Volume 13, Issue 1

In this article, we study quasiconformal curves which are a special case of quasiregular curves. Namely embeddings Ω → R m \Omega \to {{\mathbb{R}}}^{m} from some domain Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} to R m {{\mathbb{R}}}^{m} , where n ≤ m n\le m , belong in a suitable Sobolev class and satisfy a certain distortion inequality for some smooth, closed and non-vanishing n n -form in R m {{\mathbb{R}}}^{m} . These mappings can be seen as quasiconformal mappings between Ω \Omega and f ( Ω ) f\left(\Omega ) . We prove that a quasiconformal curve always satisfies the analytic definition of quasiconformal mappings and the lower half of the modulus inequality. Moreover, we give a sufficient condition for a quasiconformal curve to satisfy the metric definition of quasiconformal mappings. We also show that a quasiconformal map from Ω \Omega to f ( Ω ) ⊂ R m f\left(\Omega )\subset {{\mathbb{R}}}^{m} is a quasiconformal ω \omega curve for some form ω \omega under suitable assumptions. Finally, we show that same is true when we equip the target space f ( Ω ) f\left(\Omega ) with its intrinsic metric instead of the Euclidean one.

We study Bloom-type two-weight inequalities for commutators of the Hardy-Littlewood maximal function and sharp maximal function. Some necessary and sufficient conditions are given to characterize the two-weight inequalities for such commutators.

In this article, we revise Monti’s results on blow-ups of H-perimeter minimizing sets in H n {{\mathbb{H}}}^{n} . Monti demonstrated that the Lipschitz approximation of the blow-up, after rescaling by the square root of the excess, converges to a limit function for n ≥ 2 n\ge 2 . However, the partial differential equation he derived for this limit function φ \varphi through contact variation is incorrect. Instead, the limit function solves the following equation weakly ∂ ∂ y 1 Δ 0 φ = 0 , \frac{\partial }{\partial {y}_{1}}{\Delta }_{0}\varphi =0,

When dealing with certain mathematical problems, it is sometimes necessary to show that some function induces a metric on a certain space. When this function is not a well renowned example of a distance, one has to develop very particular arguments that appeal to the concrete expression of the function in order to do so. The main purpose of this work is to provide several sufficient results ensuring that a function of two variables induces a distance on the real line, as well as some necessary conditions, together with several examples that show the applicability of these results. In particular, we show how a hypothesis about the sign of the cross partial derivative of the candidate to distance is helpful for deriving such kind of results.

In this article, using the existence of infinite equidistant subsets of closed balls, we first characterize the injectivity of ultrametric spaces for finite ultrametric spaces. This method also gives characterizations of the Urysohn universal ultrametric spaces. As an application, we find that the operations of the Cartesian product and the hyperspaces preserve the structures of the Urysohn universal ultrametric spaces. Namely, let ( X , d ) \left(X,d) be the Urysohn universal ultrametric space. Then, we show that ( X × X , d × d ) \left(X\times X,d\times d) is isometric to ( X , d ) \left(X,d) , and show that the hyperspace consisting of all non-empty compact subsets of ( X , d ) \left(X,d) and symmetric products of ( X , d ) \left(X,d) are isometric to ( X , d ) \left(X,d) . We also establish that every complete ultrametric space injective for finite ultrametric space contains a subspace isometric to ( X , d ) \left(X,d) .

We consider a nilpotent Lie group with a bracket-generating distribution ℋ {\mathcal{ {\mathcal H} }} and an asymmetric left-invariant norm ∣ ⋅ ∣ K {| \cdot | }_{K} induced by a convex body K ⊆ R k K\subseteq {{\mathbb{R}}}^{k} containing 0 in its interior. In this study, we prove the existence of minimizers of the perimeter functional P K {P}_{K} associated with ∣ ⋅ ∣ K {| \cdot | }_{K} under a volume (Haar measure) constraint.

In this article, we study the convergence rate of the following Yamabe-type flow R ϕ ( t ) m = 0 in M and ∂ ∂ t g ( t ) = 2 ( h ϕ ( t ) m − H ϕ ( t ) m ) g ( t ) ∂ ∂ t ϕ ( t ) = m ( H ϕ ( t ) m − h ϕ ( t ) m ) on ∂ M {R}_{\phi \left(t)}^{m}=0\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}M\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}\left\{\begin{array}{l}\frac{\partial }{\partial t}g\left(t)=2\left({h}_{\phi \left(t)}^{m}-{H}_{\phi \left(t)}^{m})g\left(t)\\ \frac{\partial }{\partial t}\phi \left(t)=m\left({H}_{\phi \left(t)}^{m}-{h}_{\phi \left(t)}^{m})\end{array}\right.\hspace{0.33em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial M on a smooth metric measure space with boundary ( M , g ( t ) , e − ϕ ( t ) d V g ( t ) , e − ϕ ( t ) d A g ( t ) , m ) \left(M,g\left(t),{e}^{-\phi \left(t)}{\rm{d}}{V}_{g\left(t)},{e}^{-\phi \left(t)}{\rm{d}}{A}_{g\left(t)},m) , where R ϕ ( t ) m {R}_{\phi \left(t)}^{m} is the weighted scalar curvature, H ϕ ( t ) m {H}_{\phi \left(t)}^{m} is the weighted mean curvature, and h ϕ ( t ) m {h}_{\phi \left(t)}^{m} is the average of the weighted mean curvature.

We consider generic rank two distributions on five-dimensional nilmanifolds and show that the analytic torsion of their Rumin complex coincides with the Ray-Singer torsion.

In this article, we study Liouville type theorems of fully nonlinear elliptic partial differential equations on the Heisenberg group and obtain some nonexistence results of positive solutions of Heisenberg Hessian equations (and inequalities), including Heisenberg Hessian quotient equations (and inequalities) and mixed Heisenberg Hessian quotient equations (and inequalities). The proofs are based on integration by parts and the Newton-Maclaurin inequality.

We show that infinite cyclic subgroups of groups acting uniformly properly on injective metric spaces are uniformly undistorted. In the special case of hierarchically hyperbolic groups, we use this to study translation lengths for actions on the associated hyperbolic spaces. We then use quasimorphisms to produce examples where these latter results are sharp.

We establish a version of the Cauchy integral formula for holomorphic functions on a polidisc, with values in a complex Fréchet space X {\mathfrak{X}} . We prove a vector-valued analog to Weierstrass’s theorem on sequences of holomorphic functions f ν ∈ O ( Ω , X ) {f}_{\nu }\in {\mathcal{O}}\left(\Omega ,{\mathfrak{X}}) converging uniformly on compact subsets of Ω ⊂ C n \Omega \subset {{\mathbb{C}}}^{n} . We obtain a Cauchy-Kovalevskaja-type theorem, i.e., prove existence of X {\mathfrak{X}} -valued C ω {C}^{\omega } solutions to the Cauchy problem P ( x , D ) u = f P\left(x,D)u=f on U ⊂ Ω U\subset \Omega and ( ∂ j u / ∂ x n j ) ∣ x n = 0 = φ j ({\partial }^{j}u/\partial {x}_{n}^{j}){| }_{{x}_{n}=0}={\varphi }_{j} on U 0 = U ∩ { x n = 0 } {U}_{0}=U\cap \{{x}_{n}=0\} , j ∈ { 0 , 1 , … , m − 1 } j\in \left\{0,1,\ldots ,m-1\right\} , where P ( x , D ) ≡ ∑ ∣ α ∣ ≤ m a α ( x ) D α P\left(x,D)\equiv {\sum }_{| \alpha | \le m}{a}_{\alpha }\left(x){D}^{\alpha } and a α ∈ C ω ( Ω ) {a}_{\alpha }\in {C}^{\omega }\left(\Omega ) , f ∈ C ω ( Ω , X ) f\in {C}^{\omega }\left(\Omega ,{\mathfrak{X}}) and φ j ∈ C ω ( Ω 0 , X ) {\varphi }_{j}\in {C}^{\omega }\left({\Omega }_{0},{\mathfrak{X}}) , with 0 ∈ Ω ⊂ R n 0\in \Omega \subset {{\mathbb{R}}}^{n} open and Ω 0 = Ω ∩ { x n = 0 } {\Omega }_{0}=\Omega \cap \left\{{x}_{n}=0\right\} . The existence of an open set 0 ∈ U ⊂ Ω 0\in U\subset \Omega and of a solution u ∈ C ω ( U , X ) u\in {C}^{\omega }\left(U,{\mathfrak{X}}) to the Cauchy problem requires the structural condition a ( 0 , … , 0 , m ) ( 0 ) ≠ 0 {a}_{\left(0,\ldots ,0,m)}\left(0)\ne 0 and relies, as well as in the classical case of scalar valued solutions, on the Cauchy integral formula and on Weierstrass’ theorem for X {\mathfrak{X}} -valued holomorphic functions.

The cost functions considered are c ( x , y ) = h ( x − y ) c\left(x,y)=h\left(x-y) , where h ∈ C 2 ( R n ) h\in {C}^{2}\left({{\mathbb{R}}}^{n}) is homogeneous of degree p ≥ 2 p\ge 2 with a positive definite Hessian in the unit sphere. We study multivalued monotone maps with respect to that cost and establish that they are single-valued almost everywhere. Further consequences are then deduced.