Ellipticity of Fredholm Pseudo-Differential Operators on L p (ℝ n )

Abstract

Based on the works of Grieme and Schulze [8], it is shown in this paper that if a pseudo-differential operator with symbol in S m, — ∞ < m < ∞, is Fredholm on L p(ℝn), 1 < p < ∞, then the pseudo-differential operator is elliptic. The basic idea is to construct an isometric operator R gl , λ ∈ ℝ {0}, on Lp(ℝn) in order to prove the ellipticity of the Fredholm pseudo-differential operator with symbol in S 0. This result is then generalized for arbitrary symbol classes.KeywordsPseudo-differential operatorsSG pseudo-differential operatorsSobolev spacesFredholm operators and elliptic pseudo-differential operatorsMathematics Subject Classification (2000)35S0547G3047A53