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A Gagliardo–Nirenberg Type Inequality for Rapidly Decaying Functions

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Abstract

We improve the Gagliardo–Nirenberg inequality

$$\begin{aligned} \Vert \varphi \Vert _{L^q({\mathbb {R}}^n)} \le C \Vert \nabla \varphi \Vert _{L^r({\mathbb {R}}^n)} {\mathcal {L}}^{-(\frac{1}{q} - \frac{n-2}{2n})} (\Vert \nabla \varphi \Vert _{L^r({\mathbb {R}}^n)}), \end{aligned}$$

\(r=2\), \(0<q<\frac{2n}{(n-2)_+}\), \({\mathcal {L}}\) generalizing \({\mathcal {L}}(s)=\ln ^{-1}\frac{2}{s}\) for \(0<s<1\), from Fila and Winkler (Adv Math 357, 2019. https://doi.org/10.1016/j.aim.2019.106823) for rapidly decaying functions (\(\varphi \in W^{1,2}({\mathbb {R}}^n){\setminus }\{0\}\) with finite \(K=\int _{{\mathbb {R}}^n} \mathcal {L}(|\varphi |)\)) by specifying the dependence of C on K and by allowing arbitrary \(r\ge 1\).

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Acknowledgements

The first author was supported in part by the Slovak Research and Development Agency under the contract No. APVV-18-0308 and by the VEGA grant 1/0347/18.

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  1. Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, Bratislava, 84248, Slovakia

    Marek Fila & Johannes Lankeit

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  1. Marek Fila
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  2. Johannes Lankeit
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Correspondence to Marek Fila.

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Dedicated to the memory of Geneviève Raugel

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Fila, M., Lankeit, J. A Gagliardo–Nirenberg Type Inequality for Rapidly Decaying Functions. J Dyn Diff Equat 34, 2901–2912 (2022). https://doi.org/10.1007/s10884-020-09839-2

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  • DOI: https://doi.org/10.1007/s10884-020-09839-2

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