arXiv subjects

Luca Gennaioli

Publications and source records attributed to Luca Gennaioli.

11 recordsLinked to original sources

arXiv2026

Improvement of the dimension bound for unweighted RCD spaces

We show that if $(X,d,\mathscr H^n)$ is an $RCD(K,N)$ space for some $K\in\mathbb{R}$ and $N\in [1,\infty)$, then it is also a (non-collapsed) $RCD(K,n)$ space. In other words, unweighted finite-dimensional $RCD$ spaces enjoy an automatic improvement of the dimension bound to the Hausdorff dimension of the space. More generally, we prove that this holds also when $N=\infty$, provided that we assume in addition the $n$-rectifiability of the space.

math.MG↗

arXiv2026

Concentration phenomena and the Vanishing Mass Conjecture

Concentrating (that is, non-equi-integrable) sequences of functions satisfying linear PDE constraints arise in a wide variety of problems in PDE, the calculus of variations, and geometric measure theory. In 2003, Bouchitt\'{e} conjectured that such concentrations can always be represented as superpositions of ``simple'' concentrations, a claim he termed the Vanishing Mass Conjecture. We fully resolve this conjecture for all first-order constant-coefficient linear differential operators, proving that the value-distribution (Young) measure of any such sequence admits a Choquet-type decomposition into probability measures whose barycenters lie in the associated Tartar wave cone. In fact, we prove a significantly stronger statement than originally conjectured, namely that the constituent measures are themselves generated by concentrating sequences satisfying the same PDE constraint. This provides a complete structural description of concentrating sequences and yields several applications, including a new proof of a theorem of De Philippis and the second author on the singular polar of PDE-constrained measures, two types of compensated compactness results, a general lower bound related to the Optimal Light Structures Conjecture in shape optimization, and a surprising result on the support cardinality of extremal concentration Young measures. We also place our results in the context of Morrey's Conjecture, showing that its analogue for pure concentrations is false. Our proofs introduce several new techniques, most notably convexity arguments involving ``barrier functions'' and a careful smoothing procedure via the heat flow that replaces classical Fourier methods.

math.AP↗

arXiv2026

Lipschitz continuity of harmonic maps between ${\rm RCD}(K,N)$ spaces and ${\rm CAT}(\kappa)$ spaces

We are going to prove that energy minimizing harmonic maps from a domain $\Omega$ inside an ${\rm RCD}(K,N)$ whose image lies in a small ball inside a ${\rm CAT}(\kappa)$ space are locally Lipschitz continuous. This completes the picture concerning regularity of harmonic maps between singular spaces and justifies a variant of the Bochner-Eells-Sampson inequality between singular spaces.

math.AP↗

arXiv2026

Sets of finite perimeter on Riemannian manifolds and stochastic completeness

We prove a heat semigroup characterization of the total variation for compactly supported ${\rm BV}$ on arbitrary smooth complete weighted Riemannian manifolds, extending the main result in \cite{GP15}. We then provide an example of a weighted manifold where such equivalence does not hold for a large class of sets of finite perimeter.

math.AP↗

arXiv2026

A new Duhamel-type principle with applications to geometric (in)equalities

We introduce a simple new method, based on the Caffarelli-Silvestre extension and a Duhamel-type formula, to derive exact pointwise identities for fractional commutators and nonlinear compositions associated with the fractional Laplacian on general Riemannian manifolds. As applications, we obtain a pointwise fractional Leibniz rule, a fractional Bochner's formula with an explicit Ricci curvature term, apparently the first of this kind, and exact remainders in the C\'ordoba-C\'ordoba and Kato inequalities for the fractional Laplacian. All these formulas are new even in the Euclidean space.

math.AP↗

arXiv2025

Sharp conditions for the BBM formula and asymptotics of heat content-type energies

Given $p\in[1,\infty)$, we provide sufficient and necessary conditions on the non-negative measurable kernels $(\rho_t)_{t\in(0,1)}$ ensuring convergence of the associated Bourgain-Brezis-Mironescu (BBM) energies $(\mathscr{F}_{t,p})_{t\in(0,1)}$ to a variant of the $p$-Dirichlet energy on $\mathbb R^N$ as $t\to0^+$ both in the pointwise and in the $\Gamma$-sense. We also devise sufficient conditions on $(\rho_t)_{t\in(0,1)}$ yielding local compactness in $L^p(\mathbb R^N)$ of sequences with bounded BBM energy. Moreover, we give sufficient conditions on $(\rho_t)_{t\in(0,1)}$ implying pointwise and $\Gamma$-convergence and compactness of $(\mathscr{F}_{t,p})_{t\in(0,1)}$ when the limit $p$-energy is of non-local type. Finally, we apply our results to provide asymptotic formulas in the pointwise and $\Gamma$-sense for heat content-type energies both in the local and non-local settings.

math.AP↗

arXiv2024

Comments on the regularity of harmonic maps between singular spaces

In this work we are going to establish Hölder continuity of harmonic maps from an open set $Ω$ in an ${\rm RCD}(K,N)$ space valued into a ${\rm CAT}(κ)$ space, with the constraint that the image of $Ω$ via the map is contained in a sufficiently small ball in the target. Building on top of this regularity and assuming a local Lipschitz regularity of the map, we establish a weak version of the Bochner-Eells-Sampson inequality in such a non-smooth setting. Finally we study the boundary regularity of such maps.

math.AP↗

arXiv2024

Fourier transform of BV functions and applications

This paper investigates the relation between the Fourier transform of {\rm BV} (bounded variation) functions and their jump sets. We introduce the notion of $L^2$-jump product and obtain a weighted Plancherel identity for {\rm BV} functions. As a corollary, we get a newfound characterization of sets of finite perimeter in terms of their Fourier transform. Moreover, we sharpen a result of Herz on the set-theoretic derivative of the Fourier transform of characteristic functions of sets. Last, we obtain sharp bounds on the quadratic discrepancy of {\rm BV} functions, and as a consequence, we generalize the classic estimates of Beck and Montgomery.

math.CA↗

arXiv2024

Asymptotics as $s \to 0^+$ of the fractional perimeter on Riemannian manifolds

In this work we study the asymptotics of the fractional Laplacian as $s\to 0^+$ on any complete Riemannian manifold $(M,g)$, both of finite and infinite volume. Surprisingly enough, when $M$ is not stochastically complete this asymptotics is related to the existence of bounded harmonic functions on $M$. As a corollary, we can find the asymptotics of the fractional $s$-perimeter on (essentially) every complete manifold, generalising both the existing results for $\mathbb{R}^n$ and for the Gaussian space. In doing so, from many sets $E\subset M$ we are able to produce a bounded harmonic function associated to $E$, which in general can be non-constant.

math.DG↗

arXiv2022

A note about charts built by Eriksson-Bique and Soultanis on metric measure spaces

This note is motivated by recent studies by Eriksson-Bique and Soultanis about the construction of charts in general metric measure spaces. We analyze their construction and provide an alternative and simpler proof of the fact that these charts exist on sets of finite Hausdorff dimension. The observation made here offers also some simplification about the study of the relation between the reference measure and the charts in the setting of $\text{RCD}$ spaces.

math.MG↗

arXiv2022

On the Hausdorff Measure of $\R^n$ with the Euclidean Topology

In this paper we answer a question raised by David H. Fremlin about the Hausdorff measure of $\mathbb{R}^2$ with respect to a distance inducing the Euclidean topology. In particular we prove that the Hausdorff $n$-dimensional measure of $\mathbb{R}^n$ is never $0$ when considering a distance inducing the Euclidean topology. Finally, we show via counterexamples that the previous result does not hold in general if we remove the assumption on the topology.

math.MG↗