A Gagliardo-Nirenberg type inequality for rapidly decaying functions

TL;DR

This paper refines a Gagliardo-Nirenberg inequality for rapidly decaying functions, explicitly detailing how the constant depends on a specific integral and extending the range of the parameter r.

Contribution

It improves the inequality by specifying the dependence of the constant on the integral K and allows for any r ≥ 1, broadening its applicability.

Findings

Derived a more precise inequality with explicit constant dependence.

Extended the inequality to arbitrary r ≥ 1.

Applicable to functions with rapid decay and finite integral K.

Abstract

We improve the Gagliardo-Nirenberg inequality \[ \|\varphi\|_{L^q(\mathbb{R}^n)} \le C \|\nabla\varphi\|_{L^r(\mathbb{R}^n)} \mathcal{L}^{-(\frac 1q - \frac{n-r}{rn})} (\|\nabla\varphi\|_{L^r(\mathbb{R}^n)}), \] $r=2$, $0<q<\frac{rn}{(n-r)_+}$, $\mathcal{L}$ generalizing $\mathcal{L}(s)=\ln^{-1}\frac 2s$ 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 ($\varphi\in W^{1,r}(\mathbb{R}^n)\setminus\{0\}$ with finite $K=\int_{\mathbb{R}^n} \mathcal{L}(\varphi)$) by specifying the dependence of $C$ on $K$ and by allowing arbitrary $r\ge1$.