Complement to the Hölder inequality for multiple integrals. I

This article is the first part of the work, the main result of which is the statement that if for functions $\gamma_1 \in L^{p_1} (\mathbb{R}^n), \ldots, \gamma_m \in L^{p_m}(\mathbb{R}^n)$, where $m \geqslant 2$ and the numbers $p_1, \ldots, p_m \in (1, +\infty]$ are such that $1/p_1 + \ldots + 1/p_m<1$, a non-resonant condition is met (the concept introduced by the author for functions from $L^{p} (\mathbb{R}^n), p \in (1, +\infty])$, then $\sup_{a,b\in \mathbb{R}^n}|\int\limits_{[a,b]} \prod_{k=1}^m[\gamma_k(\tau) + \Delta\gamma_k(\tau)]d\tau|\leqslant C\prod_{k=1}^m||\gamma_k + \Delta\gamma_k||_{L_{h_k}^{p_k}(\mathbb{R}^n)}$, where $[a, b]$ is an $n$-dimensional parallelepiped, the constant $C > 0$ does not depend on functions $\Delta\gamma_k\in L_{h_k}^{p_k}(\mathbb{R}^n)$, and $L_{h_k}^{p_k}(\mathbb{R}^n) \in L^{p_k}(\mathbb{R}^n), 1\leqslant k\leqslant m$, are specially constructed normalized spaces. In the article, for any spaces $L^p(\mathbb{R}^n)$, $p_0$, $p \in (1,+\infty]$ and any function $\gamma \in L^{p_0} (\mathbb{R}^n)$ the concept of a set of resonant points of a function $\gamma$ with respect to the $L^p(\mathbb{R}^n)$ is introduced. This set is a subset of ${ \mathbb{R}^1 \cup {\infty}}^n$ for any trigonometric polynomial of n variables with respect to any $L^p(\mathbb{R}^n)$ represents the spectrum of the polynomial in question. Theorems are written on the representation of each function $\gamma \in L^{p_0}(\mathbb{R}^n)$ with a nonempty resonant set as the sum of two functions such that the first of them belongs to the $L^{p_0}(\mathbb{R}^n) \cap L^q(\mathbb{R}^n)$, $1/p + 1/q = 1$, and the carrier of the Fourier transform of the second is centered in the neighborhood of the resonant set.