Let \(q\in [1,\infty )\) , \(d\in \{0,1,\ldots \}\) , \(\theta _0\in (0,\infty )\) , A be a general expansive matrix, and X be a ball quasi-Banach function space on \({\mathbb {R}}^n\) satisfying some mild assumptions. In this article, the authors first introduce the modified anisotropic Calderón–Zygmund operator \(\widetilde{T}\) of the anisotropic Calderón–Zygmund operator T. Then the authors prove that \(\widetilde{T}\) is bounded on the ball anisotropic Campanato-type function space if and only if T satisfies the well-known vanishing condition that \(T^*(x^{\gamma })=0\) . Moreover, the authors show that \(\widetilde{T}\) is just the adjoint operator of T on Hardy-type space \(H_X^A(\mathbb {R}^n)\) [the predual space of ], which strengthens the rationality of the definition of \(\widetilde{T}\) . All these results are new even when they are applied, respectively, to anisotropic weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces, mixed-norm Lebesgue spaces, and Lorentz spaces. To obtain these results, the authors fully use the duality , atomic characterizations of \(H_X^A(\mathbb {R}^n)\) , and a specific method for decomposing molecules into a summation of atoms.
