Criteria for the compactness and the Fredholm property of pseudodifferential operators in Hoelder-Zygmund weighted spaces

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