AbstractPseudodifferential operators with non‐smooth symbols on a manifoldMwith Lipschitz boundary are considered. Theorems about order reduction and localization of such operators in Bessel potentialH spand Hölder‐Zygmund Z αp(IRn) spaces are proved. A pseudodifferential operatorAwith locally sectorial matrix symbol is proved to be Fredholm in the spaceH s2(M) and Ind A = 0 where s depends onA.Application to a boundary value problem for an elastic body with crack is discussed in