Abstract: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.
| Subjects: | Metric Geometry (math.MG); Functional Analysis (math.FA) |
| MSC classes: | 26D15, 46E30 |
| Cite as: | arXiv:2605.25366 [math.MG] |
| (or arXiv:2605.25366v1 [math.MG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25366 arXiv-issued DOI via DataCite (pending registration) |
From: Gangsong Leng [view email]
[v1]
Mon, 25 May 2026 02:44:53 UTC (5 KB)
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