Pseudodifferential operators with discontinuous symbols on manifolds with conical singularities

Abstract

In this Chapter we consider various algebras of pseudodifferential operators. The first paragraph is devoted to the study of pseudodifferential operators on the spaces H β s (ℝ n ). Symbols of ‘canonical’ pseudodifferential operators of order a will be functions (ℝ n \0) × (ℝ n \0) ∋ (x, ξ)↦F(x, ξ), homogeneous of degree a both in x and in ξ. A ‘general’ pseudodifferential operator A: H β s (ℝ n ) →H β s-Rea (ℝ n ) of order a can be expressed in terms of a meromorphic pseudodifferential operator ц by formula (1.1) (compare with (2.3.6) and (2.3.7)). We derive from the results of Chapter 3 concerning meromorphic pseudodifferential operators a symbolic calculus for pseudodifferential operators on ℝ n (modulo operators acting from H β s (ℝ n ) to H β s-Rea (ℝ n ) for arbitrary N). Similar questions are considered for pseudodifferential operators with ‘periodic’ (in the sense of §7, Chapter 3) symbols.