A priori estimates and the Fredholm property for a class of pseudodifferential operators

Pseudodifferential operators with symbols $A(x,\xi)$ satisfying
\begin{equation} |D^\beta_xD_\xi^\alpha A(x,\xi)|\leqslant C^A_{\alpha,\beta}(1+|\xi'|)^{m'-|\alpha'|}(1+|\xi''|)^{m''-|\alpha''|} \end{equation}
for all multi-indices $\alpha$, $\beta$, where $\xi=(\xi',\xi'')$ and $\alpha=(\alpha',\alpha'')$, are considered.
For operators of this class a priori estimates (in part as well as all of the variables) are established. Necessary and sufficient conditions are found for some classes of pseudodifferential operators with symbols satisfying (1) to have the Fredholm property.
Bibliography: 11 titles.