Logarithmic Sobolev inequalities on Lie groups

In this paper we show a number of logarithmic inequalities on several classes of Lie groups: log-Sobolev inequalities on general Lie groups, logSobolev (weighted and unweighted), log-Gagliardo-Nirenberg and log-CaffarelliKohn-Nirenberg inequalities on graded Lie groups. Furthermore, on stratified groups, we show that one of the obtained inequalities is equivalent to a Grosstype log-Sobolev inequality with the horizontal gradient. As a result, we obtain the Gross log-Sobolev inequality on general stratified groups but, very interestingly, with the Gaussian measure on the first stratum of the group. Moreover, our methods also yield weighted versions of the Gross log-Sobolev inequality. In particular, we also obtain new weighted Gross-type log-Sobolev inequalities on R for arbitrary choices of homogeneous quasi-norms. As another consequence we derive the Nash inequalities on graded groups and an example application to the decay rate for the heat equations for sub-Laplacians on stratified groups. We also obtain weighted versions of log-Sobolev and Nash inequalities for general Lie groups.

[1]  Michael Ruzhansky,et al.  Hardy Inequalities on Homogeneous Groups , 2019, Progress in Mathematics.

[2]  Non-uniform bound and finite time blow up for solutions to a drift–diffusion equation in higher dimensions , 2016 .

[3]  W. Beckner Inequalities in Fourier analysis , 1975 .

[4]  Daniel W. Stroock,et al.  The logarithmic Sobolev inequality for continuous spin systems on a lattice , 1992 .

[5]  William Beckner,et al.  Pitt’s inequality and the uncertainty principle , 1995 .

[6]  Marco M. Peloso,et al.  The Sobolev embedding constant on Lie groups , 2020, Nonlinear Analysis.

[7]  Michael Ruzhansky,et al.  Anisotropic Shannon inequality , 2021 .

[8]  Boguslaw Zegarlinski,et al.  Coercive Inequalities and U-Bounds , 2021, 2105.01759.

[9]  Michael Ruzhansky,et al.  $L^p$-$L^q$ multipliers on locally compact groups , 2015, 1510.06321.

[10]  P. Patel-Schneider,et al.  OWL 2 Web Ontology Language , 2009 .

[11]  Ivan Gentil,et al.  Logarithmic Sobolev inequality for diffusion semigroups , 2010, Optimal Transport.

[12]  Nicholas T. Varopoulos,et al.  Analysis and Geometry on Groups , 1993 .

[13]  A. Levine,et al.  New estimates of the storage permanence and ocean co-benefits of enhanced rock weathering , 2023, PNAS nexus.

[14]  J. Dolbeault,et al.  A logarithmic Hardy inequality , 2009, 0912.0590.

[15]  W. Hebisch,et al.  Spectral multipliers for sub-Laplacians with drift on Lie groups , 2005 .

[16]  B. Helffer,et al.  THE LOG-SOBOLEV INEQUALITY FOR UNBOUNDED SPIN SYSTEMS , 1999 .

[17]  Elias M. Stein,et al.  Hardy spaces on homogeneous groups , 1982 .

[18]  W. Beckner A generalized Poincaré inequality for Gaussian measures , 1989 .

[19]  Giuseppe Toscani,et al.  Sur l'inégalité logarithmique de Sobolev , 1997 .

[20]  T. Aubin,et al.  Problèmes isopérimétriques et espaces de Sobolev , 1976 .

[21]  D. S. Mitrinovic,et al.  Classical and New Inequalities in Analysis , 1992 .

[22]  Manuel del Pino,et al.  The optimal Euclidean Lp-Sobolev logarithmic inequality , 2003 .

[23]  J. Nash Continuity of Solutions of Parabolic and Elliptic Equations , 1958 .

[24]  W. Beckner Pitt's inequality and the fractional Laplacian: Sharp error estimates , 2007, math/0701939.

[25]  Michael Ruzhansky,et al.  Fractional logarithmic inequalities and blow-up results with logarithmic nonlinearity on homogeneous groups , 2019, Nonlinear Differential Equations and Applications NoDEA.

[26]  William Beckner,et al.  Geometric asymptotics and the logarithmic Sobolev inequality , 1999 .

[27]  Fred B. Weissler,et al.  Logarithmic Sobolev inequalities for the heat-diffusion semigroup , 1978 .

[28]  Daniel W. Stroock,et al.  Moment estimates derived from Poincar'e and log-arithmic Sobolev inequalities , 1994 .

[29]  M. Peloso,et al.  Sobolev spaces on Lie groups: Embedding theorems and algebra properties , 2018, Journal of Functional Analysis.

[30]  P. Meyer,et al.  Sur les inegalites de Sobolev logarithmiques. I , 1982 .

[31]  J. Merker Generalizations of logarithmic Sobolev inequalities , 2008 .

[32]  J. Qiao,et al.  Several logarithmic Caffarelli–Kohn–Nirenberg inequalities and applications☆ , 2018 .

[33]  Michael Ruzhansky,et al.  Quantization on Nilpotent Lie Groups , 2016 .

[34]  J. Rosen Sobolev Inequalities for Weight Spaces and Supercontractivity , 1976 .

[35]  D. Bakry,et al.  Weighted Nash Inequalities , 2010, 1004.3456.

[36]  F. Clarke,et al.  GROSS'S LOGARITHMIC SOBOLEV INEQUALITY: A SIMPLE PROOF , 1979 .

[37]  Alice Guionnet,et al.  Lectures on Logarithmic Sobolev Inequalities , 2003 .

[38]  B. Zegarliński,et al.  Coercive Inequalities on Metric Measure Spaces , 2009, 0905.1713.

[39]  William Beckner Geometric proof of Nash's inequality , 1998 .

[40]  Ujjal Das On weighted logarithmic-Sobolev & logarithmic-Hardy inequalities , 2020, Journal of Mathematical Analysis and Applications.

[41]  G. Talenti,et al.  Best constant in Sobolev inequality , 1976 .

[42]  N. Garofalo,et al.  Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups , 2000 .

[43]  Francesco Uguzzoni,et al.  Stratified Lie groups and potential theory for their sub-Laplacians , 2007 .

[44]  Michael Ruzhansky,et al.  L-L multipliers on locally compact groups , 2020 .

[45]  S. Bobkov,et al.  From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities , 2000 .

[46]  Esther Bou Dagher,et al.  Coercive inequalities in higher-dimensional anisotropic heisenberg group , 2021, Analysis and Mathematical Physics.

[47]  Michael Ruzhansky,et al.  Hypoelliptic functional inequalities , 2018, 1805.01064.

[48]  Esther Bou Dagher Note on the $q$-Logarithmic Sobolev and $p$-Talagrand Inequalities on Carnot Groups , 2021 .

[49]  Michael Ruzhansky,et al.  Sobolev spaces on graded lie groups , 2017 .

[50]  M. Ledoux,et al.  Logarithmic Sobolev Inequalities , 2014 .

[51]  R. Adams General logarithmic Sobolev inequalities and Orlicz imbeddings , 1979 .

[52]  Michael Ruzhansky,et al.  Best constants in Sobolev and Gagliardo–Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations , 2017, Calculus of Variations and Partial Differential Equations.

[53]  E. Carlen Superadditivity of Fisher's information and logarithmic Sobolev inequalities , 1991 .

[54]  D. Suragan,et al.  Fractional Hardy–Sobolev Inequalities and Existence Results for Fractional Sub-Laplacians , 2020, Journal of Mathematical Sciences.

[55]  P. Borwein Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series) Vol. 61, D. S. Mitrinovic J. E. Pecaric and A. M. Fink, Kluwer Academic, 1993, xvii + 740 pp. , 1994 .

[56]  J. Dolbeault,et al.  A variational proof of Nash’s inequality , 2018, Rendiconti Lincei - Matematica e Applicazioni.

[57]  T. Ogawa,et al.  Logarithmic Sobolev and Shannon's inequalities and an application to the uncertainty principle , 2018 .

[58]  Boguslaw Zegarlinski,et al.  Coercive Inequalities on Carnot Groups: Taming Singularities , 2021 .

[59]  E. Carlen,et al.  Sharp constant in Nash's inequality , 1993 .