Weighted variable anisotropic Hardy spaces

In this paper, we introduce the weighted variable anisotropic Hardy spaces \(H_{\omega ,A}^{p(\cdot )}\left( \mathbb {R}^n\right) \) $H_{\omega ,A}^{p(·)}\left({\mathbb{R}}^n\right)$ via the nontangential grand maximal function. We also establish the atomic decompositions for the weighted variable anisotropic Hardy spaces \(H_{\omega ,A}^{p(\cdot )}\left( \mathbb {R}^n\right) \) $H_{\omega ,A}^{p(·)}\left({\mathbb{R}}^n\right)$ . In addition, we obtain the duality between \(H_{\omega ,A}^{p(\cdot )}\left( \mathbb {R}^n\right) \) $H_{\omega ,A}^{p(·)}\left({\mathbb{R}}^n\right)$ and the weighted anisotropic Campanato spaces with variable exponents. We also obtain equivalent characterizations of the weighted variable anisotropic Hardy spaces by means of the anisotropic Lusin area function, the Littlewood–Paley g-function and the Littlewood–Paley \(g_\lambda ^*\) $g_{\lambda }^{\ast }$ -function. As applications, we study the boundedness of Calderón–Zygmund singular integral operators on the weighted variable anisotropic Hardy spaces.