Let $\mathcal{M}$ be a von Neumann algebra equipped with a normal semifinite faithful trace $\tau$, and let $A$ be an expansive dilation on $\mathbb{R}^d$. In this paper, the authors first investigate a class of local operator-valued anisotropic Hardy spaces, denoted by $\mathbf{h}_p^A(\mathbb{R}^d,\mathcal{M})$, and identify their dual spaces. This further develops the seminal work of T. Mei [MR2327840: Mem. Amer. Math. Soc., (2007)]. In addition, we also show that $\mathbf{h}_1^A(\mathbb{R}^d,\mathcal{M})$ and $\mathrm{bmo}_A(\mathbb{R}^d,\mathcal{M})$ serve as natural endpoint spaces for the noncommutative $L_p$-spaces $L_p(L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M})$, and these endpoint spaces are then used to establish the $L_p$-estimates for certain operators via interpolation. Finally, as applications, an atomic decomposition for local Hardy spaces and Hardy–Lebesgue estimates for a local class of noncommutative Calderón–Zygmund operators are also established. To the best of our knowledge, these results are new in the operator-valued setting even for diagonal anisotropic dilations (when $A:=\mathrm{diag}(\lambda_1, \lambda_2, \ldots, \lambda_d)$ with $|\lambda_j| \gt 1$, $j=1,2,\ldots,d$).

Let µ be Radon measure on Rd which may be non doubling. The only condition that µ must satisfy is the size condition µ(B(x, r)) ≤ Crn for some fixed n є (0, d). Recently, Tolsa introduced the spaces RBMO(µ) and Hatb1.∞ (µ), which, in some ways, play the role of the classical spaces BMO and H1 in case u is a doubling measure. In this paper, the author considers the local versions of the spaces RBMO(µ) and Hatb1.∞ (µ) in the sense of Goldberg and establishes the connections between the spaces RBMO(µ) and Hatb1.∞ (µ) with their local versions. An interpolation result of linear operators is also given.