ON SOME ADDITION TO THE HÖLDER INEQUALITY. I
Let m ;: 2, numbers p1,..., pm ∈ (1, +∞] satisfy inequality1 1+... +p1 pm< 1,and functions γ1 ∈ Lp1 (R1),..., γm ∈ 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 b m m (pkpksupТТ [γk(τ ) + ∆γk(τ )] dτ C ТТ lγk + ∆γklL(R1 ), a,b∈R1 ak=1akk=1where constant C > 0 is independent on functions ∆γk ∈ Lpk (R1) and Lpk (R1) ⊂ Lpk (R1), 1 k mak akare 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.