A Littlewood–Paley type theorem and a corollary

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.