On the sharp constant in the Bianchi–Egnell stability inequality

By Tobias König. Bulletin of the London Mathematical Society, 2022

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This note is concerned with the Bianchi–Egnell inequality, which quantifies the stability of the Sobolev inequality, and its generalization to fractional exponents s∈(0,d2)$s \in (0, \frac{d}{2})$ . We prove that in dimension d⩾2$d \geqslant 2$ the best constant cBE(s)=inff∈Ḣs(Rd)∖M∥(−Δ)s/2f∥L2(Rd)2−Sd,s∥f∥L2∗(Rd)2distḢs(Rd)(f,M)2$$\begin{equation*} c_{BE}(s) = \inf _{f \in \dot{H}^s(\mathbb {R}^d) \setminus \mathcal {M}} \frac{\Vert (-\Delta )^{s/2} f\Vert _{L^2(\mathbb {R}^d)}^2 - S_{d,s} \Vert f\Vert _{L^{2^*}(\mathbb {R}^d)}^2}{\operatorname{dist}_{\dot{H}^s(\mathbb {R}^d)}(f, \mathcal {M})^2} \end{equation*}$$is strictly smaller than the spectral gap constant 4sd+2s+2$\frac{4s}{d+2s+2}$ associated to sequences that converge to the manifold M$\mathcal {M}$ of Sobolev optimizers. In particular, cBE(s)$c_{BE}(s)$ 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 M$\mathcal {M}$ .

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d360ed191792a4c8e24af698a7f8118978559213

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Citations in the corpus

Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence (2022, background): “In [51] he shows that the upper bound in (3) is strict and in [52] that the infimum defining cBE is attained.”