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Andrea Pinamonti

Publications and source records attributed to Andrea Pinamonti.

At least 55 records · Page 3Linked to original sources
arXiv2023

Weighted Sobolev Spaces on Metric Measure Spaces

We investigate weighted Sobolev spaces on metric measure spaces $(X,d,m)$. Denoting by $ρ$ the weight function, we compare the space $W^{1,p}(X,d,ρm)$ (which always concides with the closure $H^{1,p}(X,d,ρm)$ of Lipschitz functions) with the weighted Sobolev spaces $W^{1,p}_ρ(X,d,m)$ and $H^{1,p}_ρ(X,d,m)$ defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that $W^{1,p}(X,d,ρm)=H^{1,p}_ρ(X,d, m)$. We also adapt results by Muckenhoupt and recent work by Zhikov to the metric measure setting, considering appropriate conditions on $ρ$ that ensure the equality $W^{1,p}_ρ(X,d,m)=H^{1,p}_ρ(X,d,m)$.

math.AP↗
arXiv2023

Nguyen's approach to Sobolev spaces in metric measure spaces with unique tangents

We extend Nguyen's characterization of Sobolev spaces $W^{1,p}$ to the setting of PI-metric measure spaces such that at a.e. point the tangent space (in the Gromov-Hausdorff sense) is unique and Euclidean with a fixed dimension. We also generalize [CLL14] to PI-metric measure spaces such that at a.e. point the tangent space is unique and equal to the Heisenberg group with a fixed homogeneous dimension. The approach is easier and completely different from the one in [Ngu06, Ngu08].

math.FA↗
arXiv2023

The asymptotic $p$-Poisson equation as $p \to \infty$ in Carnot-Carathéodory spaces

In this paper we study the asymptotic behavior of solutions to the subelliptic $p$-Poisson equation as $p\to +\infty$ in Carnot Carathéodory spaces. In particular, introducing a suitable notion of differentiability, we extend the celebrated result of Bhattacharya, DiBenedetto and Manfredi [Rend. Sem. Mat. Univ. Politec. Torino, 1989, Special Issue, 15-68] and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the $\infty-$Laplacian and the Eikonal equation.

math.AP↗
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↗
arXiv2022

On the converse of Pansu's Theorem

We provide a suitable generalisation of Pansu's differentiability theorem to general Radon measures on Carnot groups and we show that if Lipschitz maps between Carnot groups are Pansu-differentiable almost everywhere for some Radon measures $μ$, then $μ$ must be absolutely continuous with respect to the Haar measure of the group.

math.MG↗
arXiv2022

Maximal Directional Derivatives in Laakso Space

We investigate the connection between maximal directional derivatives and differentiability for Lipschitz functions defined on Laakso space. We show that maximality of a directional derivative for a Lipschitz function implies differentiability only for a $σ$-porous set of points. On the other hand, the distance to a fixed point is differentiable everywhere except for a $σ$-porous set of points. This behavior is completely different to the previously studied settings of Euclidean spaces and Carnot groups.

math.FA↗
arXiv2022

A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces

We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincaré inequality. Compared with previous works, we consider more general functionals. We also give a counterexample in the case $p=1$ demonstrating that unlike in Euclidean spaces, in metric measure spaces the limit of the nonlocal functions is only comparable, not necessarily equal, to the variation measure $\| Df\|(Ω)$.

math.FA↗
arXiv2022

A C^k Lusin Approximation Theorem For Real-Valued Functions on Carnot Groups

We study the Lusin approximation problem for real-valued measurable functions on Carnot groups. We prove that k-approximate differentiability almost everywhere is equivalent to admitting a Lusin approximation by $C^{k}_{\mathbb{G}}$ maps. We also prove that existence of an approximate (k-1)-Taylor polynomial almost everywhere is equivalent to admitting a Lusin approximation by maps in a suitable Lipschitz function space.

math.FA↗
arXiv2021

Multiple solutions for nonlinear boundary value problems of Kirchhoff type on a double phase setting

This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti-Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools.

math.AP↗
arXiv2021

On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: Bourgain-Brezis-Mironescu's theorem revisited

We generalize Bourgain-Brezis-Mironescu's asymptotic formula for fractional Sobolev functions, in the setting of abstract metric measure spaces, under the assumption that at almost every point the tangent space in the measured Gromov-Hausdorff sense is a finite dimensional Banach space or a Carnot group. Our result not only covers the known results concerning Euclidean spaces, weighted Riemannian manifolds and finite dimensional Banach spaces, but also extends Bourgain-Brezis-Mironescu's formula to $\rcdkn$ spaces and sub-Riemannian manifolds.

math.FA↗
arXiv2021

New Integral Estimates in Substatic Riemannian Manifolds and the Alexandrov Theorem

We derive new integral estimates on substatic manifolds with boundary of horizon type, naturally arising in General Relativity. In particular, we generalize to this setting an identity due to Magnanini-Poggesi leading to the Alexandrov Theorem in the Euclidean space and improve on a Heintze-Karcher type inequality due to Li-Xia. Our method relies on the introduction of a new vector field with nonnegative divergence, generalizing to this setting the P-function technique of Weinberger.

math.DG↗
arXiv2021

$Γ$-convergence for functionals depending on vector fields. II. Convergence of minimizers

Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study integral functionals depending on $X$. Using the results in \cite{MPSC1}, we study the convergence of minima, minimizers and momenta of those functionals. Moreover, we apply these results to the periodic homogenization in Carnot groups and to prove a $H$-compactness theorem for linear differential operators of the second order depending on $X$.

math.AP↗
arXiv2021

Loomis-Whitney inequalities in Heisenberg groups

This note concerns Loomis-Whitney inequalities in Heisenberg groups $\mathbb{H}^n$: $$|K| \lesssim \prod_{j=1}^{2n}|π_j(K)|^{\frac{n+1}{n(2n+1)}}, \qquad K \subset \mathbb{H}^n.$$ Here $π_{j}$, $j=1,\ldots,2n$, are the vertical Heisenberg projections to the hyperplanes $\{x_j=0\}$, respectively, and $|\cdot|$ refers to a natural Haar measure on either $\mathbb{H}^n$, or one of the hyperplanes. The Loomis-Whitney inequality in the first Heisenberg group $\mathbb{H}^1$ is a direct consequence of known $L^p$ improving properties of the standard Radon transform in $\mathbb{R}^2$. In this note, we show how the Loomis-Whitney inequalities in higher dimensional Heisenberg groups can be deduced by an elementary inductive argument from the inequality in $\mathbb{H}^1$. The same approach, combined with multilinear interpolation, also yields the following strong type bound: $$\int_{\mathbb{H}^n} \prod_{j=1}^{2n} f_j(π_j(p))\;dp\lesssim \prod_{j=1}^{2n} \|f_j\|_{\frac{n(2n+1)}{n+1}}$$ for all nonnegative measurable functions $f_1,\ldots,f_{2n}$ on $\mathbb{R}^{2n}$. These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis-Whitney inequalities in $\mathbb{H}^n$, we mention the following sharper version of the classical geometric Sobolev inequality in $\mathbb{H}^n$: $$\|u\|_{\frac{2n+2}{2n+1}} \lesssim \prod_{j=1}^{2n}\|X_ju\|^{\frac{1}{2n}}, \qquad u \in BV(\mathbb{H}^n),$$ where $X_j$, $j=1,\ldots,2n$, are the standard horizontal vector fields in $\mathbb{H}^n$.

math.CA↗
arXiv2021

A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group

We prove a $C^m$ Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous horizontal curve whose horizontal velocity is $m-1$ times $L^1$ differentiable almost everywhere coincides with a $C^m$ horizontal curve except on a set of small measure. Conversely, we show that the result no longer holds if $L^1$ differentiability is replaced by approximate differentiability. This shows our result is optimal and highlights differences between the Heisenberg and Euclidean settings.

math.MG↗
arXiv2020

Lipschitz minimizers for a class of integral functionals under the bounded slope condition

We consider the functional $\int_Ωg(\nabla u+\textbf X^\ast)d\mathscr L^{2n}$ where $g$ is convex and $\textbf X^\ast(x,y)=2(-y,x)$ and we study the minimizers in $BV(Ω)$ of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on $g$, there exists a unique minimizer which is also Lipschitz continuous.

math.AP↗