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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.

##### MSC:
 43A80 Analysis on other specific Lie groups 42B30 $$H^p$$-spaces
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##### References:
 [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
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