In the Euclidean setting the celebrated Aleksandrov-Busemann-Feller theorem states that convex functions are a.e. twice differentiable. In this paper we prove that a similar result holds in the Heisenberg group, by showing that every continuous H-convex function belongs to the class of functions whose second order horizontal distributional derivatives are Radon measures. Together with a recent result by Ambrosio and Magnani, this proves the existence a.e. of second order horizontal derivatives for the class of continuous H-convex functions in the Heisenberg group.
This paper introduces in a natural way a notion of horizontal convex envelopes of continuous functions in the Heisenberg group. We provide a convexification process to find the envelope in a constructive manner. We also apply the convexification process to show h-convexity of viscosity solutions to a class of fully nonlinear elliptic equations in the Heisenberg group satisfying a certain symmetry condition. Our examples show that in general one cannot expect h-convexity of solutions without the symmetry condition.
We prove a Hardy-type inequality for the gradient of the Heisenberg Laplacian on open bounded convex polytopes on the first Heisenberg Group. The integral weight of the Hardy inequality is given by the distance function to the boundary measured with respect to the Carnot-Carathéodory metric. The constant depends on the number of hyperplanes, given by the boundary of the convex polytope, which are not orthogonal to the hyperplane $x_3=0$.
In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexamples show that in general such properties that are well-known for semilinear and fully nonlinear parabolic equations in the Euclidean spaces do not hold in the Heisenberg group.
Geometric Hardy inequalities on the Heisenberg groups via convexity
National and Kapodistrian University of Athens（雅典国立卡波迪斯特里安大学）；Ghent University（根特大学）；University of Crete（克里特大学）；Institute of Applied and Computational Mathematics, FORTH（FORTH应用计算数学研究所）
We present a proof of the local Hölder regularity of the horizontal derivatives of weak solutions to the $p$-Laplace equation in the Heisenberg group $\mathbb{H}^1$ for $p>4 $.
In this paper we solve a problem raised by Gutiérrez 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 …
We prove geometric $L^p$ versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains $Ω$ in the Heisenberg group $\mathbb{H}^n$, where convex is meant in the Euclidean sense. When $p=2$ and $Ω$ is the half-space given by $\langle ξ, ν\rangle > d$ this generalizes an inequality previously obtained by Luan and Yang. For such $p$ and $Ω$ the inequality is sharp and takes the form \begin{equation} \int_Ω|\nabla_{\mathbb{H}^n}u|^2 \, dξ\geq \frac{1}{4}\int_Ω \sum_{i=1}^n\frac{\langle X_i(ξ), ν\rangle^2+\langle Y_i(ξ), ν\rangle^2}{\textrm{dist}(ξ, \partial Ω)^2}|u|^2\, dξ, \end{e…
In this paper we prove that every collection of measurable functions $f_α$, $|α|=m$ coincides a.e. with $m$th order derivatives of a function $g\in C^{m-1}$ whose derivatives of order $m-1$ may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.
We consider a one-Laplace equation perturbed by $p$-Laplacian with $1<p<\infty$. We prove that a weak solution is continuously differentiable ($C^{1}$) if it is convex. Note that similar result fails to hold for the unperturbed one-Laplace equation. The main difficulty is to show $C^{1}$-regularity of the solution at the boundary of a facet where the gradient of the solution vanishes. For this purpose we blow-up the solution and prove that its limit is a constant function by establishing a Liouville-type result, which is proved by showing a strong maximum principle. Our argument is rather elem…
This paper is devoted to the derivation of $L^2$ - $L^2$ decay estimates for the solution of the homogeneous linear damped wave equation on the Heisenberg group $\mathbf{H}_n$, for its time derivative and for its horizontal gradient. Moreover, we consider the improvement of these estimates when further $L^1(\mathbf{H}_n)$ regularity is required for the Cauchy data. Our approach will rely strongly on the group Fourier transform of $\mathbf{H}_n$ and on the properties of the Hermite functions that form a maximal orthonormal system for $L^2(\mathbb{R}^n)$ of eigenfunctions of the harmonic oscilla…
A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on Sobolev spaces and the loss of derivatives may be controled by the order of the operator. Although a large number of works have been devoted in the past to the construction and the study of algebras of variable-coefficient operators, including some very interesting works on the Heisenberg group, our approach is different, and in particular puts into light microloc…
This paper has been withdrawn by the authors. A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on Sobolev spaces and the loss of derivatives may be controled by the order of the operator. Although a large number of works have been devoted in the past to the construction and the study of algebras of variable-coefficient operators, including some very interesting works on the Heisenberg group, our approach is differe…
We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.
This paper is concerned with a PDE-based approach to the horizontally quasiconvex (h-quasiconvex for short) envelope of a given continuous function in the Heisenberg group. We provide a characterization for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to a first-order nonlocal Hamilton-Jacobi equation. We also construct the corresponding envelope of a continuous function by iterating the nonlocal operator. One important step in our arguments is to prove the uniqueness and existence of viscosity solutions to the Dirichlet boundary problems for the nonlocal…
We show that for any uniformly elliptic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term one can find an approximating equation which has a unique continuous and having the second derivatives locally bounded solution in a given smooth domain with smooth boundary data. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its derivatives.
We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in the whole space or in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a continuous solution with the first and the second spatial derivatives under control: bounded in the case of the whole space and locally bounded in case of equations in cylinders. The approximating equation is constructed in such a way that it modifies the original one only for large values of the second spatial derivatives o…
We prove the optimal $W^{2,\infty}$ regularity for variational problems with convex gradient constraints. We do not assume any regularity of the constraints; so the constraints can be nonsmooth, and they need not be strictly convex. When the domain is smooth enough, we show that the optimal regularity holds up to the boundary. In this process, we also characterize the set of singular points of the viscosity solutions to some Hamilton-Jacobi equations. Furthermore, we obtain an explicit formula for the second derivative of these viscosity solutions; and we show that the second derivatives satis…
We study various notions of the Schwarzian derivative for contact mappings in the Heisenberg group $\mathbb{H}_1$ and introduce two definitions: (1) the CR Schwarzian derivative based on the conformal connection approach studied by Osgood and Stowe and, recently, by Son; (2) the classical type Schwarzian refering to the well-known complex analytic definition. In particular, we take into consideration the effect of conformal rigidity and the limitations it imposes. Moreover, we study the kernels of both Schwarzians and the cocycle conditions. Our auxiliary results include a characterization of …
We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group $\mathbb{H}_1$. Several auxiliary properties of quasiconformal mappings between subdomains of $\mathbb{H}_1$ are proven, including distortion of balls estimates and local BMO-estimates for the logarithm of the Jacobian of a quasiconformal mapping. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of integrals of the average derivative and the operator norm of the horizontal differential, as well as the study o…
In this paper we present $L^2$ and $L^p$ versions of the geometric Hardy inequalities in half-spaces and convex domains on stratified (Lie) groups. As a consequence, we obtain the geometric uncertainty principles. We give examples of the obtained results for the Heisenberg and the Engel groups.
We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold $M$ with a distribution $H\subseteq TM$ analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this paper is to provide insight in the intrinsic geometry that underlies these coordinate formulas. First, we introduce `parabolic arrows' as a generalization of tangent vectors. The definition of parabolic arrows involves a mix of first and second order derivatives. Parabolic arrows …
