Plancherel-type estimates and sharp spectral multipliers
General spectral multiplier theorems for self-adjoint positive definite operators on \(L^{2}\left( X,\mu \right)\) are investigated considering an arbitrary open subset \(X\) of a space of homogeneous type. The spectral multipliers are studied in a general setting of abstract operators rather than in a specific setting of group invariant operators acting on Lie groups. It is shown that the sharp Hörmander-type spectral multiplier theorems follow from the appropriate estimates of the \(L^{2}\) norm of the kernel of spectral multipliers and the Gaussian bounds of the corresponding heat kernel. The critical exponent for the Riesz summability is also studied. Several examples including elliptic differential operators on compact manifolds, the Hermite and Laguerre expansions and scattering-type operators on \(R^{3}\) are given in order to motivate the introduction of Plancherel-type estimates.
scientific article; zbMATH DE number 1888377
scientific article