On some addition to the Hölder inequality. II

Abstract

If 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

Zb a mY k=1 [ k( ) + k( )] d

6 C mY k=1 k k + kkL pk ak (R1), where constant C > 0 is independent of functions k ∈ Lpk ak (R1) and Lpk ak (R1) ⊂ Lpk (R1), 1 6 k 6 m are specially constructed normed spaces. Besides, we give a boundedness condition for integral of product of functions over a subset of R1. Refs 3.