A Gagliardo-Nirenberg type inequality for rapidly decaying functions

We improve the Gagliardo-Nirenberg inequality $$\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^q({\mathbb{R}}^n)}\le C\mathrm{\parallel }\mathrm{\nabla }\phi {\mathrm{\parallel }}_{L^r({\mathbb{R}}^n)}{\mathcal{L}}^{-(\frac{1}{q}-\frac{n-r}{rn})}(\mathrm{\parallel }\mathrm{\nabla }\phi {\mathrm{\parallel }}_{L^r({\mathbb{R}}^n)}),$$ $r=2$, $0<q<\frac{rn}{(n-r{)}_{+}}$, $\mathcal{L}$ 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 ($\varphi\in W^{1,r}($\mathbb{R}$^n)\setminus{0}$withfinite$K=\int_{$\mathbb{R}$^n} $\mathcal{L}$(\varphi)$)byspecifyingthedependenceof$C$on$K$ and by allowing arbitrary $r\ge 1$.