A Gagliardo-Nirenberg type inequality for rapidly decaying functions

Abstract

We improve the Gagliardo-Nirenberg inequality \[ \|φ\|_{L^q(\mathbb{R}^n)} \le C \|\nablaφ\|_{L^r(\mathbb{R}^n)} \mathcal{L}^{-(\frac 1q - \frac{n-r}{rn})} (\|\nablaφ\|_{L^r(\mathbb{R}^n)}), \] $r=2$, $0<q<\frac{rn}{(n-r{)}_{+}}$, $\mathcal{L}$ generalizing $\mathcal{L}(s)={\ln }^{-1}\frac{2}{s}$ for $0<s<1$, from [M. Fila and M. Winkler: A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation, Adv. Math., 357 (2019), https://doi.org/10.1016/j.aim.2019.106823] for rapidly decaying functions ($\phi \in W^{1,r}({\mathbb{R}}^n)\setminus \{0\}$ with finite $K={\int }_{{\mathbb{R}}^n}\mathcal{L}(\phi )$) by specifying the dependence of $C$ on $K$ and by allowing arbitrary $r\ge 1$.