arXiv subjects

Pietro Wald

Publications and source records attributed to Pietro Wald.

4 recordsLinked to original sources

arXiv2026

Planar lamplighter is not of negative type

The lamplighter group over the planar integer grid is proved to not be bi-Lipschitz equivalent to any metric space of negative type, so in particular it does not admit a bi-Lipschitz embedding into $L_1$. This shows the existence of finitely generated metabelian groups on which word metrics are never comparable up to constant factors to conditionally negative definite (CND) kernels, and that the property of admitting a word metric-comparable CND kernel is not preserved by wreath products.

math.MG↗

arXiv2025

Shortcut Laakso spaces, pure PI unrectifiability and differentiability of Lipschitz functions

We construct a family of purely PI unrectifiable Lipschitz differentiability spaces and investigate the possible of Banach spaces targets for which Lipschitz differentiability holds. We provide a general investigation into the geometry of \emph{shortcut} metric spaces and characterise when such spaces are PI rectifiable, and when they are $Y$-LDS, for a given $Y$. The family of spaces arises as an example of our characterisations. Indeed, we show that Laakso spaces satisfy the required hypotheses.

math.FA↗

arXiv2025

Structure of Metric $1$-currents: approximation by normal currents and representation results

We prove the $1$-dimensional flat chain conjecture in any complete and quasiconvex metric space, namely that metric $1$-currents can be approximated in mass by normal $1$-currents. The proof relies on a new Banach space isomorphism theorem, relating metric $1$-currents and their boundaries to the Arens-Eells space. As a by-product, any metric $1$-current in a complete and separable metric space can be represented as the integral superposition of oriented $1$-rectifiable sets, thus dropping a finite dimensionality condition from previous results of Schioppa [Schioppa Adv. Math. 2016, Schioppa J. Funct. Anal. 2016]. The connection between the flat chain conjecture and the representation result is provided by a structure theorem for metric $1$-currents in Banach spaces, showing that any such current can be realised as the restriction to a Borel set of a boundaryless normal $1$-current. This generalizes, to any Banach space, the $1$-dimensional case of a recent result of Alberti-Marchese in Euclidean spaces [Alberti-Marchese 2023]. The argument of Alberti-Marchese requires the strict polyhedral approximation theorem of Federer for normal $1$-currents, which we obtain in Banach spaces.

math.MG↗

arXiv2022

Kakeya maximal inequality in the Heisenberg group

We define the Heisenberg Kakeya maximal functions $M_δf$, $0<δ<1$, by averaging over $δ$-neighborhoods of horizontal unit line segments in the Heisenberg group $\mathbb{H}^1$ equipped with the Korányi distance $d_{\mathbb{H}}$. We show that $$ \|M_δf\|_{L^3(S^1)}\leq C(\varepsilon)δ^{-1/3-\varepsilon}\|f\|_{L^3(\mathbb{H}^1)},\quad f\in L^3(\mathbb{H}^1),$$ for all $\varepsilon>0$. The proof is based on a recent variant, due to Pramanik, Yang, and Zahl, of Wolff's circular maximal function theorem for a class of planar curves related to Sogge's cinematic curvature condition. As an application of our Kakeya maximal inequality, we recover the sharp lower bound for the Hausdorff dimension of Heisenberg Kakeya sets of horizontal unit line segments in $(\mathbb{H}^1,d_{\mathbb{H}})$, first proven by Liu.

math.CA↗