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Moser inequalities in Gauss space

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Abstract

The sharp constants in a family of exponential Sobolev type inequalities in Gauss space are exhibited. They constitute the Gaussian analogues of the Moser inequality in the borderline case of the Sobolev embedding in the Euclidean space. Interestingly, the Gaussian results have features in common with the Euclidean ones, but also reveal marked diversities.

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References

  1. Adams, D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. (2) 128(2), 385–398 (1988)

    MathSciNet  MATH  Google Scholar 

  2. Adams, R.A.: General logarithmic Sobolev inequalities and Orlicz imbeddings. J. Funct. Anal. 34(2), 292–303 (1979)

    MathSciNet  MATH  Google Scholar 

  3. Aida, S., Masuda, T., Shigekawa, I.: Logarithmic Sobolev inequalities and exponential integrability. J. Funct. Anal. 126(1), 83–101 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Alberico, A.: Moser type inequalities for higher-order derivatives in Lorentz spaces. Potential Anal. 28(4), 389–400 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Alvino, A., Ferone, V., Trombetti, G.: Moser-type inequalities in Lorentz spaces. Potential Anal. 5(3), 273–299 (1996)

    MathSciNet  MATH  Google Scholar 

  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(1), 35–49 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Barthe, F., Cattiaux, P., Roberto, C.: Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoam. 22(3), 993–1067 (2006)

    MathSciNet  MATH  Google Scholar 

  8. Barthe, F., Kolesnikov, A.V.: Mass transport and variants of the logarithmic Sobolev inequality. J. Geom. Anal. 18(4), 921–979 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Beckner, W.: Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality. Ann. Math. (2) 138(1), 213–242 (1993)

    MathSciNet  MATH  Google Scholar 

  10. Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics, vol. 129. Academic Press Inc, Boston (1988)

    MATH  Google Scholar 

  11. Bobkov, S.G., Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163(1), 1–28 (1999)

    MathSciNet  MATH  Google Scholar 

  12. Bobkov, S.G., Houdré, C.: Some connections between isoperimetric and Sobolev-type inequalities. Mem. Am. Math. Soc. 129(616), viii+111 (1997)

    MathSciNet  MATH  Google Scholar 

  13. Bobkov, S.G., Ledoux, M.: On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156(2), 347–365 (1998)

    MathSciNet  MATH  Google Scholar 

  14. Borell, C.: The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30(2), 207–216 (1975)

    MathSciNet  MATH  Google Scholar 

  15. Brandolini, B., Chiacchio, F., Trombetti, C.: Hardy type inequalities and Gaussian measure. Commun. Pure Appl. Anal. 6(2), 411–428 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Branson, T.P., Fontana, L., Morpurgo, C.: Moser–Trudinger and Beckner–Onofri’s inequalities on the CR sphere. Ann. Math. (2) 177(1), 1–52 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Carlen, E.A., Kerce, C.: On the cases of equality in Bobkov’s inequality and Gaussian rearrangement. Calc. Var. Partial Differ. Equ. 13(1), 1–18 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Černý, R., Mašková, S.: A sharp form of an embedding into multiple exponential spaces. Czechoslovak Math. J. 60(3), 751–782 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Cianchi, A.: Moser–Trudinger inequalities without boundary conditions and isoperimetric problems. Indiana Univ. Math. J. 54(3), 669–705 (2005)

    MathSciNet  MATH  Google Scholar 

  20. Cianchi, A.: Moser–Trudinger trace inequalities. Adv. Math. 217(5), 2005–2044 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Cianchi, A., Fusco, N., Maggi, F., Pratelli, A.: On the isoperimetric deficit in Gauss space. Am. J. Math. 133(1), 131–186 (2011)

    MathSciNet  MATH  Google Scholar 

  22. Cianchi, A., Pick, L.: Optimal Gaussian Sobolev embeddings. J. Funct. Anal. 256(11), 3588–3642 (2009)

    MathSciNet  MATH  Google Scholar 

  23. Cipriani, F.: Sobolev–Orlicz imbeddings, weak compactness, and spectrum. J. Funct. Anal. 177(1), 89–106 (2000)

    MathSciNet  MATH  Google Scholar 

  24. Cohn, W.S., Lu, G.: Best constants for Moser–Trudinger inequalities on the Heisenberg group. Indiana Univ. Math. J. 50(4), 1567–1591 (2001)

    MathSciNet  MATH  Google Scholar 

  25. Ehrhard, A.: Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes. Ann. Sci. École Norm. Sup. (4) 17(2), 317–332 (1984)

    MathSciNet  MATH  Google Scholar 

  26. Feissner, G.F.: Hypercontractive semigroups and Sobolev’s inequality. Trans. Am. Math. Soc. 210, 51–62 (1975)

    MathSciNet  MATH  Google Scholar 

  27. Fontana, L.: Sharp borderline Sobolev inequalities on compact Riemannian manifolds. Comment. Math. Helv. 68(3), 415–454 (1993)

    MathSciNet  MATH  Google Scholar 

  28. Fontana, L., Morpurgo, C.: Adams inequalities on measure spaces. Adv. Math. 226(6), 5066–5119 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Fontana, L., Morpurgo, C.: Sharp Moser–Trudinger inequalities for the Laplacian without boundary conditions. J. Funct. Anal. 262(5), 2231–2271 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Fontana, L., Morpurgo, C.: Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on \({\mathbb{R}}^n\). Nonlinear Anal. 167, 85–122 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Fujita, Y.: An optimal logarithmic Sobolev inequality with Lipschitz constants. J. Funct. Anal. 261(5), 1133–1144 (2011)

    MathSciNet  MATH  Google Scholar 

  32. Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)

    MathSciNet  MATH  Google Scholar 

  33. Hencl, S.: A sharp form of an embedding into exponential and double exponential spaces. J. Funct. Anal. 204(1), 196–227 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Ishiwata, M.: Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in \({\mathbb{R}}^N\). Math. Ann. 351(4), 781–804 (2011)

    MathSciNet  MATH  Google Scholar 

  35. Karmakar, D., Sandeep, K.: Adams inequality on the hyperbolic space. J. Funct. Anal. 270(5), 1792–1817 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Lam, N., Lu, G.: A new approach to sharp Moser–Trudinger and Adams type inequalities: a rearrangement-free argument. J. Differ. Equ. 255(3), 298–325 (2013)

    MathSciNet  MATH  Google Scholar 

  37. Leckband, M.: Moser’s inequality on the ball \(B^n\) for functions with mean value zero. Commun. Pure Appl. Math. 58(6), 789–798 (2005)

    MathSciNet  MATH  Google Scholar 

  38. Ledoux, M.: Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter. J. Math. Kyoto Univ. 35(2), 211–220 (1995)

    MathSciNet  MATH  Google Scholar 

  39. Li, Y., Liu, P.: A Moser–Trudinger inequality on the boundary of a compact Riemann surface. Math. Z. 250(2), 363–386 (2005)

    MathSciNet  MATH  Google Scholar 

  40. Li, Y., Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \({\mathbb{R}}^n\). Indiana Univ. Math. J. 57(1), 451–480 (2008)

    MathSciNet  MATH  Google Scholar 

  41. Masmoudi, N., Sani, F.: Trudinger–Moser inequalities with the exact growth condition in \(\mathbb{R}^N\) and applications. Commun. Partial Differ. Equ. 40(8), 1408–1440 (2015)

    MATH  Google Scholar 

  42. Milman, E.: On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177(1), 1–43 (2009)

    MathSciNet  MATH  Google Scholar 

  43. Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970/71)

  44. Pelliccia, E., Talenti, G.: A proof of a logarithmic Sobolev inequality. Calc. Var. Partial Differ. Equ. 1(3), 237–242 (1993)

    MathSciNet  MATH  Google Scholar 

  45. Pohozhaev, S.I.: On the imbedding Sobolev theorem for \(pl=n\). In: Doklady Conference, Section Math. Moscow Power Inst., vol. 165, pp. 158–170 (1965) (Russian)

  46. Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. Marcel Dekker Inc, New York (1991)

    Google Scholar 

  47. Rothaus, O.S.: Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities. J. Funct. Anal. 64(2), 296–313 (1985)

    MathSciNet  MATH  Google Scholar 

  48. Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \({\mathbb{R}}^2\). J. Funct. Anal. 219(2), 340–367 (2005)

    MathSciNet  MATH  Google Scholar 

  49. Sudakov, V.N., Cirel’son, B.S.: Extremal properties of half-spaces for spherically invariant measures. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Problems in the theory of probability distributions, II, vol. 41, no. 165, pp. 14–24 (1974) (Russian)

  50. Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)

    MathSciNet  MATH  Google Scholar 

  51. Yang, Y.: Trudinger–Moser inequalities on complete noncompact Riemannian manifolds. J. Funct. Anal. 263(7), 1894–1938 (2012)

    MathSciNet  MATH  Google Scholar 

  52. Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Sov. Math. Dokl. 2, 746–749 (1961). (Russian)

    MATH  Google Scholar 

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Acknowledgements

We wish to thank Debabrata Karmakar for pointing out several misprints in a preliminary version of our paper. We also thank the referee for his careful reading of the paper and for his valuable comments.

Funding

This research was partly funded by: (i) Research Project 2015HY8JCC of the Italian Ministry of University and Research (MIUR) Prin 2015 “Partial differential equations and related analytic-geometric inequalities”; (ii) GNAMPA of the Italian INdAM—National Institute of High Mathematics (Grant number not available); (iii) Grant P201-18-00580S of the Czech Science Foundation; (iv) Grant SVV-2017-260455 of the Charles University.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica “Ulisse Dini”, University of Florence, Viale Morgagni 67/A, 50134, Florence, Italy

    Andrea Cianchi & Vít Musil

  2. The Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 115 67, Prague 1, Czech Republic

    Vít Musil

  3. Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Vít Musil & Luboš Pick

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  1. Andrea Cianchi
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  2. Vít Musil
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Correspondence to Luboš Pick.

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Cianchi, A., Musil, V. & Pick, L. Moser inequalities in Gauss space. Math. Ann. 377, 1265–1312 (2020). https://doi.org/10.1007/s00208-020-01956-z

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