A Littlewood-Paley type theorem on orthoprojectors onto mutually orthogonal subspaces of piecewise polynomial functions and its corollary

The article proves an assertion analogous to the Littlewood-Paley theorem for the orthoprojectors onto mutually orthogonal subspaces of piecewise polynomial functions on the cube $I^d\mathrm{.}$ This assertion provides an upper estimate for the norms of the functions in $L_p(I^d)$ via corresponding norms of projections onto subspaces of piecewise polynomial multivariable functions. These relationships are used to obtain upper estimates of the Kolmogorov widths of Besov classes of non-periodic functions meeting the mixed Hoelder conditions.