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Julian Weigt

Publications and source records attributed to Julian Weigt.

14 recordsLinked to original sources
arXiv2026

Sharp higher order regularity of discrete maximal functions

We derive sharp $\ell^p(\mathbb{Z})$ bounds for the $k$th derivative of the discrete uncentered maximal operator applied to characteristic functions $f:\mathbb{Z}\to\{0,1\}$ in the cases $k=0,1,2$. When $k=1,2$ these are the first sharp bounds for derivatives of a Hardy-Littlewood maximal function in continuous or discrete settings when $1<p<\infty$. We also establish several lower bounds for $k\geq 3$ and for general functions $f:\mathbb{Z}\to\mathbb{R}$.

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arXiv2025

The centered maximal operator removes the non-concave Cantor part from the gradient

We study regularity of the centered Hardy--Littlewood maximal function $M f$ of a function $f$ of bounded variation in $\mathbb R^d$, $d\in \mathbb N$. In particular, we show that at $|D^c f|$-a.e. point $x$ where $f$ has a non-concave blow-up, it holds that $M f(x)>f^*(x)$. We further deduce from this that if the variation measure of $f$ has no jump part and its Cantor part has non-concave blow-ups, then BV regularity of $M f$ can be upgraded to Sobolev regularity.

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arXiv2025

A Vitali-type lemma for the boundary

Take a set of balls in $\mathbb R^d$. We find a subset of pairwise disjoint balls whose combined perimeter controls the perimeter of the union of the original balls. This can be seen as a boundary version of the Vitali covering lemma. We further prove combined volume-perimeter results and counterexamples and apply them to find short proofs of some regularity statements for maximal functions.

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arXiv2025

Alberti representations, rectifiability of metric spaces and higher integrability of measures satisfying a PDE

We give a sufficient condition for a Borel subset $E\subset X$ of a complete metric space with $\mathcal{H}^n(E)<\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an Alberti representation. Precisely, we show that if $\mathcal{H}^n|_E$ has $n$ independent Alberti representations, then $E$ is $n$-rectifiable. This is a sharp strengthening of prior results of Bate and Li. It has been known for some time that such a result answers many open questions concerning rectifiability in metric spaces, which we discuss. An important step of our proof is to establish the higher integrability of measures on Euclidean space satisfying a PDE constraint. These results provide a quantitative generalisation of recent work of De Philippis and Rindler and are of independent interest.

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arXiv2024

Sobolev bounds and counterexamples for the second derivative of the maximal function in one dimension

We investigate the question whether the $L^1(\mathbb R)$-norm of the second derivative of the uncentered Hardy-Littlewood maximal function can be bounded by a constant times the $L^1(\mathbb R)$-norm of the function itself. We give a positive answer for a class of functions that contains Sobolev functions on the real line which are decreasing away from the origin and even, and we provide a counterexample which is also decreasing away from the origin but not even.

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arXiv2023

Weighted fractional Poincaré inequalities via isoperimetric inequalities

Our main result is a weighted fractional Poincaré-Sobolev inequality improving the celebrated estimate by Bourgain-Brezis-Mironescu. This also yields an improvement of the classical Meyers-Ziemer theorem in several ways. The proof is based on a fractional isoperimetric inequality and is new even in the non-weighted setting. We also extend the celebrated Poincaré-Sobolev estimate with $A_p$ weights of Fabes-Kenig-Serapioni by means of a fractional type result in the spirit of Bourgain-Brezis-Mironescu. Examples are given to show that the corresponding $L^p$-versions of weighted Poincaré inequalities do not hold for $p>1$.

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arXiv2022

Variation of the uncentered maximal characteristic Function

Let $\mathcal M$ be the uncentered Hardy-Littlewood maximal operator or the dyadic maximal operator and $d\geq1$. We prove that for a set $E\subset\mathbb R^d$ of finite perimeter the bound $\operatorname{var}\mathcal M1_E\leq C_d\operatorname{var}1_E$ holds. We also prove this for the local maximal operator.

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arXiv2021

The Variation of the Uncentered Maximal Operator with respect to Cubes

We consider the maximal operator with respect to uncentered cubes on Euclidean space with arbitrary dimension. We prove that for any function with bounded variation, the variation of its maximal function is bounded by the variation of the function times a dimensional constant. We also prove the corresponding result for maximal operators with respect to collections of more general sets than cubes. The sets are required to satisfy a certain inner cone star condition and in addition the collection must enjoy a tiling property which for example the collection of all cubes does enjoy and the collection of all Euclidean balls does not.

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arXiv2021

Continuity of the gradient of the fractional maximal operator on $W^{1,1}(\mathbb{R}^d)$

We establish that the map $f\mapsto |\nabla \mathcal{M}_αf|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$, where $α\in (0,d)$, $q=\frac{d}{d-α}$ and $\mathcal{M}_α$ denotes either the centered or non-centered fractional Hardy--Littlewood maximal operator. In particular, we cover the cases $d >1$ and $α\in (0,1)$ in full generality, for which results were only known for radial functions.

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arXiv2020

Variation of the dyadic maximal function

We prove that for the dyadic maximal operator $\mathrm M$ and every locally integrable function $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ with bounded variation, also $\mathrm M f$ is locally integrable and $\mathop{\mathrm{var}}\mathrm M f\leq C_d\mathop{\mathrm{var}} f$ for any dimension $d\geq1$. It means that if $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ is a function whose gradient is a finite measure then so is $\nabla \mathrm M f$ and $\|\nabla \mathrm M f\|_{L^1(\mathbb R^d)}\leq C_d\|\nabla f\|_{L^1(\mathbb R^d)}$. We also prove this for the local dyadic maximal operator.

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arXiv2020

Weak differentiability for fractional maximal functions of general $L^{p}$ functions on domains

Let $Ω\subset \mathbb{R}^{n}$ be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to $Ω$ maps $L^{p}(Ω) \to W^{1,p}(Ω)$ for all $p > 1$, when the smoothness index $α\geq 1$. In particular, the results are valid in the range $p \in (1, n/(n-1)]$ that was previously unknown. As an application, we prove an endpoint regularity result in the domain setting.

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arXiv2019

Almost-Orthogonality of Restricted Haar-Functions

We consider the Haar functions $h_I$ on dyadic intervals. We show that if $p>\frac23$ and $E\subset[0,1]$ then the set of all functions $\|h_I1_E\|_2^{-1}h_I1_E$ with $|I\cap E|\geq p|I|$ is a Riesz sequence. For $p\leq\frac23$ we provide a counterexample.

math.FA↗