Refined Hardy inequalities
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 3, pp. 375-391.

The aim of this article is to present “refined” Hardy-type inequalities. Those inequalities are generalisations of the usual Hardy inequalities, their additional feature being that they are invariant under oscillations: when applied to highly oscillatory functions, both sides of the refined inequality are of the same order of magnitude. The proof relies on paradifferential calculus and Besov spaces. It is also adapted to the case of the Heisenberg group.

Classification : 43A80, 42B99 
@article{ASNSP_2006_5_5_3_375_0,
     author = {Bahouri, Hajer and Chemin, Jean-Yves and Gallagher, Isabelle},
     title = {Refined {Hardy} inequalities},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {375--391},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {3},
     year = {2006},
     mrnumber = {2274784},
     zbl = {1121.43006},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2006_5_5_3_375_0/}
}
TY  - JOUR
AU  - Bahouri, Hajer
AU  - Chemin, Jean-Yves
AU  - Gallagher, Isabelle
TI  - Refined Hardy inequalities
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2006
SP  - 375
EP  - 391
VL  - 5
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2006_5_5_3_375_0/
LA  - en
ID  - ASNSP_2006_5_5_3_375_0
ER  - 
%0 Journal Article
%A Bahouri, Hajer
%A Chemin, Jean-Yves
%A Gallagher, Isabelle
%T Refined Hardy inequalities
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2006
%P 375-391
%V 5
%N 3
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2006_5_5_3_375_0/
%G en
%F ASNSP_2006_5_5_3_375_0
Bahouri, Hajer; Chemin, Jean-Yves; Gallagher, Isabelle. Refined Hardy inequalities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 3, pp. 375-391. http://archive.numdam.org/item/ASNSP_2006_5_5_3_375_0/

[1] H. Bahouri, J.-Y. Chemin and C.-J. Xu, Trace and trace lifting theorems in weighted Sobolev spaces, J. Inst. Math. Jussieu 4 (2005), 509-552. | MR | Zbl

[2] H. Bahouri, P. Gérard and C.-J. Xu, Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg, J. Anal. Math. 82 (2000), 93-118. | MR | Zbl

[3] H. Bahouri and I. Gallagher, Paraproduit sur le groupe de Heisenberg et applications, Rev. Mat. Iberoamericana 17 (2001), 69-105. | MR | Zbl

[4] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. 14 (1981), 209-246. | Numdam | MR | Zbl

[5] C. E. Cancelier, J.-Y. Chemin and C.-J. Xu, Calcul de Weyl-Hörmander et opérateurs sous-elliptiques, Ann. Inst. Fourier (Grenoble) 43 (1993), 1157-1178. | Numdam | MR | Zbl

[6] J.-Y. Chemin, “Fluides Parfaits Incompressibles”, Astérisque, Vol. 230, 1995. | Numdam | MR | Zbl

[7] J.-Y. Chemin and C.-J. Xu, Inclusions de Sobolev en calcul de Weyl-Hörmander et champs de vecteurs sous-elliptiques, Ann. Sci. École Norm. Sup. 30 (1997), 719-751. | Numdam | MR | Zbl

[8] J. Faraut and) K. Harzallah, “Deux Cours d'Analyse Harmonique”, École d'Été d'analyse harmonique de Tunis, 1984. Progress in Mathematics, Birkha ¨user. | MR | Zbl

[9] D. Geller, Fourier analysis on the Heisenberg groups, Proc. Natl. Acad. Sciences U.S.A, 74 (1977), 1328-1331. | MR | Zbl

[10] P. Gérard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées, Séminaire EDP, École Polytechnique, France, Décembre 1996. | Numdam | MR | Zbl

[11] G. H. Hardy, Note on a theorem of Hilbert, Math. Zeit., 6 (1920), 314-317. | EuDML | JFM | MR

[12] G. H. Hardy, An inequality between integrals, Messenger of Maths. 54 (1925), 150-156. | JFM

[13] D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's conditions, Duke Math. J., 53 (1986), 503-523. | MR | Zbl

[14] D. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, I, J. Funct. Anal. 43 (1981), 97-142. | MR | Zbl

[15] D. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, II, J. Funct. Anal. 43 (1981), 224-257. | MR | Zbl

[16] A. I. Nachman, The Wave Equation on the Heisenberg Group, Comm. Partial Differential Equations 7 (1982), 675-714. | MR | Zbl

[17] L. Rothschild and E. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320. | MR | Zbl

[18] E.M. Stein, “Harmonic Analysis”, Princeton University Press, 1993. | MR | Zbl

[19] M. E. Taylor, “Noncommutative Harmonic Analysis”, Mathematical Surveys and Monographs, Vol. 22, AMS, Providence, Rhode Island, 1986. | MR | Zbl