Smoothness of means of the spectral resolutions corresponding to elliptic pseudodifferential operators in Hölder classes (Q1778300)

This paper deals with the smoothness in Hölder classes of the following operator: \[ p(tA)f(x)= \int^\infty_{-\infty}\widehat p(z) e^{iztA}f\,dz,\quad f\in C^\alpha_0(\Omega), \] where \(A\) is an elliptic positive selfadjoint pseudodifferential operator of order \(m\) with scalar homogeneous symbol. Under three assumptions imposed on \(p(\lambda)\) (one of them is: \(p^{(n)}(0)\neq 0\) for some integer \(n> 1\)) it is proved in Theorem 2 that if \(u(x)\in C^\alpha_0\), \(0<\alpha<1\), then the function \(v(t,x)= p(tA)u(x)\in C^{\alpha/m,\alpha}(\overline{\mathbb{R}}_+\times M)\), for any compact \(M\subset\Omega\).