Optimal stability of Hardy-Littlewood-Sobolev and Sobolev inequalities of arbitrary orders with dimension-dependent constants

Recently, Dolbeault-Esteban-Figalli-Frank-Loss [20] established the optimal stability of the first-order $L^2$-Sobolev inequality with dimension-dependent constant. Subsequently, Chen-Lu-Tang [18] obtained the optimal stability for the $L^2$ fractional Sobolev inequality of order $s$ when $0<s<1$.This paper considers the remaining case $1<s<\frac{n}{2}$. Our strategy is to first establish the optimal stability for the HLS inequality directly without using the stability of the Sobolev inequality. The main difficulty lies in establishing the optimal local stability of HLS inequality when $1<s<\frac{n}{2}$. The loss of the Hilbert structure of the distance appearing in the stability of the HLS inequality brings challenge in establishing the desired stability. To achieve our goal, we develop a new strategy based on the $H^{-s}-$decomposition instead of $L^{\frac{2n}{n+2s}}-$decomposition to obtain the local stability of the HLS inequality with $L^{\frac{2n}{n+2s}}-$distance. However, new difficulties arise to deduce the global stability from the local stability because of the non-uniqueness and non-continuity of $\|r\|_{\frac{2n}{n+2s}}$ for the rearrangement flow. As an important application of the optimal stability of the HLS inequality together with the duality theory of the stability developed initially by Carlen [11] and further improved in [17], we deduce the optimal stability of the $L^2$-Sobolev inequality of order s when $1\le s<\frac{n}{2}$ and the non-Hilbertian $L^{\frac{2n}{n+2s}}$-Sobolev inequality with the dimension-dependent constants. As another application, we can derive the optimal stability of Beckner's [5] restrictive Sobolev inequality on the flat sub-manifold $\mathbb{R}^{n-1}$ and the sphere $\mathbb{S}^{n-1}$ with dimension-dependent constants.