Abstract This paper is devoted to the derivation of L 2 - L 2 decay estimates for the solution of the homogeneous linear damped wave equation on the Heisenberg group H n , for its time derivative and for its horizontal gradient. Moreover, we consider the improvement of these estimates when further L 1 ( H n ) regularity is required for the Cauchy data. Our approach will rely strongly on the group Fourier transform of H n and on the properties of the Hermite functions that form a maximal orthonormal system for L 2 ( R n ) of eigenfunctions of the harmonic oscillator.

中文翻译：

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海森堡群线性阻尼波动方程的衰减估计

摘要 本文致力于推导 Heisenberg 群 H n 上齐次线性阻尼波动方程解的 L 2 - L 2 衰减估计、时间导数和水平梯度。此外，当柯西数据需要进一步的 L 1 ( H n ) 正则性时，我们会考虑这些估计的改进。我们的方法将强烈依赖于 H n 的群傅立叶变换和 Hermite 函数的性质，这些 Hermite 函数形成了谐振子的本征函数 L 2 ( R n ) 的最大正交系统。