Remainder terms of a nonlocal Sobolev inequality

Shengbing Deng, Xingliang Tian, Minbo Yang and Shunneng Zhao

Mathematische Nachrichten, 2024, vol. 297, issue 5, 1652-1667

Abstract: In this note, we study a nonlocal version of the Sobolev inequality ∫RN|∇u|2dx≥SHLS∫RN|x|−α*u2α*u2α*dx12α*,∀u∈D1,2(RN),$$\begin{equation*} \int _{\mathbb {R}^N}|\nabla u|^2 dx \ge S_{\text{HLS}}{\left(\int _{\mathbb {R}^N}{\left(|x|^{-\alpha} \ast u^{2_\alpha ^{\ast}}\right)}u^{2_\alpha ^{\ast}} dx\right)}^{\frac{1}{2_\alpha ^{\ast}}}, \quad \forall u\in \mathcal {D}^{1,2}(\mathbb {R}^N), \end{equation*}$$where SHLS$S_{\text{HLS}}$ is the best constant, *$\ast$ denotes the standard convolution and D1,2(RN)$\mathcal {D}^{1,2}(\mathbb {R}^N)$ denotes the classical Sobolev space with respect to the norm ∥u∥D1,2(RN)=∥∇u∥L2(RN)$\Vert u\Vert _{\mathcal {D}^{1,2}(\mathbb {R}^N)}=\Vert \nabla u\Vert _{L^2(\mathbb {R}^N)}$. By using the nondegeneracy property of the extremal functions, we prove that the existence of the gradient type remainder term and a reminder term in the weak LNN−2$L^{\frac{N}{N-2}}$‐norm of above inequality for all 0

Date: 2024
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