Complement to the Hlder 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 ∈ Lp1 (Rn), . . . , γm ∈ Lpm(Rn), where m 2 and the numbers p1, . . . , pm ∈ (1,+∞] are such that 1 p1 + . . . + 1 pm < 1, a non-resonant condition is met (the concept introduced by the author for functions from Lp(Rn), p ∈ (1,+∞]), then supa,b∈Rn

[a,b] m k=1 [γk(τ) +Δγk(τ )] dτ

C m k=1 γk +Δγk L pk hk (Rn), where [a, b] is an n-dimensional parallelepiped, the constant C > 0 does not depend on functions Δγk ∈ Lpk hk (Rn), and Lpk hk (Rn) ⊂ Lpk (Rn), 1 k m, are specially constructed normalized spaces. In the article, for any spaces Lp0 (Rn), Lp(Rn), p0, p ∈ (1,+∞] and any function γ ∈ Lp0 (Rn) the concept of a set of resonant points of a function γ with respect to the Lp(Rn) is introduced. This set is a subset of {R1 ∪{∞}}n for any trigonometric polynomial of n variables with respect to any Lp(Rn) represents the spectrum of the polynomial in question. Theorems are written on the representation of each function γ ∈ Lp0 (Rn) with a nonempty resonant set as the sum of two functions such that the first of them belongs to the Lp0 (Rn) ∩ Lq(Rn), 1 p + 1 q = 1, and the carrier of the Fourier transform of the second is centered in the neighborhood of the resonant set.