Hardy不等式中位数版本

AI总结：

针对Duanyang Zhang提出的离散不等式问题，证明Hardy不等式中位数版本，给出最佳常数，并包含离散形式及其最优性。

摘要：

受Duanyang Zhang在2022年春季NSMO中作为问题6提出的离散不等式问题启发，我们证明了Hardy不等式的中间值版本。对于非负函数$f\in L^p(0,\infty)$，$p>1$，令$A(t)$为$f$在$(0,t)$上的平均值，$M(t)$为$f$在$(0,t)$上的下中位数。我们证明\[ \int_0^\infty |M(t)-A(t)|^p\,dt \leq 2^{1-p}\left(\frac p{p-1}\right)^p \int_0^\infty f(t)^p\,dt, \]并且该常数是最优的。证明基于由中位数的半测度性质导出的逐点重排估计，随后应用经典Hardy不等式。还包含了离散形式及其最优性。

英文摘要：

Motivated by a discrete inequality problem proposed by Duanyang Zhang as Problem 6 of the 2022 Spring NSMO, we prove a median version of Hardy's inequality. For a nonnegative function $f\in L^p(0,\infty)$, $p>1$, let $A(t)$ be the average of $f$ over $(0,t)$, and let $M(t)$ be the lower median of $f$ over $(0,t)$. We show that \[ \int_0^\infty |M(t)-A(t)|^p\,dt \leq 2^{1-p}\left(\frac p{p-1}\right)^p \int_0^\infty f(t)^p\,dt, \] and that the constant is best possible. The proof is based on a pointwise rearrangement estimate coming from the half-measure property of the median, followed by the classical Hardy inequality. A discrete form and its sharpness are also included.

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