LAUSR

Abstract Let p ∈ [1,∞], q ∈ (1,∞), s ∈ Z+ := N∪{0}, and α ∈ R. In this article, the authors introduce a reasonable version T̃ of the Calderón–Zygmund operator T on JN (p,q,s)α (R), the special John–Nirenberg–Campanato space via congruent cubes, which coincides with the Campanato space Cα,q,s(R) when p = ∞. Then the authors prove that T̃ is bounded on JN (p,q,s)α (R) if and only if, for any γ ∈ Z+ with |γ| ≤ s, T ∗(xγ) = 0, which is a well-known assumption. To this end, the authors find an equivalent version of this assumption. Moreover, the authors show that T can be extended to a unique continuous linear operator on the Hardykind space HK (p,q,s)α (R), the predual space of JN (p′ ,q′,s)α (R) with 1 p + 1 p′ = 1 = 1 q + 1 q′ , if and only if, for any γ ∈ Z+ with |γ| ≤ s, T ∗(xγ) = 0. The main interesting integrands in the latter boundedness are that, to overcome the difficulty caused by that ‖ · ‖HKcon (p,q,s)α (Rn) is no longer concave, the authors first find an equivalent norm of ‖·‖HKcon (p,q,s)α (Rn), and then establish a criterion for the boundedness of linear operators on HK (p,q,s)α (R) via introducing molecules of HK (p,q,s)α (R), using the boundedness of T̃ on JN (p,q,s)α (R), and skillfully applying the dual relation (HK (p,q,s)α (Rn))∗ = JN (p′ ,q′,s)α (R).