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Atomic and subatomic decompositions in anisotropic function spaces. (English) Zbl 0954.46021
Let $$1< p< \infty$$ and $$\overline s= (s_1,\dots, s_n)\in \mathbb{N}^n$$. Then the anisotropic Sobolev space $$W^{\overline s}_p(\mathbb{R}^n)$$ in $$\mathbb{R}^n$$ is the collection of all $$f\in L_p(\mathbb{R}^n)$$ such that $${\partial^{s_k}f\over\partial x^{s_k}_k}\in L_p(\mathbb{R}^n)$$, where $$k= 1,\dots, n$$. Anisotropic function spaces such as classical and fractional Sobolev spaces $$H^{\overline s}_p(\mathbb{R}^n)$$ and Besov spaces $$B^{\overline s}_{pq}(\mathbb{R}^n)$$ have been treated, especially by the Russian school, since the early sixties parallel to the isotropic spaces. In the eighties and nineties new far-reaching tools for isotropic spaces $$B^s_{pq}(\mathbb{R}^n)$$ and $$F^s_{pq}(\mathbb{R}^n)$$, now for all $$s\in\mathbb{R}$$, $$0< p\leq\infty$$, $$0< q\leq\infty$$, have been developed. The key words are local means, atomic and, most recently, quarkonial decompositions. The aim of the paper is to raise the theory of anisotropic spaces at the same level as it is now available for isotropic spaces.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 42B25 Maximal functions, Littlewood-Paley theory
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