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The paper is devoted to prove Allard-Michael-Simon-type $L^p$-Sobolev $(p>1)$ inequalities with explicit constants on Euclidean submanifolds of any codimension. Such inequalities contain, beside the Dirichlet $p$-energy, a term involving the mean curvature of the submanifold. Our results require separate discussions for the cases $p\geq 2$ and $1

We study $p$-Wasserstein spaces over the branching spaces $\mathbb{R}^2$ and $[0,1]^2$ equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all $p\geq1,$ meaning that all isometries of these spaces are induced by isometries of the underlying space via the push-forward operation. This is in contrast to the case of the Euclidean metric since with that distance the $2$-Wasserstein space over $\mathbb{R}^2$ is not rigid. Also, we highlight that $1$-Wasserstein space is not rigid over the interval $[0,1]$, while according to our result, its two-dimensional analog with the more complicated geodesic structure is rigid. To the best of our knowledge, our results are the first ones concerning the structure of isometries of Wasserstein spaces over branching spaces. Accordingly, our proof techniques substantially deviate from those used for non-branching underlying spaces., Comment: 23 pages, 3 figures

The paper is devoted to provide Michael-Simon-type $L^p$-logarithmic-Sobolev inequalities on complete, not necessarily compact $n$-dimensional submanifolds $\Sigma$ of the Euclidean space $\mathbb R^{n+m}$. Our first result, stated for $p=2$, is sharp, it is valid on general submanifolds, and it involves the mean curvature of $\Sigma$. It implies in particular the main result of S. Brendle [Comm. Pure Appl. Math.}, 2022]. In addition, it turns out that equality can only occur if and only if $\Sigma$ is isometric to the Euclidean space $\mathbb R^{n}$ and the extremizer is a Gaussian. The second result is a general $L^p$-logarithmic-Sobolev inequality for $p\geq 2$ on Euclidean submanifolds with constants that are codimension-free in case of minimal submanifolds. In order to prove the above results - especially, to deal with the equality cases - we elaborate the theory of optimal mass transport on submanifolds between measures that are not necessarily compactly supported. Applications are provided to sharp hypercontractivity estimates of Hopf-Lax semigroups on submanifolds. The first hypercontractivity estimate is for general submanifolds with bounded mean curvature vector, the second one is for self-similar shrinkers endowed with the natural Gaussian measure. The equality cases are characterized here as well., Comment: 41 pages

We consider the affine-additive group as a metric measure space with a canonical left-invariant measure and a left-invariant sub-Riemannian metric. We prove that this metric measure space is locally 4-Ahlfors regular and it is hyperbolic, meaning that it has a non-vanishing 4-capacity at infinity. This implies that the affine-additive group is not quasiconformally equivalent to the Heisenberg group or to the roto-translation group in contrast to the fact that both of these groups are globally contactomorphic to the affine-additive group. Moreover, each quasiregular map, from the Heisenberg group to the affine-additive group must be constant., Comment: 17 pages

This paper aims to study isometries of the $1$-Wasserstein space $\mathcal{W}_1(\mathbf{G})$ over Carnot groups endowed with horizontally strictly convex norms. Well-known examples of horizontally strictly convex norms on Carnot groups are the Heisenberg group $\mathbb{H}^n$ endowed with the Heisenberg-Kor\'anyi norm, or with the Naor-Lee norm; and $H$-type Iwasawa groups endowed with a Kor\'anyi-type norm. We prove that on a general Carnot group there always exists a horizontally strictly convex norm. The main result of the paper says that if $(\mathbf{G},N_{\mathbf{G}})$ is a Carnot group where $N_{\mathbf{G}}$ is a horizontally strictly convex norm on $\mathbf{G}$, then the Wasserstein space $\mathcal{W}_1(\mathbf{G})$ is isometrically rigid. That is, for every isometry $\Phi:\mathcal{W}_1(\mathbf{G})\to\mathcal{W}_1(\mathbf{G})$ there exists an isometry $\psi:\mathbf{G}\to \mathbf{G}$ such that $\Phi=\psi_{\#}$., Comment: 20 pages. arXiv admin note: text overlap with arXiv:2303.15095

Our purpose in this paper is to study isometries and isometric embeddings of the $p$-Wasserstein space $\mathcal{W}_p(\mathbb{H}^n)$ over the Heisenberg group $\mathbb{H}^n$ for all $p>1$ and for all $n\geq 1$. First, we create a link between optimal transport maps in the Euclidean space $\mathbb{R}^{2n}$ and the Heisenberg group $\mathbb{H}^n$. Then we use this link to understand isometric embeddings of $\mathbb{R}$ and $\mathbb{R}_+$ into $\mathcal{W}_p(\mathbb{H}^n)$ for $p>1$. That is, we characterize complete geodesics and geodesic rays in the Wasserstein space. Using these results we determine the metric rank of $\mathcal{W}_p(\mathbb{H}^n)$. Namely, we show that $\mathbb{R}^k$ can be embedded isometrically into $\mathcal{W}_p(\mathbb{H}^n)$ for $p>1$ if and only if $k\leq n$. As a consequence, we conclude that $\mathcal{W}_p(\mathbb{R}^k)$ and $\mathcal{W}_p(\mathbb{H}^k)$ can be embedded isometrically into $\mathcal{W}_p(\mathbb{H}^n)$ if and only if $k\leq n$. In the second part of the paper, we study the isometry group of $\mathcal{W}_p(\mathbb{H}^n)$ for $p>1$. We find that these spaces are all isometrically rigid meaning that for every isometry $\Phi:\mathcal{W}_p(\mathbb{H}^n)\to\mathcal{W}_p(\mathbb{H}^n)$ there exists a $\psi:\mathbb{H}^n\to\mathbb{H}^n$ such that $\Phi=\psi_{\#}$., Comment: 29 pages. v2: a part of this preprint discussing the isometric rigidity of the 1-Wasserstein space has been moved to the preprint "Isometric rigidity of the Wasserstein space $\mathcal{W}_1(\mathbb{G})$ over Carnot groups" by the same authors + minor improvements. v3: proof of Proposition 3.3 revised and streamlined

Given $p,N>1,$ we prove the sharp $L^p$-log-Sobolev inequality on noncompact metric measure spaces satisfying the ${\sf CD}(0,N)$ condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in ${\sf CD}(0,N)$ spaces, symmetrisation, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf-Lax semigroup in ${\sf CD}(0,N)$ spaces. The proof of this result uses Hamilton-Jacobi inequality and Sobolev regularity properties of the Hopf-Lax semigroup, which turn out to be essential in the present setting of nonsmooth and noncompact spaces. Furthermore, a sharp Gaussian-type $L^2$-log-Sobolev inequality is also obtained in ${\sf RCD}(0,N)$ spaces. Our results are new, even in the smooth setting of Riemannian/Finsler manifolds. In particular, an extension of the celebrated rigidity result of Ni (J. Geom. Anal., 2004) on Riemannian manifolds will be a simple consequence of our sharp log-Sobolev inequality., Comment: Published in J. Funct. Anal

We prove Gagliardo-Nirenberg inequalities with three weights -- verifying a joint concavity condition -- on open convex cones of $\mathbb R^n$. If the weights are equal to each other the inequalities become sharp and we compute explicitly the sharp constants. For a certain range of parameters we can characterize the class of extremal functions; in this case, we also show that the sharpness in the main three-weighted Gagliardo-Nirenberg inequality implies that the weights must be equal up to some constant multiplicative factors. Our approach uses optimal mass transport theory and a careful analysis of the joint concavity condition of the weights. As applications we establish sharp weighted $p$-log-Sobolev, Faber-Krahn and isoperimetric inequalities with explicit sharp constants., Comment: 34 pages

By using optimal mass transport theory, we provide a direct proof to the sharp $L^p$-log-Sobolev inequality $(p\geq 1)$ involving a log-concave homogeneous weight on an open convex cone $E\subseteq \mathbb R^n$. The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the $L^p$-log-Sobolev inequality. The characterization of the equality cases is new for $p\geq n$ even in the unweighted setting and $E=\mathbb R^n$. As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases., Comment: 36 pages; to appear in Transactions of the American Mathematical Society

By using optimal mass transport theory we prove a sharp isoperimetric inequality in ${\sf CD} (0,N)$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of $n$-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of S. Brendle (2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space., Comment: 23 pages; the paper has been shortened, changing also its title. To appear in Mathematische Annalen

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form \[\left(\int_E |u(x)|^q\,\omega(x) \,dx\right)^{1/q}\leq K_0\,\left(\int_E |\nabla u(x)|^p\,\sigma(x)\,dx\right)^{1/p},\ \ u\in C_0^\infty(\mathbb R^n),\ \ \ \ \ \ {\rm (WSI)}\] where $p\geq 1$ and $q>0$ is the corresponding Sobolev critical exponent. Here $E\subseteq \mathbb R^n$ is an open convex cone, and $\omega,\sigma:E\to (0,\infty)$ are two homogeneous weights verifying a general concavity-type structural condition. The constant $K_0= K_0(n, p, q, \omega, \sigma) >0$ is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that $K_0$ is optimal in (WSI) if and only if $\omega$ and $\sigma$ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to PDEs are also provided., Comment: 35 pages; some references are updated. To appear in the Proceedings of the London Mathematical Society

We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of hyperbolic $n$-space by a geometric argument. Moreover, we obtain a Besicovitch-Federer type characterization of purely unrectifiable sets in terms of these hyperbolic orthogonal projections., Comment: 6 pages, 2 figures

We consider Marstrand type projection theorems for closest-point projections in the normed space $\mathbb{R}^2$. We prove that if a norm on $\mathbb{R}^2$ is regular enough, then the analogues of the well-known statements from the Euclidean setting hold, while they fail for norms whose unit balls have corners. We establish our results by verifying Peres and Schlag's transversality property and thereby also obtain a Besicovitch-Federer type characterization of purely unrectifiable sets., Comment: 20 pages, 6 figures. To appear in Journal of Fractal Geometry

We study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitrary left-invariant homogeneous distance that is not necessarily sub-Riemannian. We show that if all infinite geodesics in the target are straight lines, then such an embedding must be a homogeneous homomorphism. We discuss a necessary and certain sufficient conditions for the target space to have this `geodesic linearity property', and we provide various examples., Comment: 29 pages

By using optimal mass transportation and a quantitative H\"older inequality, we provide estimates for the Borell-Brascamp-Lieb deficit on complete Riemannian manifolds. Accordingly, equality cases in Borell-Brascamp-Lieb inequalities (including Brunn-Minkowski and Pr\'ekopa-Leindler inequalities) are characterized in terms of the optimal transport map between suitable marginal probability measures. These results provide several qualitative applications both in the flat and non-flat frameworks. In particular, by using Caffarelli's regularity result for the Monge-Amp\`ere equation, we {give a new proof} of Dubuc's characterization of the equality in Borell-Brascamp-Lieb inequalities in the Euclidean setting. When the $n$-dimensional Riemannian manifold has Ricci curvature ${\rm Ric}(M)\geq (n-1)k$ for some $k\in \mathbb R$, it turns out that equality in the Borell-Brascamp-Lieb inequality is expected only when a particular region of the manifold between the marginal supports has constant sectional curvature $k$. A precise characterization is provided for the equality in the Lott-Sturm-Villani-type distorted Brunn-Minkowski inequality on Riemannian manifolds. Related results for (not necessarily reversible) Finsler manifolds are also presented., Comment: 28 pages (with 1 figure); to appear in Advances in Mathematics

We establish a weighted pointwise Jacobian determinant inequality on corank 1 Carnot groups related to optimal mass transportation akin to the work of Cordero-Erausquin, McCann and Schmuckenschl\"ager. In this setting, the presence of abnormal geodesics does not allow the application of the general sub-Riemannian optimal mass transportation theory developed by Figalli and Rifford and we need to work with a weaker notion of Jacobian determinant. Nevertheless, our result achieves a transition between Euclidean and sub-Riemannian structures, corresponding to the mass transportation along abnormal and strictly normal geodesics, respectively. The weights appearing in our expression are distortion coefficients that reflect the delicate sub-Riemannian structure of our space. As applications, entropy, Brunn-Minkowski and Borell-Brascamp-Lieb inequalities are established on Carnot groups., Comment: to appear in Journal of Functional Analysis

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $\mathbb H^n$. Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott-Villani and Sturm and also a geodesic version of the Borell-Brascamp-Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschl\"ager. The latter statement implies sub-Riemannian versions of the geodesic Pr\'ekopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of $\mathbb H^n$ developed by Ambrosio and Rigot. These results refute a general point of view, according to which no geometric inequalities can be derived by optimal mass transportation on singular spaces., Comment: to appear in Calculus of Variations and Partial Differential Equations (42 pages, 1 figure)

We construct a quasiconformal mapping of $n$-dimensional Euclidean space, $n \geq 2$, that simultaneously distorts the Hausdorff dimension of a nearly maximal collection of parallel lines by a given amount. This answers a question of Balogh, Monti, and Tyson., Comment: 12 pages

The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy Mean Value Theorem is taken at a point which has a well-determined position in the interval. As an application of this result, a partial answer is given to a question posed by Sahoo and Riedel., Comment: 14 pages, 3 figures

We apply the theory of Peres and Schlag to obtain estimates for generic Hausdorff dimension distortion under orthogonal projections on simply connected two dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain appropriate versions of Marstrand theorem, Kaufman's theorem and Falconer's theorem in the above geometrical settings., Comment: 11 pages. Version 2: corrected typos, added references, changed title, improved structure in Section 3. This corresponds to the published version

Let k>n be positive integers. We consider mappings from a subset of k-dimensional Euclidean space R^k to the Heisenberg group H^n with a variety of metric properties, each of which imply that the mapping in question satisfies some weak form of the contact equation arising from the sub-Riemannian structure of the Heisenberg group. We illustrate a new geometric technique that shows directly how the weak contact equation greatly restricts the behavior of the mappings. In particular, we provide a new and elementary proof of the fact that the Heisenberg group H^n is purely k-unrectifiable. We also prove that for an open set U in R^k, the rank of the weak derivative of a weakly contact mapping in the Sobolev space W^{1,1}_{loc}(U;R^{2n+1}) is bounded by $n$ almost everywhere, answering a question of Magnani. Finally we prove that if a mapping from U to H^n is s-H\"older continuous, s>1/2, and locally Lipschitz when considered as a mapping into R^{2n+1}, then the mapping cannot be injective. This result is related to a conjecture of Gromov., Comment: 28 pages

In this paper we solve a problem raised by Guti\'errez and Montanari about comparison principles for $H-$convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous $H-$convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples., Comment: Published in: J. Funct. Anal. 269 (2015), no. 9, 2669-2708

We study the behavior of Sobolev mappings defined on the Heisenberg groups with respect to a foliation by left cosets of a horizontal homogeneous subgroup. We quantitatively estimate, in terms of Euclidean Hausdorff dimension, the size of the set of cosets that are mapped onto sets of high dimension. The proof of our main result combines ideas of Gehring and Mostow about the absolute continuity of quasiconformal mappings with Mattila's projection and slicing machinery., Comment: 22 pages, 3 figures

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincar\'e inequality. For foliations of a metric space X defined by a David--Semmes regular mapping we quantitatively estimate, in terms of Hausdorff dimension, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups. Examples are given demonstrating the extent to which our results are sharp. In fact, we show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space; the latter result is new even in Euclidean space., Comment: Some proofs simplified and typos removed from earlier versions