Fourier integral operators on noncompact symmetric spaces of real rank one
Let \(X=G/K\) be a noncompact symmetric space of rank one, \(g\) and \(k\) be the Lie algebras of \(G\) and \(K\), respectively, \(g=k\oplus p\) the Cartan decomposition, \(a\) be a maximal abelian subspace of \(p\), \(a^*_c\) the complex dual of \(a\). A Fourier transform on the symmetric space \(X\) associates to a smooth compactly supported function \(f\) on \(X\) a function \(\widetilde f\) on \(a^*_c \times K/M\), where \(M\) is the centralizer of \(a\) in \(K\). Then every bounded function \(m:R\to C\) defines a bounded operator \(T_m\) given by \(\widetilde T_mf(\lambda,b) =m(\lambda) \widetilde f(\lambda,b)\). Using explicit expressions of the Harish-Chandra function \(c(\lambda)\) and elementary spherical functions \(\Phi_\lambda(s)\) on the symmetric spaces of rank one, the author obtains \(L^p\) boundedness properties of radial Fourier integral operators defined by the multipliers \(m_\tau (\lambda) \cos( \lambda \tau)\) and \(m_\tau(\lambda) \lambda^{-1} \sin(\lambda \tau)\), from which he derives an estimate of the solution \(u_\tau\) of the wave equation on \(X\) by its initial datas \(f\) and \(g\): \[ \|u_\tau\|_{l^p (X)}\leq c_p(\tau) \bigl(\|f\|_{L^p_{b_p (X)}}+ (1+\tau) \|g\|_{L^p_{b_p-1}(X)} \bigr). \] The second part deals with an analog of E. M. Stein's maximal spherical averages, i.e., let \(d\sigma_\tau\) be a normalized spherical measure, the author then obtains exponential estimates on the \(L^p\) norm of the maximal operator \(\sup_{T\leq\tau \leq(T+1)} |f*d \sigma_\tau (z)|\) for a compactly supported function \(f\).
scientific article; zbMATH DE number 1496042
scientific article