Criteria for the Compactness and the Fredholm Property of Pseudodifferential Operators in H_lder-Zygmund Weighted Spaces

摘要

We obtain necessary and sufficient conditions for the complete continuity (the Fredholm property) in H_lder_Zygmund spaces on R~n whose weight has a power-law behavior at infinity for pseudodifferential operators with symbols in the H_rmander class S_(1,0)~m, 0 ≤δ< 1(slowly varying symbols in the class S_(1,0)~m). We show that such operators are compact operators or Fredholm operators in weighted H_lder_Zygmund spaces if and only if they are compact operators or Fredholm operators, respectively, in Sobolev spaces.