Gagliardo–Nirenberg type inequalities for smoothness spaces related to Morrey spaces over n-dimensional torus

Abstract

In the paper, the Gagliardo–Nirenberg type inequalities for smoothness spaces B_pq^sτ(Tⁿ) of Nikol’skii–Besov type and spaces F_pq^sτ(Tⁿ) of Lizorkin–Triebel type both related to Morrey spaces over n-dimensional torus for some range of the parameters s, p, q, τ were proved. These spaces are natural analogues of the spaces B_pq^sτ(Rⁿ) and F_pq^sτ(Rⁿ) in the case of multidimensional torus Tⁿ. The main results of the article are two theorems, each of which proves the Gagliardo–Nirenberg type inequality for the Lizorkin–Triebel type spaces or the Nikol’skii–Besov type spaces respectively.