Weighted and Logarithmic Caffarelli-Kohn-Nirenberg type inequalities on stratified groups and applications
DOI:
https://doi.org/10.70474/se1et136Keywords:
Caffarelli-Kohn-Nirenberg inequality, logarithmic Caffarelli-Kohn-Nirenberg inequality, Stratified Lie groupAbstract
The classical Caffarelli–Kohn–Nirenberg inequalities, originally established in Euclidean space in the 1980s, provide a unified framework for interpolation between Sobolev and Hardy inequalities. Their extension to stratified (or homogeneous Carnot) Lie groups began in the early 2000s, motivated by subelliptic analysis and geometric measure theory, revealing rich interactions between group structure, dilation symmetry, and functional inequalities.
In this paper, we establish the weighted and logarithmic Caffarelli–Kohn–Nirenberg type inequalities on a stratified Lie group. As a consequence, we can apply them to prove the weighted ultracontractivity of positive strong solutions to the equation:
dα · ∂u/∂t = ℒₚ((dα · u)m),
where ℒₚ f = ∇H ( |∇H f|p−2 · ∇H f ) is a p-sub-Laplacian, d is a homogeneous norm associated with a fundamental solution of the sub-Laplacian, α ∈ ℝ, and 1 < p < Q.
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D'Ambrosio L. Hardy-type inequalities related to degenerate elliptic differential operators Ann. Scuola Norm. Sup. Pisa CI. Sci. 5 2005 451-486
Bonfiglioli A. Lanconelli E. Uguzzoni F. Stratified Lie Groups and Potential Theory for their Sub-Laplacians Springer-Verlag Berlin-Heidelberg 2007
Bouchez V. Willem M. Extremal functions for the Caffarelli-Kohn-Nirenberg inequalities: a simple proof of symmetry J. Math. Anal. Appl. 352 1 2009 293-300
Caffarelli L. Kohn R. Nirenberg L. First order interpolation inequalities with weights Compositio Mathematica 53 25927 1984
Catrina F. Wang Z. On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions Comm. Pure Appl. Math. 54 2 2001 229-258 DOI: https://doi.org/10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I
Chou K. Chu C. On the best constant for a weighted Sobolev-Hardy inequality J. London Math. Soc. 48 1 1993 137-151
Del Pino M. Dolbeault J. Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions J. Math. Pures Appl. 81 9 2002 847-875 DOI: https://doi.org/10.1016/S0021-7824(02)01266-7
Dolbeault J. Esteban M.J. Laptev A. Loss M. One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: remarks on duality and flows J. Lond. Math. Soc. 90 2 2014 525-550
Garofalo N. Lanconelli E. Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation Ann. Inst. Fourier (Grenoble) 40 2 1990 313-356 DOI: https://doi.org/10.5802/aif.1215
Goldstein J. Kombe I. Yener A. A unified approach to weighted Hardy type inequalities on Carnot groups Discrete and Continuous Dynamical Systems 37 4 2017 2009-2021
Han Y. Weighted Caffarelli-Kohn-Nirenberg type inequality on the Heisenberg group Indian J. Pure Appl. Math. 46 2 2015 147-161 DOI: https://doi.org/10.1007/s13226-015-0118-7
Han Y.Z. Niu P.C. Zhang T. On first order interpolation inequalities with weights on the Heisenberg group Acta Mathematica Sinica, English series 27 12 2011 2493-2506
Feng T. Niu P. Qiao J. Several Logarithmic Caffarelli-Kohn-Nirenberg inequalities and Applications Journal of Mathematical Analysis and Applications 457 1 2018 822-840
Folland G.B. Sten E.M. Estimates for the $bar{partial}_b$-complex and analysis on the Heisenberg group Comm. Pure Appl. Math. 27 1974 429-522
Folland G.B. Subelliptic estimates and function spaces on nilpotent groups Ark. Mat. 13 1975 161-207 DOI: https://doi.org/10.1007/BF02386204
Fischer V. Ruzhansky M. Quantization on nilpotent Lie groups Birkh"auser 314 2016 book
Lin C. Wang Z. Symmetry of extremal functions for the Caffarrelli-Kohn-Nirenberg inequalities Proc. Amer. Math. Soc. 132 6 2004 1685-1691
Liu H. Zhao L. Nontrivial solutions for a class of critical elliptic equations of Caffarelli-Kohn-Nirenberg type J. Math. Anal. Appl. 404 2 2013 317-325
Merker J. Generalizations of logarithmic Sobolev inequalities Discrete and Continuous Dynamical Systems Series S 1 2 2008 329-338 DOI: https://doi.org/10.3934/dcdss.2008.1.329
Ozawa T. Ruzhansky M. Suragan D. Lp-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups Quart. J. Math. 70 2019 305-318
Ruzhansky M. Sabitbek B. Suragan D. Weighted anisotropic Hardy and Rellich type inequalities for general vector fields NoDEA Nonlinear Differential Equations Appl. 26 2 2019 26:13
Ruzhansky M. Suragan D. Hardy inequalities on homogeneous groups Progress in Math. 327 2019 588 book
Ruzhansky M. Suragan D. Anisotropic L2-weighted Hardy and $L^2$-Caffarelli-Kohn-Nirenberg inequalities Commun. Contemp. Math. 19 2017 DOI: https://doi.org/10.1142/S0219199717500146
Ruzhansky M. Suragan D. Yessirkegenov N. Caffarelli-Kohn-Nirenberg and Sobolev type inequalities on stratified Lie groups Nonlinear Differential Equations and Applications 24 56 2017
Ruzhansky M. Suragan D. Yessirkegenov N. Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability and superweights for Lp-weighted Hardy inequalities Trans. Amer. Math. Soc. Ser. B 5 2018 32-62
Ruzhansky M. Suragan D. On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and p-sub-Laplacian inequalities on stratified groups J. Differential Equations 262 2017 1799-1821 DOI: https://doi.org/10.1016/j.jde.2016.10.028
Sabitbek B. Suragan D. Horizontal Weighted Hardy–Rellich Type inequality on Stratified Lie groups Complex Anal. Oper. Theory 12 6 2018 1469-1480 DOI: https://doi.org/10.1007/s11785-017-0650-z
Zhang S. Han Y. Dou J. A class of Caffarelli-Kohn-Nirenberg type inequalities on the H-type group Rend. Semin. Mat. Univ. Padova 132 2014 249-266
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