A T1 theorem and Calderón–Zygmund operators in Campanato spaces on domains

Given a Lipschitz domain D⊂Rd, a Calderón–Zygmund operator T and a modulus of continuity ω(x), we solve the problem when the truncated operator TDf=T(fχD)χD sends the Campanato space Cω(D) into itself. The solution is a T1 type sufficient and necessary condition for the characteristic function χD of D: (TχD)χD∈Cω∼(D),whereω∼(x)=ω(x)1+∫x1ω(t)dt/t. To check the hypotheses of T1 theorem we need extra restrictions on both the boundary of D and the operator T. It is proved that the truncated Calderón–Zygmund operator TD with an even kernel is bounded on Cω(D), provided D is a C1,ω∼‐smooth domain.