Complement to the Hölder inequality for multiple integrals. II

Abstract

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 γ1 ∈ Lp1 (Rn), . . . , γm ∈ Lpm(Rn), where m 2 and the numbers p1, . . . , pm ∈ (1,+∞] are such that 1 p1 + . . . + 1 pm < 1 the “non-resonant” condition is fulfilled (the concept introduced by the author in the previous work for functions from spaces 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] — n-dimensional parallelepiped, the constant C > 0 does not depend on functions of Δγk ∈ Lpk hk (Rn) and Lpk hk (Rn) ⊂ Lpk (Rn), 1 k 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 Rn.