Complement to the Hölder inequality for multiple integrals. II

This article is the second and final part of the author's work published in the previous issue of the journal. The main result of the article is the statement that if for functions $\gamma_1 \in L^p_1(\mathrm{R}^n)$, where $m \geqslant 2$ and the numbers $p_1,\ldots , p_m \in (1, + \infty]$ are such that $1/p_1 + \ldots + 1/p_m < 1$ the "non-resonant" condition is fulfilled (the concept introduced by the author in the previous work for functions from spaces $L^p(\mathrm{R}^n)$, $p \in (1, +\infty])$, then: $\sup_{a,b \in {\mathrm{R}^n}}\left|\int_{[a,b]}\prod_{k=1}^m [\gamma_k(\tau)+\Delta_k(\tau)]d\tau\right| \leqslant \mathrm{C}\prod_{k=1}^m||\gamma_k+\Delta_{\gamma_k}||_{L_{h_k}^{p_k}(\mathrm{R}^n)}$, where $[a, b]$ - $n$-dimensional parallelepiped, the constant $C > 0$ does not depend on functions of $\Delta_{\gamma_k} \in L_{h_k}^{p_k}(\mathrm{R}^n)$, and $L_{h_k}^{p_k}(\mathrm{R}^n) \subset L^{p_k}(\mathrm{R}^n), 1 \leqslant k \leqslant m$ - are some specially constructed normalized spaces. In addition, in terms of the fulfillment of some non-resonant condition, the paper gives a test of a boundedness of the integral from the product of functions when integrating over a subset of $\mathrm{R}^n$.