The paper aims to provide necessary and sufficient conditions for the boundedness of inhomogeneous Calderón-Zygmund singular integral operators on the local Hardy spaces \(h^p(\mathbb{R}^n)\) and their duals. An inhomogeneous Calderón-Zygmund singular integral operator is a non-convolution integral operator \(T\) with a kernel \(\mathcal{K}(x,y)\) satisfying \begin{align*} |\mathcal{K}(x,y)| \leq C \min \left\{\frac{1}{|x-y|^n}, \frac{1}{|x-y|^{n +\delta}} \right\} \quad \text{for \(x \neq y\)} \end{align*} and \begin{align*} |\mathcal{K}(x,y) -\mathcal{K}(x,y^\prime)| + |\mathcal{K}(y,x) -\mathcal{K}(y^\prime,x)| \leq C \frac{|y-y^\prime|^\varepsilon}{|x-y|^{n+\varepsilon}}, \quad \text{whenever \(|y -y^\prime| <\frac{|x-y|}{2}\)}. \end{align*} where \(\varepsilon\) and \(\delta\) are constants such that \(\delta >0\) and \(0< \varepsilon \leq 1\). Let \(T\) be an inhomogeneous Calderón-Zygmund singular integral operator and \(\max \left\{\frac{n}{n+\varepsilon}, \frac{n}{n +\delta}\right\} < p \leq 1\). The authors in this paper prove that

(i)

if \(T^\ast (1) \in \tilde{\Lambda}_{n(1/p -1)}\) where \(\tilde{\Lambda}_{n(1/p -1)}\) is the homogeneous Lipschitz space, then \(T\) is bounded on \(h^p(\mathbb{R}^n)\);

(ii)

if \(T\) is bounded on \(h^p(\mathbb{R}^n)\), then \(T^\ast (1) \in \Lambda_{n(1/p -1)}\) where \(\Lambda_{n(1/p -1)}\) is the inhomogeneous Lipschitz space.

A similar result for the boundedness of \(T\) on the duals of \(h^p(\mathbb{R}^n)\) is also proved in the paper. Since pseudo-differential operators with symbols in \(S^0_{1, 0}\) are examples of inhomogeneous Calderón-Zygmund singular integral operators, the main results in the paper extend the corresponding result in D. Goldberg [Duke Math. J. 46, No. 1, 27–42 (1979; Zbl 0409.46060)] concerning boundedness of pseudo-differential operators on \(h^p(\mathbb{R}^n)\).