\(L^ \infty\)-decay estimations of the spherical mean value on symmetric spaces
Let \(M_t[\varphi] (x)\) be the spherical mean value operator applied to a function \(\varphi\) on a symmetric Riemannian space \(M\) of the non- compact type. The following inequality is proved: For \(\forall t > 0\), \(\forall x \in M\), \[ |(\partial^k / \partial t^k) M_t [\varphi] |_l (x) \leq cM_t [|\varphi|_{l + k}] (x). \] Here \(|\cdot |_l\) denotes a suitable pointwise norm for the spatial Lie derivatives of order \(\leq l\). For rank one spaces the inequality is proved also for the covariant derivatives. Then it is shown that \(M_t [\varphi] (x)\) has decay rate \(1/(1 + m(t)^\sigma)\), \(\sigma \in (0,1]\), provided that \(\varphi\) has a decay rate \(1/(1 + m(t)^\rho)\) with \(\rho > \sigma\), where \(m(t)\) denotes the area of the sphere of radius \(t\). For spaces where Huygens principle hold as were discovered by Helgason, this gives the decay estimate for the solution of the Cauchy problem of the wave equation. For real hyperbolic space \(H^n (\mathbb{R})\) of odd dimension, it is shown that the decay estimate can be improved to an exponential one.
scientific article; zbMATH DE number 729755
scientific article