Geodesic
A Littlewood--Paley type theorem and a~corollary
Izvestiya. Mathematics , Tome 77 (2013) no. 6, pp. 1155-1194

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We prove an analogue of the Littlewood–Paley theorem for orthoprojectors onto mutually orthogonal subspaces of piecewise-polynomial functions on the cube $I^d$. This yields upper bounds for the norms of functions in $L_p(I^d)$ in terms of the corresponding norms of the projections to subspaces of piecewise-polynomial functions of several variables. We use these results to obtain upper bounds for the Kolmogorov widths of Besov classes of (non-periodic) functions satisfying mixed Hölder conditions.
Keywords: orthoprojector, mutually orthogonal subspaces, piecewise-polynomial functions, Littlewood–Paley theorem, width. 
@article{IM2_2013_77_6_a3,
     author = {S. N. Kudryavtsev},
     title = {A {Littlewood--Paley} type theorem and a~corollary},
     journal = {Izvestiya. Mathematics },
     pages = {1155--1194},
     publisher = {mathdoc},
     volume = {77},
     number = {6},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2013_77_6_a3/}
}
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S. N. Kudryavtsev. A Littlewood--Paley type theorem and a~corollary. Izvestiya. Mathematics , Tome 77 (2013) no. 6, pp. 1155-1194. http://geodesic.mathdoc.fr/item/IM2_2013_77_6_a3/