Regularity properties of wave propagation on conic manifolds and applications to spectral multipliers (Q5946163)

Let \(({\mathcal N},g)\) be a compact Riemannian manifold of dimension \(n-1\), \(n\geq 2\), possibly with boundary. Denote by \(\Gamma({\mathcal N})\) the space \(R^+\times{\mathcal N}\) with metric \(dr^2+ r^2g\), and by \(\Delta\) the Laplacian on \(\Gamma({\mathcal N})\). If \(h\) is a suitable function on \(\Gamma({\mathcal N})\), put NEWLINE\[NEWLINE\|h\|_{p,2}= \Biggl(\int^\infty_0\|h(r,\cdot)\|^p_{L^2({\mathcal N})} r^{n-1} dr\Biggr)^{1/p}.NEWLINE\]NEWLINE The authors prove various estimates for \(Z(t) f\), in particular for NEWLINE\[NEWLINE\Biggl({1\over T} \int^T_0\|Z(t) f\|^p_{p, 2} dt\Biggr)^{1/p},NEWLINE\]NEWLINE where \(Z(t)= \cos(t\sqrt{-\Delta})\), \({J_{\alpha-{1\over 2}}(t\sqrt{-\Delta})\over (t\sqrt{-\Delta})^{\alpha-{1\over 2}}}\) (where \(J_{\alpha-{1\over 2}}\) is the Bessel function of order \(\alpha-{1\over 2})\), \((I- t^2\Delta)^{-\alpha/2}\cos(t\sqrt{- \Delta})\). This gives estimates for solutions to the wave equation \(u_{tt}-\Delta u= 0\).