On spectral expansions of piecewise smooth functions depending on the geodesic distance (Q610363)

Let \(\Omega \) be a domain in a Riemannian symmetric space of rank one, and dimension \(n\geq 3\). For a selfadjoint extension of the Beltrami-Laplace operator \(B\), let \(E_{\lambda}\) be the corresponding resolution of the identity: \[ B=\int _0^{\infty } \lambda dE_{\lambda}. \] The author analyzes the convergence of the spectral expansion of a piecewise smooth function. Let \(f(x)=F(r)\) be a radial function with respect to a point \(a\) \((r=d(x,a))\), with compact support. Assume that \(F\) has discontinuities at \(R_1,\dots ,R_m\). On the interval \(]R_{j-1},R_j[\), \(F\) agrees with a smooth function on \([R_{j-1},R_j]\), and is of class \({\mathcal C}^{\ell }\), with \(\ell \geq {n-1\over 2}\). The main result of the paper states that there exists \(\gamma >0\) such that, if \(\sum R_j<\gamma \), and \(\lim _{\lambda \to \infty }(E_{\lambda} f)(a)=f(a)\), then, for \(s\leq {n-3\over 2}\), \[ D^sF(R_j-0)=D^sF(R_j+0)\quad (j=1,\ldots ,m). \]