The goal of this note is to investigate quantitative stability properties of the critical Sobolev inequality in ${\sf CD}(N-1,N)$ metric measure spaces. Assuming that the optimal constant for the inequality is almost the same as the one of the round sphere, we show that the cumulative distribution of any almost extremal function is close, in Wasserstein distance, to the one of an Aubin-Talenti bubble on the round sphere. We obtain similar results for the log Sobolev inequality and the spectral gap under various curvature and dimension assumptions. In all cases we obtain a quantitative stability with sharp exponent.
Let \(n\ge3\) and \(1<p<n\). We prove a quantitative stability estimate for the critical Sobolev trace inequality on the upper half-space. More precisely, the Sobolev trace deficit controls the \(\max\{2,p\}\)-th power of the gradient distance to the manifold of trace bubbles. A central part of the proof is the spectra...
Xi-Nan Ma, Yi-Tian Zhang, Yang Zhou· 2 citations· ⚡1
We prove a sharp isoperimetric inequality for the harmonic mean of the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, endowed with radial log-concave measures. The main novelty is that we establish the sharp harmonic mean inequality under general radi...
Trace theorems are an indispensable tool for the analysis of kinetic equations. They have been established in wide generality by Cessenat and co-workers for radiative transfer and related applications. A variety of trace estimates have been established for the kinetic Fokker-Planck and Kolmogorov equation, typically re...
We study the problem of stable spectral differentiation of functions in Sobolev spaces from noisy data. We introduce a class of admissible Fourier multipliers under simple and directly verifiable conditions and show that the corresponding regularized differentiation operators achieve minimax optimal stability rates. Th...
It is well known the local integral inequality which is called Poincar\'e--Sobolev inequality on each domain cube (or ball). After that many authors investigated the generalization for this inequality with Muckenhoupt-type weights or global weights which are independent of the domain cube. In this paper, we investigate...
We prove the existence of stability optimizers for the Yamabe inequality on closed Riemannian manifolds of dimension at least three with positive Yamabe invariant that satisfy two threshold conditions. Remarkably, the compactness threshold we uncover is different from the special case of the round sphere treated previo...
J. Andrade, Tobias König, J. Ratzkin et al.· 0 citations
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