On the uniform convergence and basis property of the means of spectral expansions corresponding to elliptic pseudodifferential operators for continuous functions in the Liouville and Nikolskij-Besov classes (Q1778281)

This paper deals with the convergence property of the operator \[ p(tA)f(x)= \int^\infty_{-\infty}\widehat p(z) e^{iztA}f(x)\,dz, \] for \(t\to 0\), where \(A\) is an elliptic positive selfadjoint scalar pseudodifferential operator with homogeneous symbol of order \(m\) and constant coefficients. The author introduces the classes \(\mathring L^\alpha_p(\Omega)\) of compactly supported Liouville type functions and \(\mathring B^\alpha_{pq}(\Omega)\) containing the compactly supported functions that belongs to the Nikolskii-Besov spaces \(B^\alpha_{pq}(\Omega)\). In the theorems 1--4 results are proved concerning the uniform convergence \(\lim_{t\to 0} p(tA)u(x)= u(x)\) and the convergence \(\lim_{t\to 0}\| p(tA)u- u\|= 0\) in \(L^\alpha_p(M)\) and \(B^\alpha_{pq}(M)\), \(u\in \mathring L^\alpha_p(\Omega)\), \(M\Subset\Omega\).