and the Minkowski problem in centroaffine geometry. Adv. Math. 205, 33–83 (2006)
Colesanti A., Salani P.: The Brunn-Minkowski inequality for p-capacity of convex bodies. Math. Ann. 327, 459–479 (2003)
Cordero-Erausquin D., Nazaret D., Villani C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 307–332 (2004)
Cwickel M., Pustylnik E.: Sobolev type embeddings in the limiting case. J. Fourier Anal. Appl. 4, 433–446 (1998)
Danielli, D., Garofalo, N., Phuc, N.C.: 2009 Inequalities of Hardy-Sobolev type in Carnot-Caratheodory spaces. In: Maz′ya, V. (ed.) Sobolev Spaces in Mathematics I. Sobolev Type Inequalities. International Mathematical Series, vol. 8, pp. 117–151 (2009)
Del Pino M., Dolbeault J.: Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81, 847–875 (2002)
Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Federer H., Fleming W.: Normal and integral currents. Ann. Math. 72, 458–520 (1960)
Frank R.L., Seiringer R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255, 3407–3430 (2008)
Gardner R.: The Brunn-Minkowski inequality. Bull. Am. Math. Soc. (N.S.) 39, 355–405 (2002)
Gardner, R.: Geometric tomography, 2nd edn. In: Encyclopedia of Mathematics and its Applications, vol. 58. Cambridge University Press, Cambridge (2006)
Grigor’yan, A.: Isoperimetric inequalities and capacities on Riemannian manifolds. In: The Maz’ya anniversary collection, vol. 1 (Rostock, 1998), pp. 139–153. Oper. Theory Adv. Appl., 109. Birkhäuser, Basel (1999)
Guan, P., Li, C.S.: On the equation det(u ij + δ ij u) = u p f on S n. No. 2000-7, NCTS in Tsing-Hua University
Haberl, C.: Blaschke valuations. Am. J. Math. (in press)
Haberl C., Ludwig M.: A characterization of L p intersection bodies. Int. Math. Res. Not. 10548, 1–29 (2006)
Haberl C., Schuster F.: Asymmetric affine L p Sobolev inequalities. J. Funct. Anal. 257, 641–658 (2009)
Haberl C., Schuster F.: General L p affine isoperimetric inequalities. J. Differ. Geom. 83, 1–26 (2009)
Haberl, C., Schuster, F., Xiao, J.: An asymmetric affine Pólya-Szegö principle. Preprint
Hardy G.H., Littlewood J.E., Polya G.: Some simple inequalities satisfied by convex functions. Messenger Math. 58, 145–152 (1929)
Hardy G.H., Littlewood J.E., Polya G.: Inequalities. Cambridge Univeristy Press, Cambridge (1952)
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)
Hug D., Lutwak E., Yang D., Zhang G.: On the L p Minkowski problem for polytopes. Discrete Comput. Geom. 33, 699–715 (2005)
Ludwig M.: Projection bodies and valuations. Adv. Math. 172, 158–168 (2002)
Ludwig M.: Ellipsoids and matrix-valued valuations. Duke Math. J. 119, 159–188 (2003)
Ludwig M.: Intersection bodies and valuations. Am. J. Math. 128, 1409–1428 (2006)
Ludwig, M., Reitzner, M.: A classification of SL(n) invariant valuations. Ann. Math. (in press)
Lutwak E.: The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)
Lutwak E.: The Brunn-Minkowski-Firey theory. Part II: affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)
Lutwak E.: Selected affine isoperimetric inequalities. In: Gruber, P.M., Wills, J.M. (eds) Handbook of Convex Geometry, vol. A, pp. 151–176. Elsevier, Horth-Holland (1993)
Lutwak E., Oliker V.: On the regularity of solutions to a generalization of the Minkowski problem. J. Differ. Geom. 41, 227–246 (1995)
Lutwak E., Yang D., Zhang G.: L p Affine isoperimetric inequalities. J. Differ. Geom. 56, 111–132 (2000)
Lutwak E., Yang D., Zhang G.: Sharp affine L p Sobolev inequalities. J. Differ. Geom. 62, 17–38 (2002)
Lutwak E., Yang D., Zhang G.: The Cramer-Rao inequality for star bodies. Duke Math. J. 112, 59–81 (2002)
Lutwak E., Yang D., Zhang G.: On the L p -Minkowski problem. Trans. Am. Math. Soc. 356, 4359–4370 (2004)
Lutwak, E., Yang, D., Zhang, G.: Optimal Sobolev norms and the L p Minkowski problem. Int. Math. Res. Not. 21. Art. ID 62987 (2006)
Malý J., Pick L.: An elementary proof of sharp Sobolev embeddings. Proc. Am. Math. Soc. 130, 555–563 (2002)
Matskewich T., Sobolevskii P.E.: The sharp constant in Hardy’s inequality for complement of bounded domain. Nonlinear Anal. 33, 105–120 (1998)
Maz′ya V.G.: Classes of domains and imbedding theorems for function spaces. Soviet Math. Dokl. 1, 882–885 (1960)
Maz′ya, V.G.: Certain integral inequalities for functions of many variables. Problems in Mathematical Analysis. Leningrad: LGU, No. 3, 33–68 (1972) (Russian). English translation: J. Soviet Math. 1, 205–234 (1973)
Maz′ya V.G.: Sobolev Spaces. Springer-Verlag, Berlin (1985)
Maz′ya V.G.: Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces. Contemp. Math. 338, 307–340 (2003)
Maz′ya V.G.: Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev-type imbeddings. J. Funct. Anal. 224, 408–430 (2005)
Maz′ya, V.G.: Integral and isocapacitary inequalities. Linear and Complex Analysis, pp. 85–107. Amer. Math. Soc. Transl. Ser. 2, 226. Amer. Math. Soc., Providence (2009)
Maz′ya V.G., Verbitsky I.E.: The Schrödinger operator on the energy space: boundedness and compactness criteria. Acta Math. 188, 263–302 (2002)
Maz′ya V.G., Verbitsky I.E.: Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator. Invent. Math. 162, 81–136 (2005)
Maz′ya V.G., Verbitsky I.E.: Form boundedness of the general second-order differential operator. Comm. Pure Appl. Math. 59, 1286–1329 (2006)
Meyer M., Werner E.: On the p-affine surface area. Adv. Math. 152, 288–313 (2000)
Milman E.: On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177, 1–43 (2009)
O’Neil R.: Convolution operators and L (p,q) spaces. Duke Math. J. 30, 129–142 (1963)
Peetre J.: Espaces d’interpolation et theorème de Soboleff. Ann. Inst. Fourier 16, 279–317 (1966)
Petty, C.M.: Isoperimetric problems. In: Proceedings of the Conference on Convexity and Combinatorial Geometry (Univ. Oklahoma, Norman, Okla., 1971), pp. 26–41. Dept. Math., Univ. Oklahoma, Norman, Oklahoma (1971)
Pólya, G., Szegö, G.: Isoperimetric inequalities in mathematical physics. Annals of Mathematics Studies, no. 27. Princeton University Press, Princeton (1951)
Ryabogin D., Zvavitch A.: The Fourier transform and Firey projections of convex bodies. Indiana Univ. Math. J. 53, 667–682 (2004)
Schneider, R.: Convex bodies: the Brunn-Minkowski theory. In: Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1993)
Schütt C., Werner E.: Surface bodies and p-affine surface area. Adv. Math. 187, 98–145 (2004)
Stancu A.: The discrete planar L 0-Minkowski problem. Adv. Math. 167, 160–174 (2002)
Stancu A.: On the number of solutions to the discrete two-dimensional L 0-Minkowski problem. Adv. Math. 180, 290–323 (2003)
Talenti G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)
Taylor J.: Crystalline variational problems. Bull. Am. Math. Soc. 84, 568–588 (1978)
Umanskiy V.: On solvability of two dimensional L p -Minkowski problem. Adv. Math. 180, 176–186 (2003)
Xiao J.: The sharp Sobolev and isoperimetric inequalities split twice. Adv. Math. 211, 417–435 (2007)
Xiao, J.: The p-Faber-Krahn inequality noted. Topics Around Research of Vladimir Maz′ya, I. Function Spaces, pp. 373–390. Springer (2010)
Zhang G.: The affine Sobolev inequality. J. Differ. Geom. 53, 183–202 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ludwig, M., Xiao, J. & Zhang, G. Sharp convex Lorentz–Sobolev inequalities. Math. Ann. 350, 169–197 (2011). https://doi.org/10.1007/s00208-010-0555-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-010-0555-x