On some addition to the Hölder inequality. I

Abstract

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 the “non-resonance” condition holds (both concepts have been defined by the author for functions from Lp(R1), p ∈ (1,+∞]), then sup a,b2R1

b Z a m Yk=1 [ k( ) + k( )] d 6 C m Yk=1 k k + kkL pk ak (R1), where constant C > 0 is independent on functions k ∈ L pk ak (R1) and L pk ak (R1) ⊂ Lpk (R1), 1 6 k 6 m are specially constructed normed spaces. Besides, we give a condition of integral boundedness from the product of functions when integrating over a subset of R1. Refs 12.