Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.39178

Sharp derivative inequalities for maximal functions

Sung-Yi Liao·José Madrid·Eyvindur Ari Palsson
View original source ↗Read source document ↗Export metadata ↓

Abstract

We determine the optimal constant in the $L^p$-derivative inequality for the uncentered Hardy--Littlewood maximal operator on the real line for every $1<p<\infty$. It is the unique number $γ_p\in [0,1]$ satisfying $(2p-1)γ_p+(p-1)γ_p^{(2p-1)/(p-1)}=p$. As $p$ goes from $1$ to $+\infty$, $γ_p$ decreases from $1$ to $\sqrt2-1$, and at the endpoints the limiting arguments recover the classical total-variation and Lipschitz inequalities, with the qualitative regularity ingredient at the $BV$ endpoint supplied by Aldaz and Pérez Lázaro.

Explore related subjects

math.CAmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sung-Yi Liao, José Madrid, Eyvindur Ari Palsson. 2026-10-29. Sharp derivative inequalities for maximal functions. https://arxiv.org/abs/2610.39178

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hausdorff dimension of the graph of a random vector-valued Weierstrass function

Let $\{θ_n,λ_n:n\geq0\}$ be mutually independent random variables, each uniformly distributed on $[0,1]$, and write $Θ=(θ_n)_{n\geq0}$ and $Λ=(λ_n)_{n\geq0}$. The random vector-valued Weierstrass function is given by $$f_{Θ,Λ}(x)= \left( \sum_{n=0}^{\infty} a^n\cos\bigl(2π(b^n x+θ_n)\bigr),\ \sum_{n=0}^{\infty} a^n\sin\bigl(2π(b^n x+λ_n)\bigr) \right), \; x\in[0,1],$$ where $0 1$. The Hausdorff dimension of the graph of this function is proved to be $$\dim_H G(f_{Θ,Λ}) = \min\left\{-\frac{\log b}{\log a}, \, 3 +2\frac{\log a}{\log b}\right\} \quad \text{a.s}.$$

math.CA↗
arXiv2026

Characterization of the boundedness of the segment multiplier on rearrangement-invariant spaces

Given rearrangement-invariant (r.i.) spaces $X$ and $Y$, we show that the segment multiplier, the truncated Hilbert transform, and the discrete Hilbert transform (on the associated discretized space) are bounded from $X$ to $Y$ simultaneously. Moreover, when $X = Y$ this boundedness is characterized by a condition on a pair of Boyd-type indices. We also exhibit an r.i. space on which the segment multiplier is bounded while the (non-truncated) Hilbert transform fails to be bounded.

math.CA↗
arXiv2026

Endpoint regularity of Radon transforms associated with real analytic phases in the plane

In this paper, we consider the endpoint Sobolev regularity of Radon transforms associated with real-analytic functions in the plane. We establish all of the failure of endpoint regularity for real-analytic phases. On the one hand, the critical endpoint regularity $L^p(\mathbb{R}^2)\rightarrow L^p_s(\mathbb{R}^2)$, corresponding to extreme points of the reduced Newton polyhedron of phase function, is true if only and only if $p=2$. For generic extreme points, this failure of endpoint regularity generalizes a result of Christ \cite{christ}. On the other hand, we find that the endpoint Sobolev regularity also fails on a bounded segment of the reduced Newton polyhedron when the bisectrix $y=x$ intersects one of unbounded faces at a relative interior point. In particular, our results imply that the Radon transform is not bounded from $L^p$ into $L^p_{1/n}$ with $p=n/(n-1)$ or $p=n$, where the phase functions are real-valued homogeneous polynomials $S(x,y)=a_1x^{n-1}y+a_2x^{n-2}y^2+\cdots+a_{n-1}xy^{n-1}$ with $n\geq 3$ and $a_1a_{n-1}\neq0$. This answers a problem of Phong and Stein in 1994. Moreover, the main results of this paper also give examples which show failure of critical Sobolev regularity for degenerate Fourier integral operators satisfying finite type conditions formulated in \cite{Seeger93,seeger2}. Based on earlier results and our proof of several remaining cases, we conclude that all of other critical endpoint Sobolev estimates are true for Radon transforms associated with real-analytic phases in the plane. Therefore, all failures of endpoint regularity estimates are clearly determined.

math.CA↗