| Summary: | Abstract Anisotropy is a common attribute of Nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics, can be expressed by a fairly general discrete group of dilations { A k : k ∈ Z } $\{A^{k}: k\in\mathbb{Z}\}$ , where A is a real n × n $n\times n$ matrix with all its eigenvalues λ satisfy | λ | > 1 $|\lambda|>1$ . Let φ : R n × [ 0 , ∞ ) → [ 0 , ∞ ) $\varphi: \mathbb{R}^{n}\times[0, \infty)\to[0,\infty)$ be an anisotropic Musielak-Orlicz function such that φ ( x , ⋅ ) $\varphi(x,\cdot)$ is an Orlicz function and φ ( ⋅ , t ) $\varphi(\cdot,t)$ is a Muckenhoupt A ∞ ( A ) $\mathbb {A}_{\infty}(A)$ weight. The aim of this article is to obtain two anisotropic interpolation theorems of Musielak-Orlicz type, which are weighted anisotropic extension of Marcinkiewicz interpolation theorems. The above results are new even for the isotropic weighted settings.
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