arXiv subjects

Stefano Pigola

Publications and source records attributed to Stefano Pigola.

At least 19 recordsLinked to original sources

arXiv2026

Isoperimetric and Michael-Simon inequalities on manifolds with asymptotically nonnegative curvature

We establish the validity of the isoperimetric inequality (or equivalently, an $L^1$ Euclidean-type Sobolev inequality) on manifolds with asymptotically non-negative sectional curvature. Unlike previous results in the literature, our approach does not require the negative part of the curvature to be globally small. Furthermore, we derive a Michael-Simon inequality on manifolds whose curvature is non-negative outside a compact set. The proofs employ the ABP method for isoperimetry, initially introduced by Cabré in the Euclidean setting and subsequently extended and skillfully adapted by Brendle to the challenging context of non-negatively curved manifolds. Notably, we show that this technique can be localized to appropriate regions of the manifold. Additional key elements of the argument include the geometric structure at infinity of asymptotically non-negatively curved manifolds, their spectral properties - which ensure the non-negativity of a Bakry-Émery Ricci tensor on a conformal deformation of each end - and a result that deduces the validity of the isoperimetric inequality on the entire manifold, provided it holds outside a compact set.

math.DG↗

arXiv2025

Lower bounds on the normal injectivity radius of hypersurfaces and bounded geometries on manifolds with boundary

We prove for the first time a pointwise lower estimate of the normal injectivity radius of an embedded hypersurface in an arbitrary Riemannian manifold. Main applications include: (i) a pointwise lower estimate of the graphing radius of a properly embedded hypersurface; (ii) the construction of metrics of bounded geometry on arbitrary manifolds with boundary; (iii) the equivalence of the classical (topological) notion of orientation with that of the geometric notion (in the sense of metric measure spaces) on arbitrary Riemannian manifolds with boundary. In addition, we prove that every manifold with boundary admits a metric with bounded geometry such that the boundary becomes convex. This result strengthens the justification of a recent notion of orientation on finite dimensional RCD spaces.

math.DG↗

arXiv2025

Hardy--Littlewood maximal operators on certain manifolds with bounded geometry

In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on certain Riemannian manifolds with bounded geometry. Our results complement those of various authors. We show that, under mild assumptions, $L^p$ estimates for the centred operator are ``stable'' under conformal changes of the metric, and prove sharp~$L^p$ estimates for the centred operator on Riemannian models with pinched negative scalar curvature. Furthermore, we prove that the centred operator is of weak type $(1,1)$ on the connected sum of two space forms with negative curvature, whereas the uncentred operator is, perhaps surprisingly, bounded only on $L^\infty$. We also prove that if two locally doubling geodesic metric measure spaces enjoying the uniform ball size condition are strictly quasi-isometric, then they share the same boundedness properties for both the centred and the uncentred maximal operator. Finally, we discuss some $L^p$ mapping properties for the centred operator on a specific Riemannian surface introduced by Strömberg, providing new interesting results.

math.FA↗

arXiv2024

Unique continuation at infinity: Carleman estimates on general warped cylinders

We obtain a vanishing result for solutions of the inequality $|Δu|\le q_1|u|+q_2|\nabla u|$ that decay to zero along a very general warped cylindrical end of a Riemannian manifold. The appropriate decay condition at infinity on $u$ is related to the behavior of the potential functions $q_1$ and $q_2$ and to the asymptotic geometry of the end. The main ingredient is a new Carleman estimate of independent interest. Geometric applications to conformal deformations and to minimal graphs are presented.

math.AP↗

arXiv2024

A new notion of subharmonicity on locally smoothing spaces, and a conjecture by Braverman, Milatovic, Shubin

Given a strongly local Dirichlet space and $λ\geq 0$, we introduce a new notion of $λ$--subharmonicity for $L^1_\loc$--functions, which we call \emph{local $λ$--shift defectivity}, and which turns out to be equivalent to distributional $λ$--subharmonicity in the Riemannian case. We study the regularity of these functions on a new class of strongly local Dirichlet, so called locally smoothing spaces, which includes Riemannian manifolds (without any curvature assumptions), finite dimensional RCD spaces, Carnot groups, and Sierpinski gaskets. As a byproduct of this regularity theory, we obtain in this general framework a proof of a conjecture by Braverman, Milatovic, Shubin on the positivity of distributional $L^q$-solutions of $Δf\leq f$ for complete Riemannian manifolds.

math.AP↗

arXiv2023

$L^p_{loc}$ positivity preservation and Liouville-type theorems

On a complete Riemannian manifold $(M,g)$, we consider $L^{p}_{loc}$ distributional solutions of the the differential inequality $-Δu + λu \geq 0$ with $λ>0$ a locally bounded function that may decay to $0$ at infinity. Under suitable growth conditions on the $L^{p}$ norm of $u$ over geodesic balls, we obtain that any such solution must be nonnegative. This is a kind of generalized $L^{p}$-preservation property that can be read as a Liouville type property for nonnegative subsolutiuons of the equation $Δu \geq λu$. An application of the analytic results to $L^{p}$ growth estimates of the extrinsic distance of complete minimal submanifolds is also given.

math.AP↗

arXiv2023

Approximation, regularity and positivity preservation on Riemannian manifolds

The paper focuses on the $L^{p}$-Positivity Preservation property ($L^{p}$-PP for short) on a Riemannian manifold $(M,g)$. It states that any $L^p$ function $u$ with $1<p<+\infty$, which solves $(-Δ+ 1)u\ge 0$ on $M$ in the sense of distributions must be non-negative. Our main result is that the $L^{p}$-PP holds if (the possibly incomplete) $M$ has a finite number of ends with respect to some compact domain, each of which is $q$-parabolic for some, possibly different, values $2p/(p-1) < q \leq +\infty$. When $p=2$, since $\infty$-parabolicity coincides with geodesic completeness, our result settles in the affirmative a conjecture by M. Braverman, O. Milatovic and M. Shubin in 2002. On the other hand, we also show that the $L^{p}$-PP is stable by removing from a complete manifold a possibly singular set with Hausdorff co-dimension strictly larger than $2p/(p-1)$ or with a uniform Minkowski-type upper estimate of order $2p/(p-1)$. The threshold value $2p/(p-1)$ is sharp as we show that when the Hausdorff co-dimension of the removed set is strictly smaller, then the $L^{p}$-PP fails. This gives a rather complete picture. The tools developed to carry out our investigations include smooth monotonic approximation and consequent regularity results for subharmonic distributions, a manifold version of the Brezis-Kato inequality, Liouville-type theorems in low regularity, removable singularities results for $L^{p}$-subharmonic distributions and a Frostman-type lemma. Since the seminal works by T. Kato, the $L^{p}$-PP has been linked to the spectral theory of Schrödinger operators with singular potentials $Δ- V$. Here we present some applications of the main results of this paper to the case where $V\in L^p_{loc}$, addressing the essential self-adjointness of the operator when $p=2$ and whether or not $C^\infty_c(M)$ is an operator core for $Δ-V$ in $L^p$.

math.AP↗

arXiv2023

$L^{p}$ Positivity Preserving and a conjecture by M. Braverman, O. Milatovic and M. Shubin

In this paper we prove that a complete Riemannian manifold is $L^p$-positivity preserving for any $p\in(1,\infty)$. This means that any $L^p$ function which solves $(-Δ+ 1)u\ge 0$ in the sense of distributions is necessarily non-negative. In particular, the case $p=2$ of our result answers in the affermative a conjecture formulated by M. Braverman, O. Milatovic and M. Shubin in 2002. The two main ingredients are a new a-priori regularity result for positive subharmonic distributions, which in turn permits to prove a Liouville type theorem, and a Brezis-Kato inequality on Riemannian manifolds. Both these results rely on a smooth monotonic approximation of distributional solutions of $Δu \ge λ(x) u$ of independent interest.

math.AP↗

arXiv2022

Gradient estimates under integral Ricci bounds

In this paper we study $W^{1,p}$ global regularity estimates for solutions of $Δu = f$ on Riemannian manifolds. Under integral (lower) bounds on the Ricci tensor we prove the validity of $L^p$-gradient estimates of the form $|| \nabla u ||_{L^p} \le C (|| u ||_{L^p} + || Δu||_{L^p})$. We also construct a counterexample which proves that the previously known constant lower bounds on the Ricci curvature are optimal in the pointwise sense. The relation between $L^p$-gradient estimates and different notions of Sobolev spaces is also investigated.

math.AP↗

arXiv2022

$L^{p}$ gradient estimates and Calderón--Zygmund inequalities under Ricci lower bounds

In this paper we investigate the validity of first and second order $L^{p}$ estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present $L^{p}$ estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain $L^p$ estimates for the second order Riesz transform (or, equivalently, the validity of $L^{p}$ Calderón--Zygmund inequalities) on the whole scale $1<p<+\infty$ by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded or non-negative in a global integral sense. When $1<p \leq 2$, analogous $L^p$ bounds on even higher order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order. In the same range of values of $p$, for manifolds with lower Ricci bounds and positive bottom of the spectrum, we show that the $L^{p}$ norm of the Laplacian controls the whole $W^{2,p}$-norm on compactly supported functions.

math.AP↗

arXiv2022

Symmetry of solutions of semilinear PDEs on Riemannian domains

This paper deals with symmetry phenomena for solutions of the Dirichlet problem involving semilinear PDEs on Riemannian domains. We shall present a rather general framework where the symmetry problem can be formulated and provide some evidence that this framework is completely natural by pointing out some results for stable solutions. The case of manifolds with density, and corresponding weighted Laplacians, is inserted in the picture from the very beginning.

math.AP↗

arXiv2021

Global Calderón-Zygmund inequalities on complete Riemannian manifolds

This paper is a survey of some recent results on the validity and the failure of global $W^{2,p}$ regularity properties of smooth solutions of the Poisson equation $Δu = f$ on a complete Riemannian manifold $(M,g)$. We review different methods developed to obtain a-priori $L^p$-Hessian estimates of the form $\| \Hess(u) \|_{L^p} \leq C_1 \| u \|_{L^p} + C_2 \| f \|_{L^p}$ under various geometric conditions on $M$ both in the case of real valued functions and for manifold valued maps. We also present explicit and somewhat implicit counterexamples showing that, in general, this integral inequality may fail to hold even in the presence of a lower sectional curvature bound. The rôle of a gradient estimate of the form $\| \nabla u \|_{L^{p}} \leq C_1 \| u \|_{L^p} + C_2 \| f \|_{L^p}$, and its connections with the $L^{p}$-Hessian estimate, are also discussed.

math.AP↗

arXiv2020

The Frankel property for self-shrinkers from the viewpoint of elliptic PDE's

We show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f-harmonic function with controlled gradient. In the immersed case, a new direct proof of the generalized half-space property is also presented.

math.DG↗

arXiv2020

Qualitative properties of bounded subsolutions of nonlinear PDEs

We study decay and compact support properties of positive and bounded solutions of $Δ_{p} u \geq Λ(u)$ on the exterior of a compact set of a complete manifold with rotationally symmetry. In the same setting, we also give a new characterization of stochastic completeness for the $p$-Laplacian in terms of a global $W^{1,p}$-regularity of such solutions. One of the tools we use is a nonlinear version of the Feller property which we investigate on general Riemannian manifolds and which we establish under integral Ricci curvature conditions.

math.AP↗

arXiv2018

Height estimates for Killing graphs

The paper aims at proving global height estimates for Killing graphs defined over a complete manifold with nonempty boundary. To this end, we first point out how the geometric analysis on a Killing graph is naturally related to a weighted manifold structure, where the weight is defined in terms of the length of the Killing vector field. According to this viewpoint, we introduce some potential theory on weighted manifolds with boundary and we prove a weighted volume estimate for intrinsic balls on the Killing graph. Finally, using these tools, we provide the desired estimate for the weighted height in the assumption that the Killing graph has constant weighted mean curvature and the weighted geometry of the ambient space is suitably controlled.

math.DG↗

arXiv2018

$L^{p}$-interpolation inequalities and global Sobolev regularity results (with an appendix by Ognjen Milatovic)

On any complete Riemannian manifold $M$ and for all $p\in [2,\infty)$, we prove a family of second order $L^{p}$-interpolation inequalities that arise from the following simple $L^{p}$-estimate valid for every $u \in C^{\infty}(M)$: $$ \|\nabla u\|_{p}^p \leq \|u Δ_{p} u\|_1\in [0,\infty], $$ where $Δ_p$ denotes the $p$-Laplace operator. We show that these inequalities, in combination with abstract functional analytic arguments, allow to establish new global Sobolev regularity results for $L^p$-solutions of the Poisson equation for all $p\in (1,\infty)$, and new global Sobolev regularity results for the singular magnetic Schrödinger semigroups.

math.AP↗

arXiv2018

Nonlinear Calderón-Zygmund inequalities for maps

Being motivated by the problem of deducing $L^p$-bounds on the second fundamental form of an isometric immersion from $L^p$-bounds on its mean curvature vector field, we prove a (nonlinear) Calderón-Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.

math.DG↗

arXiv2017

Dirichlet parabolicity and $L^1$-Liouville property under localized geometric conditions

We shed a new light on the $L^1$-Liouville property for positive, superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space. We also present examples in any dimension showing that the $L^1$-Liouville property is strictly weaker than the stochastic completeness of the manifold. The main tool in our investigations is represented by the potential theory of a manifold with boundary subject to Dirichlet boundary conditions. The paper incorporates, under a unifying viewpoint, some old and new aspects of the theory, with a special emphasis on global maximum principles and on the role of the Dirichlet Green's kernel.

math.DG↗