zbMATH — the first resource for mathematics

Refined Hardy inequalities. (English) Zbl 1121.43006
As a generalization of the classical Hardy inequality on \(\mathbb R^N\), the authors obtain the following two inequalities: For \(0<s<N/2\) there exists a constant \(C\) such that \[ \int_{\mathbb R^N}\frac{|u|^2(x)}{|x|^{2s}}dx \leq C\,\|u\|^2_{\Dot{H}^s}, \] where \(\Dot{H}^s\) is the Sobolev space of order \(s\), and for \(2\leq q<2N/(N-2s)<p\leq\infty\), let \(\alpha=pq(1/q-1/2+s/N)/(p-q)\). Then there exists a constant \(C\) such that \[ \Big(\int_{\mathbb R^N}\frac{|u|^2(x)}{|x|^{2s}}dx\Big)^{1/2} \leq C\,\|u\|^\alpha_{\Dot{B}^{s-N(1/2-1/p)}_{p,2}} \|u\|^{1-\alpha}_{\Dot{B}^{s-N(1/2-1/q)}_{q,2}}, \] where \(\Dot{B}^s_{p,q}\) is the Besov space. Moreover, they show that these inequalities also hold on the Heisenberg group. In order to prove the first one they use the Littlewood-Paley theory and prove that \(|x|^{-2s}\in\Dot{B}^{N-s}_{1,\infty}\) and \(u^2\in\Dot{B}^{2s-N/2}_{2,1}\). For the second one, they use the paraproduct algorithm introduced by J.-M. Bony. Furthermore, they deduce that the second one is invariant under oscillations, that is, both sides are of the same order of magnitude when \(u\) are highly oscillatory functions.

43A80 Analysis on other specific Lie groups
42B30 \(H^p\)-spaces
Full Text: EuDML
[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. Zbl1089.35016 MR2171730 · Zbl 1089.35016 · doi:10.1017/S1474748005000150
[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. Zbl0965.22010 MR1799659 · Zbl 0965.22010 · doi:10.1007/BF02791223
[3] H. Bahouri and I. Gallagher, Paraproduit sur le groupe de Heisenberg et applications, Rev. Mat. Iberoamericana 17 (2001), 69-105. Zbl0971.43002 MR1846091 · Zbl 0971.43002 · doi:10.4171/RMI/289 · eudml:39623
[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. Zbl0495.35024 MR631751 · Zbl 0495.35024 · numdam:ASENS_1981_4_14_2_209_0 · eudml:82073
[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. Zbl0797.35008 MR1252940 · Zbl 0797.35008 · doi:10.5802/aif.1367 · numdam:AIF_1993__43_4_1157_0 · eudml:75028
[6] J.-Y. Chemin, “Fluides Parfaits Incompressibles”, Astérisque, Vol. 230, 1995. Zbl0829.76003 MR1340046 · Zbl 0829.76003
[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. Zbl0892.35161 MR1476294 · Zbl 0892.35161 · doi:10.1016/S0012-9593(97)89937-5 · numdam:ASENS_1997_4_30_6_719_0 · eudml:82448
[8] J. Faraut and) K. Harzallah, “Deux Cours d’Analyse Harmonique”, École d’Été d’analyse harmonique de Tunis, 1984. Progress in Mathematics, Birkh\(\ddot{a}\)user. Zbl0622.43001 MR898880 · Zbl 0622.43001
[9] D. Geller, Fourier analysis on the Heisenberg groups, Proc. Natl. Acad. Sciences U.S.A, 74 (1977), 1328-1331. Zbl0351.43012 MR486312 · Zbl 0351.43012 · doi:10.1073/pnas.74.4.1328
[10] P. Gérard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées, Séminaire EDP, École Polytechnique, France, Décembre 1996. Zbl1066.46501 MR1482810 · Zbl 1066.46501 · numdam:SEDP_1996-1997____A4_0
[11] G. H. Hardy, Note on a theorem of Hilbert, Math. Zeit., 6 (1920), 314-317. MR1544414 JFM47.0207.01 · JFM 47.0207.01
[12] G. H. Hardy, An inequality between integrals, Messenger of Maths. 54 (1925), 150-156. JFM51.0192.01 · JFM 51.0192.01
[13] D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s conditions, Duke Math. J., 53 (1986), 503-523. Zbl0614.35066 MR850547 · Zbl 0614.35066 · doi:10.1215/S0012-7094-86-05329-9
[14] D. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, I, J. Funct. Anal. 43 (1981), 97-142. Zbl0493.58021 MR639800 · Zbl 0493.58021 · doi:10.1016/0022-1236(81)90040-9
[15] D. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, II, J. Funct. Anal. 43 (1981), 224-257. Zbl0493.58022 MR633978 · Zbl 0493.58022 · doi:10.1016/0022-1236(81)90031-8
[16] A. I. Nachman, The Wave Equation on the Heisenberg Group, Comm. Partial Differential Equations 7 (1982), 675-714. Zbl0524.35065 MR660749 · Zbl 0524.35065 · doi:10.1080/03605308208820236
[17] L. Rothschild and E. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320. Zbl0346.35030 MR436223 · Zbl 0346.35030 · doi:10.1007/BF02392419
[18] E.M. Stein, “Harmonic Analysis”, Princeton University Press, 1993. Zbl0821.42001 MR1232192 · Zbl 0821.42001
[19] M. E. Taylor, “Noncommutative Harmonic Analysis”, Mathematical Surveys and Monographs, Vol. 22, AMS, Providence, Rhode Island, 1986. Zbl0604.43001 MR852988 · Zbl 0604.43001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.