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

Dolbeault-Esteban-Figalli-Frank-Loss 19 and Chen-Lu-Tang 17 established the optimal asymptotic lower bound for stability of the first-order Sobolev inequality and fractional Sobolev inequality of order s for 0<s<1 respectively. However, it left the problem of the optimal lower bound for stability of high-order Sobolev inequality and high-order fractional Sobolev inequality unsolved. The purpose of this paper is to solve this problem. The main difficulty lies in establishing the optimal asymptotic behavior for the local stability of the Sobolev inequality for all 0<s<n/2. The proof of the local stability when 0<s 1 relies on ``cuttings" at various heights and this helps to split the L² integral of first order or fractional order derivative of order 0<s<1. However, this approach does not seem to work for 1<s<n/2. In order to overcome this difficulty, we directly establish the local stability for the HLS inequality with the optimal asymptotic lower bounds. To achieve our goal, we develop a new strategy based on the H^-s-decomposition instead of L^2n{n+2s}-decomposition to obtain the local stability of the HLS inequality with L^2n{n+2s}-distance. This kind of ``new local stability" also brings more difficulties to using the rearrangement flow to deduce the global stability from local stability because of the non-uniqueness of \|r\|₂₍₍+₂ₒ and non-continuity of \|r\|₂₍₍+₂ₒ norm for the rearrangement flow. We establish the norm comparison theorem for \|r\|₂₍₍+₂ₒ and "new continuity" theorem for the rearrangement flow to overcome this difficulty (see Lemma 3. 1, Lemma 3. 3 and Lemma 3. 5).