  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. $ 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.
