论文标题
在Bianchi-Egnell稳定性不平等中的急剧常数
On the sharp constant in the Bianchi-Egnell stability inequality
论文作者
论文摘要
本说明与bianchi egnell不平等有关，它量化了sobolev不平等的稳定性及其对分数指数的概括$ s \ in（0，\ frac {d} {2} {2}）$。我们证明，在尺寸$ d \ geq 2 $中，最佳常数\ [c_ {be}（s）= \ inf_ {f \ in \ dot {h}^s（\ mathbb r^d）\ setMinus \ setMinus \ setMinus \ setMincal m} m} \ frac { r^d）}^2 -s_ {d，s} \ | f \ | _ {l^{2^*}（\ Mathbb r^d）}^2}^2} {\ text {dist} _ {\ dot {\ dot {\ dot {常数$ \ frac {4S} {D+2S+2} $与序列相关联，将sobolev优化器的歧管$ \ Mathcal m $收敛。特别是，$ c_ {be}（s）$无法通过此类序列渐近地实现。我们的证明依赖于BianchiEgnell商的精确扩展，沿精心选择的测试功能融合到$ \ Mathcal M $。
This note is concerned with the Bianchi-Egnell inequality, which quantifies the stability of the Sobolev inequality, and its generalization to fractional exponents $s \in (0, \frac{d}{2})$. We prove that in dimension $d \geq 2$ the best constant \[ c_{BE}(s) = \inf_{f \in \dot{H}^s(\mathbb R^d) \setminus \mathcal M} \frac{\|(-Δ)^{s/2} f\|_{L^2(\mathbb R^d)}^2 - S_{d,s} \|f\|_{L^{2^*}(\mathbb R^d)}^2}{\text{dist}_{\dot{H}^s(\mathbb R^d)}(f, \mathcal M)^2} \] is strictly smaller than the spectral gap constant $\frac{4s}{d+2s+2}$ associated to sequences which converge to the manifold $\mathcal M$ of Sobolev optimizers. In particular, $c_{BE}(s)$ cannot be asymptotically attained by such sequences. Our proof relies on a precise expansion of the Bianchi-Egnell quotient along a well-chosen sequence of test functions converging to $\mathcal M$.
