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)


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
Full Text: DOI


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