On some addition to the H¨older inequality. Resonance case. I
Boris F. Ivanov · Vestnik of Saint Petersburg University Mathematics Mechanics Astronomy · 2018
Let m > 2, numbers p1, . . . , pm ∈ (1,+∞] satisfy inequality 1 p1 + . . . + 1 pm < 1, and functions 1 ∈ Lp1 (R1), . . . , m ∈ 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 as well then there are such arbitrary small (low norm) perturbations k ∈ Lpk (R1) that the resonance set of function k + k coincides with the resonance set of function k, 1 6 k 6 m but t Z0 m Yk=1 [ k( ) + k( )] d L∞(R1) = ∞. Concepts of “resonance point” and “resonance condition” for functions from the spaces Lp(R1), p ∈ (1,+∞] were introduced by the author in his earlier papers.