Abstract
We improve the Gagliardo–Nirenberg inequality
\(r=2\), \(0<q<\frac{2n}{(n-2)_+}\), \({\mathcal {L}}\) 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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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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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