标题： 关于与Cwikel-Lieb-Rozenblum和Lieb-Thirring不等式相关的变分问题 显示英文标题

标题： On a variational problem related to the Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities

摘要： 我们明确求解了一个与Cwikel--Lieb--Rozenblum(CLR)和Lieb--Thirring(LT)不等式中最佳常数上界相关的变分问题，该问题最近在[Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ]和[J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ]中推导出来。 我们通过函数的傅里叶变换的$L^1$范数的变分特征和对偶性来实现这一点，从中我们得到了一个关于Hadamard三线引理变体的重新表述。 通过研究复平面上条带中的全纯函数的类似Hardy空间，我们能够为极小值提供一个解析公式，并用它来得到由[Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ]和[J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ]方法可达到的CLR和LT不等式中最佳常数的最佳可能上界。 显示英文摘要

摘要： We explicitly solve a variational problem related to upper bounds on the optimal constants in the Cwikel--Lieb--Rozenblum (CLR) and Lieb--Thirring (LT) inequalities, which has recently been derived in [Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ]. We achieve this through a variational characterization of the $L^1$ norm of the Fourier transform of a function and duality, from which we obtain a reformulation in terms of a variant of the Hadamard three lines lemma. By studying Hardy-like spaces of holomorphic functions in a strip in the complex plane, we are able to provide an analytic formula for the minimizers, and use it to get the best possible upper bounds for the optimal constants in the CLR and LT inequalities achievable by the method of [Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ].