×

zbMATH — the first resource for mathematics

Besov spaces and Strichartz estimates on the Heisenberg group. (Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg.) (French) Zbl 0965.22010
The authors consider the Cauchy problem for the wave equation \[ \partial^2_tu-\triangle_{\mathbb{H}^n}u=f,\quad u|_{t=0}=u_0,\quad \partial_tu|_{t=0}=u_1 \] on the Heisenberg group \({\mathbb{H}}^n\), where \(\triangle_{{\mathbb{H}}^n}\) is the Laplace-Kohn operator on \({\mathbb{H}}^n\). The authors construct the Littlewood-Paley dyadic decomposition on the Heisenberg group \({\mathbb{H}}^n\), use this and the group Fourier transformation to introduce the Besov space on \({\mathbb{H}}^n\), then deduce the disperse inequality \[ \text{Sup}_{t\in \mathbb{R}}|t|^{1/2} \|u\|_{L^{\infty}({\mathbb{H}}^n)}<+\infty \] and the generalized Strichartz estimation for the solution of the wave equation on the Heisenberg group \({\mathbb{H}}^n:\) \[ \|u\|_{L^p([0,T]; L^q({\mathbb{H}}^n))}\leq c_q{\|f\|_{L^1([0,t]; L^2({\mathbb{H}}^n))}+E_0(u)^{1/2}}, \] where \(E_0(u)=\|u_0\|^2_{H({\mathbb{H}}^n)}+ \|u_1\|^2_{L^2({\mathbb{H}}^n)}\) is the energy of the Cauchy data, and \(p, q\) satisfy \[ \frac{1}{p}+\frac{2n+2}{q}=n. \]
Reviewer: Zhu Fulin (Hubei)

MSC:
22E30 Analysis on real and complex Lie groups
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
35L05 Wave equation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] H. Bahouri, P. Gérard et C.-J. Xu,Estimations de Strichartz généralisées sur le groupe de Heisenberg, Séminaire ”Equations aux dérivées partielles” de l’École Polytechnique, Exposé 10, 1997–1998
[2] M. Beals,Time decay of\(\cdot\)L p norms for solutions of wave equation on exterior domains, inGeometrical Optics and Related Topics (F. Colombini and N. Lerner, eds.), Progress in Nonlinear Differential Equations and their Applications32 (1997), 59–77. · Zbl 0920.35027
[3] 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.
[4] J.-Y. Chemin,Fluides parfaits incompressibles, Astérisque230 (1995). · Zbl 0829.76003
[5] J.-Y. Chemin et C.-J. Xu,Inclusion de Sobolev en calcul de Weyl-Hörmander et champs de vecteurs sous-elliptiques, Ann. Sci. École Norm. Sup.30 (1997), 719–751. · Zbl 0892.35161
[6] R. R. Coifman et Y. Meyer,Au-delà des opérateurs pseudo-différentiels, Astérisque57 (1978).
[7] D. Geller,Fourier analysis on the Heisenberg groups, Proc. Nat. Acad. Sci. U.S.A.74 (1977), 1328–1331. · Zbl 0351.43012 · doi:10.1073/pnas.74.4.1328
[8] P. Gérard, Y. Meyer et F. Oru,Inégalités de Sobolev précisées, Séminaire de EDP, Ecole Polytechnique, Exposé4, 1996–1997.
[9] J. Ginibre et G. Velo,Generalized Strichartz inequatities for the wave equations, J. Funct. Anal.133 (1995), 50–68. · Zbl 0849.35064 · doi:10.1006/jfan.1995.1119
[10] L. Hörmander,The Analysis of Linear Partial Differential Operators, I, Springer, Berlin, 1983.
[11] A. Hulanicki,A functional calculus for Rocland operators on nilpotent Lie groups, Studia Math.78 (1984), 253–266. · Zbl 0595.43007
[12] L. Kapitanski,Some generalisations of the Strichartz-Brenner inequality, Leningrad Math. J.1 (1990), 693–726. · Zbl 0732.35118
[13] M. Keel and T. Tao,Endpoint Strichartz estimates, Amer. J. Math.120 (1998), 955–980. · Zbl 0922.35028 · doi:10.1353/ajm.1998.0039
[14] N. Lohoue,Estimées L p des solutions de l’équation des ondes sur les variétés riemanniennes, les groupes de Lie et applications, Proc. Conf. Canadian Math. Soc.21 (1997), 103–126. · Zbl 0927.58014
[15] W. Magnus, F. Oberhettinger and F. G. Tricomi,Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, Toronto, London, 1953. · Zbl 0052.29502
[16] D. Müller,A restriction theorem for the Heisenberg group, Ann. of Math.131 (1990), 567–587. · Zbl 0731.43003 · doi:10.2307/1971471
[17] D. Müller and E. M. Stein,L p-estimates for the wave equation on the Heisenberg group, prépublication Université de Kiel, Juin 1997.
[18] A. I. Nachman,The wave equation on the Heisenberg group, Comm. Partial Differential Equations7 (1982), 675–714. · Zbl 0524.35065 · doi:10.1080/03605308208820236
[19] M. M. Nessibi and K. Trimeche,Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl.208 (1997), 337–363. · Zbl 0870.43004 · doi:10.1006/jmaa.1997.5299
[20] H. Pecher,Nonlinear small data scattering for the wave and Klein-Gordon equations, Math. Z.185 (1984), 261–270. · Zbl 0538.35063 · doi:10.1007/BF01181697
[21] J. Peetre,New Thoughts on Besov Spaces, Duke Univ. Math. Series, 1, 1976. · Zbl 0356.46038
[22] L. Rothschild and E. Stein,Hypoelliptic differential operators and nilpotent groups, Acta Math.137 (1977), 247–320. · Zbl 0346.35030 · doi:10.1007/BF02392419
[23] I. E. Segal,Space-time decay for solutions of wave equations, Adv. Math.22 (1976), 304–311. · Zbl 0344.35058 · doi:10.1016/0001-8708(76)90097-9
[24] E. M. Stein,Topics in Harmonic Analysis, Related to the Littlewood-Paley Theory, Princeton University Press, 1970. · Zbl 0193.10502
[25] E. M. Stein,Oscillatory integrals in Fourier analysis, inBeijing Lectures in Harmonic Analysis, Princeton University Press, 1986, pp. 307–355.
[26] E. M. Stein,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993. · Zbl 0821.42001
[27] R. S. Strichartz,A priori estimates for the wave equation and some applications, J. Funct. Anal.5 (1970), 218–235. · Zbl 0189.40701 · doi:10.1016/0022-1236(70)90027-3
[28] R. S. Strichartz,Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J.44 (1977), 705–774. · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1
[29] M. E. Taylor,Noncommutative harmonic Analysis, Mathematical Surveys and Monographs, No. 22, Amer. Math. Soc., Providence, RI, 1986.
[30] H. Triebel,Theory of Function Spaces, Birkhäuser, Basel, 1983. · Zbl 0546.46028
[31] C.-J. Xu,Semilinear subelliptic equations and Sobolev inequality for vector fields satisfying Hörmander’s condition, Chinese Ann. Math. Ser. A15 A (1994), 65–72. Version en anglais: Chinese J. Contemp. Math.15 (1994), 34–40.
[32] K. Yajima,Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys.110 (1987), 415–426. · Zbl 0638.35036 · doi:10.1007/BF01212420
[33] C. Zuily,Existence globale de solutions régulières pour l’équation des ondes non linéaires amorties sur le groupe de Heisenberg, Indiana Univ. Math. J.42 (1993), 323–360. · Zbl 0798.35109 · doi:10.1512/iumj.1993.42.42016
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.