Volume 12, Issue 1

A natural higher-order notion of C 1 , α {C}^{1,\alpha } -rectifiability, 0 < α ≤ 1 0\lt \alpha \le 1 , is introduced for subsets of the Heisenberg groups H n {{\mathbb{H}}}^{n} in terms of covering a set almost everywhere with a countable union of ( C H 1 , α , H ) \left({{\bf{C}}}_{H}^{1,\alpha },{\mathbb{H}}) -regular surfaces. Using this, we prove a geometric characterization of C 1 , α {C}^{1,\alpha } -rectifiable sets of low codimension in Heisenberg groups H n {{\mathbb{H}}}^{n} in terms of an almost everywhere existence of suitable approximate tangent paraboloids.

A Λ \Lambda -tree is a Λ \Lambda -metric space satisfying three axioms (1), (2), and (3). We give a characterization of those ordered abelian groups Λ \Lambda for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for the important class of ordered abelian groups Λ \Lambda that satisfy Λ = 2 Λ \Lambda =2\Lambda , (3) follows from (1) and (2). For some ordered abelian groups Λ \Lambda , we show that axiom (2) is independent of axioms (1) and (3) and ask whether this holds for all ordered abelian groups. Part of this work has been formalized in the proof assistant Lean {\mathsf{Lean}} .

Given a homeomorphism f : X → Y f:X\to Y between Q Q -dimensional spaces X , Y X,Y , we show that f f satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that f f belongs to the Sobolev class N loc 1 , p ( X ; Y ) {N}_{{\rm{loc}}}^{1,p}\left(X;\hspace{0.33em}Y) , where 1 ≤ p ≤ Q 1\le p\le Q , and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors Q Q -regularity, which in particular enables various weighted spaces to be included in the theory. Notably, even in the classical Euclidean setting, we are able to obtain new results using this approach. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity f ∈ N loc 1 , Q ( X ; Y ) f\in {N}_{{\rm{loc}}}^{1,Q}\left(X;\hspace{0.33em}Y) without the strong assumption of the infinitesimal distortion h f {h}_{f} belonging to L ∞ ( X ) {L}^{\infty }\left(X) .

We give a characterization of metric space-valued Sobolev maps in terms of weak* derivatives. More precisely, we show that Sobolev maps with values in dual-to-separable Banach spaces can be defined in terms of classical weak derivatives in a weak* sense. Since every separable metric space X X embeds isometrically into ℓ ∞ {\ell }^{\infty } , we conclude that Sobolev maps with values in X X can be characterized by postcomposition with such embedding and the mentioned weak gradients. A slight variation on our definition was proposed previously by Hajłasz and Tyson. However, we show that their definition does not work in the sense that for technical reasons the arising Sobolev space is essentially empty.

We completely describe the Gromov-Hausdorff closure of the class of length spaces being homeomorphic to a fixed closed surface.

In this study, we characterize the geodesic dimension N GEO {N}_{{\rm{GEO}}} and give a new lower bound to the curvature exponent N CE {N}_{{\rm{CE}}} on Sard-regular Carnot groups. As an application, we give an example of step-two Carnot group on which N CE > N GEO {N}_{{\rm{CE}}}\gt {N}_{{\rm{GEO}}} ; this answers a question posed by Rizzi ( Measure contraction properties of Carnot groups . Calc. Var. Partial Differential Equations 55 (2016), no. 3, Art. 60, 20).

We prove that any Lipschitz map that satisfies a condition inspired by the work of G. David may be decomposed into countably many bi-Lipschitz pieces.

Given a metrizable space X X , let A M ( X ) AM\left(X) be the space of continuous bounded admissible metrics on X X , which is endowed with the sup-metric. In this article, we shall investigate the Borel complexity and the complete metrizability of A M ( X ) AM\left(X) and show that a separable metrizable space X X is σ \sigma -compact if and only if A M ( X ) AM\left(X) is completely metrizable.

In the realm of sub-Riemannian manifolds, a relevant question is: what are the metric lines (isometric embedding of the real line)? The space of k k -jets of a real function of one real variable x x , denoted by J k ( R , R ) {J}^{k}\left({\mathbb{R}},{\mathbb{R}}) , admits the structure of a Carnot group. Every Carnot group is sub-Riemannian manifold, so is J k ( R , R ) {J}^{k}\left({\mathbb{R}},{\mathbb{R}}) . This study aims to present a partial result about the classification of the metric lines within J k ( R , R ) {J}^{k}\left({\mathbb{R}},{\mathbb{R}}) . The method is to use an intermediate three-dimensional sub-Riemannian space R F 3 {{\mathbb{R}}}_{F}^{3} lying between the group J k ( R , R ) {J}^{k}\left({\mathbb{R}},{\mathbb{R}}) and the Euclidean space R 2 {{\mathbb{R}}}^{2} .

We provide sufficient conditions as to when a boundary component of a cocompact convex set in a CAT ( 0 ) {\rm{CAT}}\left(0) -space is contractible. We then use this to study when the limit set of a quasi-convex, codimension one subgroup of a negatively curved manifold group is “wild” in the boundary. The proof is based on a notion of coarse upper curvature bounds in terms of barycenters and the careful study of interpolation in geodesic metric spaces.

In this mostly expository article, we give streamlined proofs of several well-known Lipschitz extension theorems. We pay special attention to obtaining statements with explicit expressions for the extension constants. One of our main results is an explicit version of a very general Lipschitz extension theorem of Lang and Schlichenmaier. A special case of the theorem reads as follows. We prove that if X X is a metric space and A ⊂ X A\subset X satisfies the condition Nagata ( n , c ) \hspace{0.1em}\text{Nagata}\hspace{0.1em}\left(n,c) , then any 1-Lipschitz map f : A → Y f:A\to Y to a Banach space Y Y admits a Lipschitz extension F : X → Y F:X\to Y whose Lipschitz constant is at most 1,000 ⋅ ( c + 1 ) ⋅ log 2 ( n + 2 ) \hspace{0.1em}\text{1,000}\hspace{0.1em}\cdot \left(c+1)\cdot {\log }_{2}\left(n+2) . By specifying to doubling metric spaces, this recovers an extension result of Lee and Naor. We also revisit another theorem of Lee and Naor by showing that if A ⊂ X A\subset X consists of n n points, then Lipschitz extensions as above exist with a Lipschitz constant of at most 600 ⋅ log n ⋅ ( log log n ) − 1 600\cdot \log n\cdot {\left(\log \log n)}^{-1} .

We derive several properties of the heat equation with the Hodge operator associated with the Rumin’s complex on Heisenberg groups and prove several properties of the fundamental solution. As an application, we use the heat kernel for Rumin’s differential forms to construct a Calderón reproducing formula on Rumin’s forms.

This article is devoted to the study of a critical Choquard-Kirchhoff p p -sub-Laplacian equation on the entire Heisenberg group H n {{\mathbb{H}}}^{n} , where the Kirchhoff function K K can be zero at zero, i.e., the equation can be degenerate, and involving a nonlinearity, which is critical in the sense of the Hardy-Littlewood-Sobolev inequality. We first establish the concentration-compactness principle for the p p -sub-Laplacian Choquard equation on the Heisenberg group, and we then prove existence results.

Our review of Liouville theorems includes a special focus on nonlinear partial differential equations and inequalities.

We consider degenerate Kolmogorov-Fokker-Planck operators ℒ u = ∑ i , j = 1 q a i j ( x , t ) u x i x j + ∑ k , j = 1 N b j k x k u x j − u t , {\mathcal{ {\mathcal L} }}u=\mathop{\sum }\limits_{i,j=1}^{q}{a}_{ij}\left(x,t){u}_{{x}_{i}{x}_{j}}+\mathop{\sum }\limits_{k,j=1}^{N}{b}_{jk}{x}_{k}{u}_{{x}_{j}}-{u}_{t}, such that the corresponding model operator having constant a i j {a}_{ij} is hypoelliptic, translation invariant w.r.t. a Lie group operation in R N + 1 {{\mathbb{R}}}^{N+1} and 2-homogeneous w.r.t. a family of nonisotropic dilations. We assume that the a i j {a}_{ij} ’s are globally bounded and Hölder continuous in x x (w.r.t. some intrinsic distance induced by ℒ {\mathcal{ {\mathcal L} }} ); the matrix { a i j } i , j = 1 q {\{{a}_{ij}\}}_{i,j=1}^{q} is symmetric and uniformly positive on R q {{\mathbb{R}}}^{q} . We prove “partial Schauder a priori estimates” on a bounded open set Ω ⊂ R N + 1 \Omega \subset {{\mathbb{R}}}^{N+1} , of the kind ∑ i , j = 1 q ( [ u x i x j ] ∗ , α , 2 x + [ u x i x j ] ∗ , 2 ) ≤ c { [ ℒ u ] ∗ , α , 2 x + [ ℒ u ] ∗ , 2 + sup Ω ∣ u ∣ } \mathop{\sum }\limits_{i,j=1}^{q}({\left[{u}_{{x}_{i}{x}_{j}}]}_{\ast ,\alpha ,2}^{x}+{\left[{u}_{{x}_{i}{x}_{j}}]}_{\ast ,2})\le c\left\{{\left[{\mathcal{ {\mathcal L} }}u]}_{\ast ,\alpha ,2}^{x}+{\left[{\mathcal{ {\mathcal L} }}u]}_{\ast ,2}+\mathop{\sup }\limits_{\Omega }| u| \right\} for suitable functions u u , where [ u ] ∗ , α , 2 x = sup ( x , t ) ≠ ( y , t ) ∈ Ω d ( x , t ) , ( y , t ) 2 + α ∣ u ( x , t ) − u ( y , t ) ∣ ‖ x − y ‖ α [ u ] ∗ , 2 = sup ξ ∈ Ω d ξ 2 ∣ u ( ξ ) ∣ . \begin{array}{rcl}{\left[u]}_{\ast ,\alpha ,2}^{x}& =& \mathop{\sup }\limits_{\left(x,t)\ne (y,t)\in \Omega }{d}_{\left(x,t),(y,t)}^{2+\alpha }\frac{| u\left(x,t)-u(y,t)| }{\Vert x-y{\Vert }^{\alpha }}\\ {{[}u]}_{\ast ,2}& =& \mathop{\sup }\limits_{\xi \in \Omega }{d}_{\xi }^{2}| u\left(\xi )| .\end{array} Here ‖ ⋅ ‖ \Vert \cdot \Vert is a homogeneous norm in R N {{\mathbb{R}}}^{N} , while d ξ = dist ( ξ , ∂ Ω ) {d}_{\xi }={\rm{dist}}\left(\xi ,\partial \Omega ) and d ξ , η = min { d ξ , d η } {d}_{\xi ,\eta }=\min \left\{{d}_{\xi },{d}_{\eta }\right\} . We also prove that the derivatives u x i x j {u}_{{x}_{i}{x}_{j}} are locally Hölder continuous in space and time.

We present two Liouville-type results for solutions to anisotropic elliptic equations that have a growth of power 2 along the first s s coordinate directions and of power p p , with 1 < p < 2 1\lt p\lt 2 along the other ( N − s ) \left(N-s) ones. First, we begin our investigation by assuming that the solution is bounded only from below, deriving a rigidity result for the range p + ( N − s ) ( p − 2 ) > 0 p+\left(N-s)\left(p-2)\gt 0 of non-degeneration, which is a purely parabolic shade. Then we break free from this constraint at the price of assuming the solution to be bounded also from above.

In this article, we review some recent results about the role of embeddability in conformal CR (Cauchy-Riemann) geometry. We will show how this condition enters in the second variation of the pseudo-hermitian counterpart of the Einstein-Hilbert action, in the positivity of the pseudohermitian mass and in the extremality properties of the CR (Cauchy-Riemann) Yamabe quotient. Given also some explicit examples at hand, this aspect shows sharp differences compared to the Riemannian case.

We study the gradient flow of an energy with mixed homogeneity, which is at the interface of Finsler and sub-Riemannian geometry.

It is known that for a possibly degenerate hypoelliptic Ornstein-Uhlenbeck (OU) operator L = 1 2 tr ( Q D 2 ) + ⟨ A x , D ⟩ = 1 2 div ( Q D ) + ⟨ A x , D ⟩ , x ∈ R N , L=\frac{1}{2}\hspace{0.1em}\text{tr}\hspace{0.1em}\left(Q{D}^{2})+\langle Ax,D\rangle =\frac{1}{2}\hspace{0.1em}\text{div}\hspace{0.1em}\left(QD)+\langle Ax,D\rangle ,\hspace{0.33em}\hspace{0.33em}x\in {{\mathbb{R}}}^{N}, all (globally) bounded solutions of L u = 0 Lu=0 on R N {{\mathbb{R}}}^{N} are constant if and only if all the eigenvalues of A A have non-positive real parts (i.e., s ( A ) ≤ 0 s\left(A)\le 0 ). We show that if Q Q is positive definite and s ( A ) ≤ 0 s\left(A)\le 0 , then any non-negative solution v v of L v = 0 Lv=0 on R N {{\mathbb{R}}}^{N} , which has at most an exponential growth, is indeed constant. Thus, under a non-degeneracy condition, we relax the boundedness assumption on the harmonic functions and maintain the sharp condition on the eigenvalues of A A . We also prove a related one-side Liouville theorem in the case of hypoelliptic OU operators.