Complement to the H¨older inequality for multiple integrals. I

Abstract

This article is the first part of the work, the main result of which is the statement that if for functions γ1 ∈ L^(p_1) (R^n), . . . , γm ∈ L^(p_m)(R^n), where m >= 2 and the numbers p_1, . . . , p_m ∈ (1,+∞] are such that 1/p_1 + ... + 1/p_m, a non-resonant condition is met (the concept introduced by the author for functions from L^p(R^n), p ∈ (1,+∞]), then sup_(a,b∈R^n) (...), where [a, b] is an n-dimensional parallelepiped, the constant C > 0 does not depend on functions Δ_γ_k ∈ L^(p_k)_(h_k) (R^n) C L^(p_k) (R^n), 1 <= k <= m, are specially constructed normalized spaces. In the article, for any spaces L^p_0 (R^n), L^p(R^n) p_0, p ∈ (1,+∞] and any function γ ∈ L^p_0 (R^n) the concept of a set of resonant points of a function γ with respect to the L^p(R^n) is introduced. This set is a subset of {R1 ∪{∞}}^n for any trigonometric polynomial of n variables with respect to any L^p(R^n) represents the spectrum of the polynomial in question. Theorems are written on the representation of each function γ ∈ L^p_0 (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 (R^n) ∩ L^q(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.