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

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Abstract Given a Lipschitz domain , a Calderón–Zygmund operator T and a modulus of continuity , we solve the problem when the truncated operator sends the Campanato space into itself. The solution is a T1 type sufficient and necessary condition for the characteristic function of D: 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 with an even kernel is bounded on , provided D is a ‐smooth domain.

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