A Median Version of Hardy's Inequality
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914599019741184 |
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| author | Leng, Gangsong |
| author_facet | Leng, Gangsong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_25366 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Median Version of Hardy's Inequality Leng, Gangsong Metric Geometry Functional Analysis 26D15, 46E30 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. |
| title | A Median Version of Hardy's Inequality |
| topic | Metric Geometry Functional Analysis 26D15, 46E30 |
| url | https://arxiv.org/abs/2605.25366 |