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Leonardo Franchi

Publications and source records attributed to Leonardo Franchi.

5 recordsLinked to original sources
arXiv2026

Osgood meets Ambrosio-DiPerna-Lions

In this note we extend the Ambrosio-DiPerna-Lions theory on the well-posedness of the transport equation to the case in which the vector field satisfies an $L^p$ Osgood condition. In particular we show that bounded distributional solutions to the transport equation are unique and thus renormalized. As opposed to the classical proof, we do not rely on the vanishing of a commutator, but we show that a weighted energy built out of the Littlewood-Paley decomposition of the solution is conserved.

math.AP↗
arXiv2026

Product-free subsets of $(0,1)$

The third problem in Ben Green's collection of 100 open problems asks whether an open subset of $(0,1)$ that does not contain $x,y,z$ with $xy=z$ must have measure at most 1/3. We give an affirmative answer to this question. As part of the proof we obtain a result of independent interest that gives a lower bound for the size of the sumset and the difference set of a set of reals in terms not just of its size but also of a parameter that measures how far it is from being an interval.

math.CO↗
arXiv2026

A unified abstract regularity lemma

The goal of this short note is to prove a unified abstract regularity lemma which recovers Szemer\'edi's graph regularity lemma, Green's arithmetic regularity lemma, and a regularity lemma for Boolean functions as direct corollaries.

math.CO↗
arXiv2026

A strengthening of Chang's lemma

We prove a strengthening of Chang's lemma for subsets of $\mathbb F_p^n$. The classical conclusion that the large spectrum is contained in a subspace of dimension at most $2\varepsilon^{-2}\log(1/\alpha)$ is refined to show that every character outside this subspace has small correlation with the set not only globally, but also on average over the cosets of the orthogonal complement, in a natural cosetwise $\ell^1$ norm. As a consequence, we obtain a localized counting lemma. We also give an extension of the argument to arbitrary finite abelian groups.

math.NT↗