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Sharp affine Trudinger–Moser inequalities: A new argument

Part of: Partial differential equations Linear function spaces and their duals

Published online by Cambridge University Press:  22 October 2020

Nguyen Tuan Duy ,
Nguyen Lam  and
Phi Long Le
Nguyen Tuan Duy
Affiliation:
Faculty of Economics and Law, University of Finance-Marketing, 2/4 Tran Xuan Soan St., Tan Thuan Tay Ward, Dist. 7, HCM City, Vietname-mail:nguyenduy@ufm.edu.vn
Nguyen Lam
Affiliation:
School of Science and the Environment, Grenfell Campus, Memorial University of Newfoundland, Corner Brook, NL A2H 5G4, Canadae-mail:nlam@grenfell.mun.ca
Phi Long Le*
Affiliation:
Institute of Research and Development, Duy Tan University, Da Nang550000, Vietnam
*
e-mail:lelongphi@duytan.edu.vn
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Abstract

We set up the sharp Trudinger–Moser inequality under arbitrary norms. Using this result and the $L_{p}$ Busemann-Petty centroid inequality, we will provide a new proof to the sharp affine Trudinger–Moser inequalities without using the well-known affine Pólya–Szegö inequality.

Keywords

Lp Busemann–Petty centroid inequalityaffine energyTrudinger–Moser inequality

MSC classification

Primary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Secondary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 765 - 778
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

P.L.L. is corresponding author.

References

Adachi, S. and Tanaka, K., Trudinger type inequalities in $\mathbb{R}{n}^N$ and their best exponents. Proc. Amer. Math. Soc. 128(2000), no. 7, 2051–2057. http://dx.doi.org/10.1090/S0002-9939-99-05180-1 CrossRefGoogle Scholar
Adams, D. R., A sharp inequality of J. Moser for higher order derivatives . Ann. of Math. (2) 128(1988), no. 2, 385–398. http://dx.doi.org/10.2307/1971445 CrossRefGoogle Scholar
Adimurthi, S. K., A singular Moser-Trudinger embedding and its applications . NoDEA Nonlinear Diff. Equat. Appl. 13(2007), no. 5–6, 585–603. http://dx.doi.org/10.1007/s00030-006-4025-9 CrossRefGoogle Scholar
Adimurthi, S. K. and Yang, Y., An interpolation of Hardy inequality and Trudinger-Moser inequality in ${\mathbb{R}}^N$ and its applications. Int. Math. Res. Not. IMRN 2010, no. 13, 2394–2426. http://dx.doi.org/10.1093/imrn/rnp194 Google Scholar
Alvino, A., Ferone, V., Trombetti, G., and Lions, P. L., Convex symmetrization and applications . Ann. Inst. H. Poincaré Anal. Non Linéaire 14(1997), no. 2, 275–293. http://dx.doi.org/10.1016/S0294-1449(97)80147-3 CrossRefGoogle Scholar
Balogh, Z. M., Manfredi, J. J., and Tyson, J. T., Fundamental solution for the Q-Laplacian and sharp Moser-Trudinger inequality in Carnot groups . J. Funct. Anal. 204(2003), no. 1, 35–49. http://dx.doi.org/10.1016/S0022-1236(02)00169-6 CrossRefGoogle Scholar
Branson, T. P., Fontana, L., and Morpurgo, C., Moser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere . Ann. of Math. (2) 177(2013), no. 1, 1–52. http://dx.doi.org/10.4007/annals.2013.177.1.1 CrossRefGoogle Scholar
Carleson, L., Chang, S.-Y. A., On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. (2) 110(1986), no. 2, 113–127.Google Scholar
Cassani, D., Ruf, B., and Tarsi, C., Best constants for Moser type inequalities in Zygmund spaces . Mat. Contemp. 36(2009), 79–90.Google Scholar
Chang, S.-Y. A. and Yang, P. C., The inequality of Moser and Trudinger and applications to conformal geometry . Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math. 56(2003), no. 8, 1135–1150. http://dx.doi.org/10.1002/cpa.3029 CrossRefGoogle Scholar
Cianchi, A., Lutwak, E., Yang, D., and Zhang, G., Affine Moser-Trudinger and Morrey-Sobolev inequalities . Calc. Var. Partial Diff. Equat. 36(2009), no. 3, 419–436. http://dx.doi.org/10.1007/s00526-009-0235-4 Google Scholar
Cohn, W. S. and Lu, G., Best constants for Moser-Trudinger inequalities on the Heisenberg group . Indiana Univ. Math. J. 50(2001), no. 4, 1567–1591. http://dx.doi.org/10.1512/iumj.2001.50.2138 CrossRefGoogle Scholar
Cohn, W. S. and Lu, G., Best constants of Moser-Trudinger inequalities, fundamental solutions and one-parameter representation formulas on groups of Heisenberg type . Acta. Math. Sin. (Engl. Ser.) 18(2002) 375–390. http://dx.doi.org/10.1007/s101140200159 CrossRefGoogle Scholar
Cohn, W. S. and Lu, G., Sharp constants for Moser-Trudinger inequalities on spheres in complex space ${\mathbb{C}}^n$ . Comm. Pure Appl. Math. 57(2004), no. 11, 1458–1493. http://dx.doi.org/10.1002/cpa.20043 CrossRefGoogle Scholar
Csató, G. and Roy, P., Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions . Calc. Var. Partial Diff. Equat. 54(2015), no. 2, 2341–2366. http://dx.doi.org/10.1007/s00526-015-0867-5 Google Scholar
de Figueiredo, D. G., do Ó, J. M., and Ruf, B., Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete Contin. Dyn. Syst. 30(2011), no. 2, 455–476. http://dx.doi.org/10.3934/dcds.2011.30.455 CrossRefGoogle Scholar
De Nápoli, P. L., Haddad, J., Jiménez, C. H., and Montenegro, M., M. The sharp affine ${L}^2$ Sobolev trace inequality and variants . Math. Ann. 370(2018), no. 1–2, 287–308. http://dx.doi.org/10.1007/s00208-017-1548-9 CrossRefGoogle Scholar
do Ó, J. M., N-Laplacian equations in ${\mathbb{R}}^N$ with critical growth. Abstr. Appl. Anal. 2(1997), no. 3–4, 301–315. http://dx.doi.org/10.1155/S1085337597000419 CrossRefGoogle Scholar
Dong, M., Lam, N., and Lu, G., Sharp weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg inequalities and their extremal functions . Nonlinear Anal. 173(2018), 75–98. http://dx.doi.org/10.1016/j.na.2018.03.006 CrossRefGoogle Scholar
Flucher, M., Extremal functions for the Trudinger-Moser inequality in 2 dimensions . Comment. Math. Helv. 67(1992), 471–497. http://dx.doi.org/10.1007/BF02566514 CrossRefGoogle Scholar
Fontana, L., Sharp borderline Sobolev inequalities on compact Riemannian manifolds . Comment. Math. Helv. 68(1993), no. 3, 415–454. http://dx.doi.org/10.1007/BF02565828 CrossRefGoogle Scholar
Fontana, L. and Morpurgo, C., Adams inequalities on measure spaces . Adv. Math. 226(2011), no. 6, 5066–5119. http://dx.doi.org/10.1016/j.aim.2011.01.003 CrossRefGoogle Scholar
Haberl, C. and Schuster, F. E., General ${L}_p$ affine isoperimetric inequalities . J. Diff. Geom. 83(2009), no. 1, 1–26.Google Scholar
Haberl, C., Schuster, F. E., and Xiao, J., J. An asymmetric affine Pólya–Szegö principle . Math. Ann. 352(2012), 517–542. http://dx.doi.org/10.1007/s00208-011-0640-9 CrossRefGoogle Scholar
Haddad, J., Jiménez, C. H., and Montenegro, M., Asymmetric Blaschke-Santaló functional inequalities . J. Funct. Anal. 278(2020), no. 2, 108319, 18 pp. http://dx.doi.org/10.1016/j.jfa.2019.108319 CrossRefGoogle Scholar
Haddad, J., Jiménez, C. H., and Montenegro, M., Sharp affine Sobolev type inequalities via the ${L}_p$ Busemann–Petty centroid inequality . J. Funct. Anal. 271(2016) 454–473. http://dx.doi.org/10.1016/j.jfa.2016.03.017 CrossRefGoogle Scholar
Lam, N. and Lu, G., Sharp Adams type inequalities in Sobolev spaces ${W}^{m,({n}{/m})}\left({\mathbb{R}}^n\right)$ for arbitrary integer m . J. Diff. Equat. 253(2012), no. 4, 1143–1171. http://dx.doi.org/10.1016/j.jde.2012.04.025 CrossRefGoogle Scholar
Lam, N. and Lu, G., Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications . Adv. Math. 231(2012), no. 6, 3259–3287. http://dx.doi.org/10.1016/j.aim.2012.09.004 CrossRefGoogle Scholar
Lam, N., Lu, G., and Zhang, L., Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities . Adv. Math. 352(2019), 1253–1298. http://dx.doi.org/10.1016/j.aim.2019.06.020 CrossRefGoogle Scholar
Lam, N. and Tang, H., Sharp constants for weighted Moser-Trudinger inequalities on groups of Heisenberg type . Nonlinear Anal. 89(2013), 95–109. http://dx.doi.org/10.1016/j.na.2013.04.017 CrossRefGoogle Scholar
Li, Y., Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds . Sci. China Ser. A 48(2005), no. 5, 618–648. http://dx.doi.org/10.1360/04ys0050 CrossRefGoogle Scholar
Li, Y., Remarks on the extremal functions for the Moser-Trudinger inequality. Acta Math. Sin. (Engl. Ser.) 22(2006), no. 2, 545–550. http://dx.doi.org/10.1007/s10114-005-0568-7 CrossRefGoogle Scholar
Li, Y. and Ruf, B., B. A sharp Trudinger-Moser type inequality for unbounded domains in ${\mathbb{R}}^n$ . Indiana Univ. Math. J. 57(2008), no. 1, 451–480. http://dx.doi.org/10.1512/iumj.2008.57.3137 CrossRefGoogle Scholar
Lin, K.-C., Extremal functions for Moser’s inequality . Trans. Amer. Math. Soc. 348(1996), 2663–2671. http://dx.doi.org/10.1090/S0002-9947-96-01541-3 CrossRefGoogle Scholar
Lutwak, E., The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem . J. Diff. Geom. 38(1993), no. 1, 131–150.Google Scholar
Lutwak, E., The Brunn-Minkowski-Firey theory. II . Affine and geominimal surface areas. Adv. Math. 118(1996), no. 2, 244–294.Google Scholar
Lutwak, E., Yang, D., and Zhang, G., ${L}_p$ affine isoperimetric inequalities. J. Diff. Geom. 56(2000), no. 1, 111–132.Google Scholar
Masmoudi, N. and Sani, F., Trudinger-Moser inequalities with the exact growth condition in ${\mathbb{R}}^N$ and applications. Comm. Partial Diff. Equat. 40(2015), no. 8, 1408–1440. http://dx.doi.org/10.1080/03605302.2015.1026775 CrossRefGoogle Scholar
Moser, J., A sharp form of an inequality by Trudinger . Indiana Univ. Math. J. 20(1970/71), 1077–1092. http://dx.doi.org/10.1512/iumj.1971.20.20101 CrossRefGoogle Scholar
Nguyen, V. H., New approach to the affine Pólya-Szegö principle and the stability version of the affine Sobolev inequality . Adv. Math. 302(2016), 1080–1110. http://dx.doi.org/10.1016/j.aim.2016.08.003 CrossRefGoogle Scholar
O’Neil, R., Convolution operators and $L(p,q)$ spaces. Duke Math. J. 30(1963), 129–142.CrossRefGoogle Scholar
Petty, C. M., Centroid surfaces . Pacific J. Math. 11(1961), 1535–1547.CrossRefGoogle Scholar
Pohozhaev, S. I., On the imbedding Sobolev theorem for $pl=n$ . (Russian) Doklady Conference, Section Math. Moscow Power Inst., 1965, pp. 158–170.Google Scholar
Ruf, B., A sharp Trudinger-Moser type inequality for unbounded domains in ${\mathbb{R}}^2$ . J. Funct. Anal. 219(2005), no. 2, 340–367. http://dx.doi.org/10.1016/j.jfa.2004.06.013CrossRefGoogle Scholar
Ruf, B. and Sani, F., Sharp Adams-type inequalities in ${\mathbb{R}}^n$ . Trans. Amer. Math. Soc. 365(2013), no. 2, 645–670. http://dx.doi.org/10.1090/S0002-9947-2012-05561-9CrossRefGoogle Scholar
Schneider, R., Convex bodies: the Brunn-Minkowski theory . 2nd expanded ed., Encyclopedia of Mathematics and its Applications, 151, Cambridge University Press, Cambridge, 2014.Google Scholar
Tarsi, C., Adams’ inequality and limiting Sobolev embeddings into Zygmund spaces . Potential Anal. 37(2012), no. 4, 353–385. http://dx.doi.org/10.1007/s11118-011-9259-4 CrossRefGoogle Scholar
Trudinger, N. S., On imbeddings into Orlicz spaces and some applications . J. Math. Mech. 17(1967), 473–483. http://dx.doi.org/10.1512/iumj.1968.17.17028 Google Scholar
Xiao, J. and Zhai, Zh., Fractional Sobolev, Moser-Trudinger Morrey-Sobolev inequalities under Lorentz norms . Problems in Mathematical Analysis, 45, J. Math. Sci. (N.Y.) 166(2010), no. 3, 357–376. http://dx.doi.org/10.1007/s10958-010-9872-6 CrossRefGoogle Scholar
Yudovic, V. I., Some estimates connected with integral operators and with solutions of elliptic equations . Dokl. Akad.Nauk SSSR, 138(1961), 805–808; English trans. in Soviet Math. Doklady 2(1961), 746–749.Google Scholar
Zhang, G., The affine Sobolev inequality . J. Diff. Geom. 53(1999), no. 1, 183–202.Google Scholar