Complement to Hölder’s Inequality for Multiple Integrals. II

Abstract

This article is the second and final part of my work published in the previous issue of the journal. The main result of the article is the statement that if the functions γ₁ ∈ ${{L}^{{{{p}_{1}}}}}({{\mathbb{R}}^{n}})$, …, γ_m ∈ ${{L}^{{{{p}_{m}}}}}({{\mathbb{R}}^{n}})$, where m $\geqslant$ 2, and the numbers p₁, …, p_m ∈ (1, +∞] are such that $\frac{1}{{{{p}_{1}}}}$ + … + $\frac{1}{{{{p}_{m}}}}$ < 1 and the nonresonance condition (the notion introduced in the previous article for functions from the spaces ${{L}^{p}}({{\mathbb{R}}^{n}})$, p ∈ (1, +∞]) is satisfied, then ${{\sup }_{{a,b \in {{\mathbb{R}}^{n}}}}}\left| {\int_{[a,b]} {\prod\nolimits_{k = 1}^m {[{{\gamma }_{k}}(\tau ) + \Delta {{\gamma }_{k}}(\tau )]d\tau } } } \right|$ $\leqslant$ $C\prod\nolimits_{k = 1}^m {{{{\left\| {{{\gamma }_{k}} + \Delta {{\gamma }_{k}}} \right\|}}_{{L_{{{{h}_{k}}}}^{{{{p}_{k}}}}({{\mathbb{R}}^{n}})}}}}$, where [a, b] is an n-dimensional parallelepiped, the constant C > 0 is independent of the functions Δγ_k ∈ $L_{{{{h}_{k}}}}^{{{{p}_{k}}}}({{\mathbb{R}}^{n}})$, and $L_{{{{h}_{k}}}}^{{{{p}_{k}}}}({{\mathbb{R}}^{n}})$ ⊂ ${{L}^{{{{p}_{k}}}}}({{\mathbb{R}}^{n}})$, 1 $\leqslant$ k $\leqslant$ m, are some specially constructed normed spaces. In addition, a boundedness criterion for the integral of the product of functions over a subset of ${{\mathbb{R}}^{n}}$ is given in terms of fulfillment of some nonresonance condition.