Geodesic
Equivalent Normings of Spaces of Functions of Variable Smoothness
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 87-95

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For the Banach spaces $B_{p,q}^a$ and $F_{p,q}^a$ of functions defined on $\mathbb R^n$ whose variable smoothness $a=a(x)$ is determined by the behavior of their differences, equivalent normings are established in terms of weighted norms of smooth dyadic decompositions of their Fourier transforms. 
@article{TM_2003_243_a7,
     author = {O. V. Besov},
     title = {Equivalent {Normings} of {Spaces} of {Functions} of {Variable} {Smoothness}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {87--95},
     publisher = {mathdoc},
     volume = {243},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2003_243_a7/}
}
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O. V. Besov. Equivalent Normings of Spaces of Functions of Variable Smoothness. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 87-95. http://geodesic.mathdoc.fr/item/TM_2003_243_a7/