On some addition to the H¨older inequality. Resonance case. II

Abstract

Let m > 2, numbers p1, . . . , pm 2 (1,+1] satisfy inequality 1 p1 + . . . + 1 pm < 1, and functions 1 2 Lp1 (R1), . . . , m 2 Lpm(R1). We prove that if the set of “resonance points” of each of these functions is not empty and so-called “resonance condition” holds too then there exist such arbitrary small (low norm) perturbations k 2 Lpk (R1) that the resonance set of the function k + k coincides with the resonance set of the function k, 1 6 k 6 m, but at the same time

t Z 0 m Yk=1 [ k( ) + k( )] d

L∞(R1) = 1. Concepts of a “resonance point” and of a “resonance condition” for functions from the spaces Lp(R1), p 2 (1,+1], were introduced by the author in his earlier papers.