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Register a new userA note on strong-form stability for the Sobolev inequality
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Authors
Robin Neumayer
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In this note, we establish a strong form of the quantitive Sobolev inequality in Euclidean space for $p in (1,n)$. Given any function $u in dot W^{1,p}(mathbb{R}^n)$, the gap in the Sobolev inequality controls $| abla u - abla v|_{p}$, where $v$ is an extremal function for the Sobolev inequality.
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We prove a strong form of the quantitative Sobolev inequality in $mathbb{R}^n$ for $pgeq 2$, where the deficit of a function $uin dot W^{1,p} $ controls $| abla u - abla v|_{L^p}$ for an extremal function $v$ in the Sobolev inequality.
Quantitative isoperimetric inequalities for anisotropic surface energies are shown where the isoperimetric deficit controls both the Fraenkel asymmetry and a measure of the oscillation of the boundary with respect to the boundary of the corresponding Wulff shape.
We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains.
The classical sharp Hardy-Littlewood-Sobolev inequality states that, for $1<p, t<infty$ and $0<lambda=n-alpha <n$ with $ 1/p +1 /t+ lambda /n=2$, there is a best constant $N(n,lambda,p)>0$, such that $$ |int_{mathbb{R}^n} int_{mathbb{R}^n} f(x)|x-y|^{-lambda} g(y) dx dy|le N(n,lambda,p)||f||_{L^p(mathbb{R}^n)}||g||_{L^t(mathbb{R}^n)} $$ holds for all $fin L^p(mathbb{R}^n), gin L^t(mathbb{R}^n).$ The sharp form is due to Lieb, who proved the existence of the extremal functions to the inequality with sharp constant, and computed the best constant in the case of $p=t$ (or one of them is 2). Except that the case for $pin ((n-1)/n, n/alpha)$ (thus $alpha$ may be greater than $n$) was considered by Stein and Weiss in 1960, there is no other result for $alpha>n$. In this paper, we prove that the reversed Hardy-Littlewood-Sobolev inequality for $0<p, t<1$, $lambda<0$ holds for all nonnegative $fin L^p(mathbb{R}^n), gin L^t(mathbb{R}^n).$ For $p=t$, the existence of extremal functions is proved, all extremal functions are classified via the method of moving sphere, and the best constant is computed.
In various analytical contexts, it is proved that a weak Sobolev inequality implies a doubling property for the underlying measure.
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