Stratified Lie Groups Research Articles

On subelliptic equations on stratified Lie groups driven by singular nonlinearity and weak L 1 data

On subelliptic equations on stratified Lie groups driven by singular nonlinearity and weak L 1 data

Results of existence and uniqueness for the Cauchy problem of semilinear heat equations on stratified Lie groups

The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups G with the homogeneous dimension N. We consider the nonlinear function behaves like |u|α or |u|α−1u(α>1) and the initial data u0 belongs to the Sobolev spaces Lsp(G) for 1<p<∞ and 0<s<N/p. Since stratified Lie groups G include the Euclidean space Rn as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on Rn to G. It should be noted that our proof is very different from it given by Ribaud on Rn. We adopt the generalized fractional chain rule on G to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on Rn. By using the generalized fractional chain rule on G, we can avoid the discussion of Fourier analysis on G and make the proof more simple.

Schatten properties of Calderón–Zygmund singular integral commutator on stratified Lie groups

We provide full characterization of the Schatten properties of [Mb,T], the commutator of Calderón–Zygmund singular integral T with symbol b(Mbf(x):=b(x)f(x)) on stratified Lie groups G. We show that, when p is larger than the homogeneous dimension Q of G, the Schatten Lp norm of the commutator is equivalent to the Besov semi-norm BpQ/p of the function b; but when p≤Q, the commutator belongs to Lp if and only if b is a constant. For the endpoint case at the critical index p=Q, we further show that the Schatten LQ,∞ norm of the commutator is equivalent to the Sobolev norm W˙1,Q of b. Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively, and hence can be applied to various settings beyond.

A multiphase eigenvalue problem on a stratified Lie group

AbstractWe consider a multiphase spectral problem on a stratified Lie group. We prove the existence of an eigenfunction of (2, q)-eigenvalue problem on a bounded domain. Furthermore, we also establish a Pohozaev-like identity corresponding to the problem on the Heisenberg group.

Gradient estimates for fundamental solutions of a Schrödinger operator on stratified Lie groups

Let L = − Δ G + Υ \mathcal L=-\Delta _{\mathbb {G}}+\Upsilon be a Schrödinger operator with a nonnegative potential Υ \Upsilon belonging to the reverse Hölder class B Q / 2 B_{Q/2} , where Q Q is the homogeneous dimension of the stratified Lie group G \mathbb {G} . Inspired by Shen’s pioneer work and Li’s work, we study fundamental solutions of the Schrödinger operator L \mathcal L on the stratified Lie group G \mathbb {G} in this paper. By proving an exponential decreasing variant of mean value inequality, we obtain the exponential decreasing upper estimates, the local Hölder estimates and the gradient estimates of the fundamental solutions of the Schrödinger operator L \mathcal L on the stratified Lie group. As two applications, we obtain the De Giorgi-Nash-Moser theory on the improved Hölder estimate for the weak solutions of the Schrödinger equation and a Liouville-type lemma for L \mathcal {L} -harmonic functions on G \mathbb {G} .

A study of a critical hypoelliptic problem in a stratified Lie group

In this paper, the existence of at least one solution to the problem (P) { − L M u = λf ( u ) e α | u | M ′ + μ | u | p − 2 u in Ω u = 0 on ∂Ω is established, where L M u := div G ⋅ ( | ∇ G u | M − 2 ∇ G u ) is the M-sub-Laplacian, 2 ≤ p < M , }0 $ ]]> α > 0 , M ′ = M M − 1 , Ω is a bounded subset of the N-dimensional stratified Lie group G with N ≥ 3 (N = 1, 2 has been avoided since they do not produce a purely stratified Lie group) and M is the homogeneous dimension of G . We will also show that in fact, the problem admits infinitely many solutions with some restrictions on the parameters involved.

Existence and Uniqueness for the Cauchy Problem of Semilinear Heat Equations on Stratified Lie Groups in the Critical Sobolev Space

Existence and Uniqueness for the Cauchy Problem of Semilinear Heat Equations on Stratified Lie Groups in the Critical Sobolev Space

Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

&lt;p style='text-indent:20px;'&gt;We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups.&lt;/p&gt;

Fujita exponent on stratified Lie groups

Fujita exponent on stratified Lie groups

Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups

Along the line of singular value estimates for commutators by Rochberg–Semmes, Lord–McDonald–Sukochev–Zanin and Fan–Lacey–Li, we establish the endpoint weak Schatten class estimate for commutators of Riesz transforms with multiplication operator Mf on Heisenberg groups via homogeneous Sobolev norm of the symbol f. The new tool we exploit is the construction of a singular trace formula on Heisenberg groups, which, together with the use of double operator integrals, allows us to bypass the use of Fourier analysis and provides a solid foundation to investigate the singular values estimates for similar commutators in general stratified Lie groups.

Some estimates for commutators of the fractional maximal function on stratified Lie groups

In this paper, the main aim is to consider the boundedness of the nonlinear commutator [b, M_{alpha}] and the maximal commutator M_{alpha ,b} on the Lebesgue spaces over some stratified Lie group mathbb{G} when the symbol b belongs to the Lipschitz space. As a result, some new characterizations of the Lipschitz spaces on Lie group via [b, M_{alpha}] and M_{alpha ,b} are given.

Functional Inequalities in Stratified Lie Groups with Sobolev, Besov, Lorentz and Morrey Spaces

Functional Inequalities in Stratified Lie Groups with Sobolev, Besov, Lorentz and Morrey Spaces

Boundary behavior of positive solutions of the heat equation on a stratified Lie group

In this article, we are concerned with a certain type of boundary behavior of positive solutions of the heat equation on a stratified Lie group at a given boundary point. We prove that a necessary and sufficient condition for the existence of the parabolic limit of a positive solution u at a point on the boundary is the existence of the strong derivative of the boundary measure of u at that point. Moreover, the parabolic limit and the strong derivative are equal. We also construct an example of a positive measure on the Heisenberg group to show that the set of all points where strong derivative exists is strictly larger than the set of Lebesgue points of the measure.

Spectral estimates and asymptotics for stratified Lie groups

We study Cwikel-type estimates for the singular values and Schatten Lp-norms of compositions of multiplication and convolution operators acting on stratified Lie groups. This enables us to obtain novel spectral asymptotic formulas for certain operators derived from sub-Laplacians.

Jet spaces over Carnot groups

Jet spaces over \mathbb{R}^n have been shown to have a canonical structure of stratified Lie groups (also known as Carnot groups). We construct jet spaces over stratified Lie groups adapted to horizontal differentiation and show that these jet spaces are themselves stratified Lie groups. Furthermore, we show that these jet spaces support a prolongation theory for contact maps, and in particular, a Bäcklund type theorem holds. A byproduct of these results is an embedding theorem that shows that every stratified Lie group of step s + 1 can be embedded in a jet space over a stratified Lie group of step s .

Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups

The purpose of this paper is twofold: first we study an eigenvalue problem for the fractional p-sub-Laplacian over the fractional Folland–Stein–Sobolev spaces on stratified Lie groups. We apply variational methods to investigate the eigenvalue problems. We conclude the positivity of the first eigenfunction via the strong minimum principle for the fractional p-sub-Laplacian. Moreover, we deduce that the first eigenvalue is simple and isolated. Secondly, utilising established properties, we prove the existence of at least two weak solutions via the Nehari manifold technique to a class of subelliptic singular problems associated with the fractional p-sub-Laplacian on stratified Lie groups. We also investigate the boundedness of positive weak solutions to the considered problem via the Moser iteration technique. The results obtained here are also new even for the case p=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=2$$\\end{document} with G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {G}$$\\end{document} being the Heisenberg group.

Boundedness of square functions related with fractional Schrödinger semigroups on stratified Lie groups

&lt;abstract&gt;&lt;p&gt;In this paper, we consider a Schrödinger operator $ L = -\Delta_{\mathbb{H}}+V $ on the stratified Lie group $ \mathbb{H} $. First, we establish fractional heat kernel estimates related to $ L^{\beta} $, $ \beta\in(0, 1) $. By utilizing kernel estimations and the fractional Carleson measure, we are able to derive a characterization of the Campanato type space $ BMO_{L}^{v}(\mathbb{H}) $. Second, we demonstrate that both Littlewood-Paley $ {\bf g} $-functions and area functions are bounded on $ BMO^{v}_{L}(\mathbb{H}) $. Finally, we also obtain that the above square functions are bounded on the Morrey space $ L^{\gamma, \theta}_{p, \kappa}(\mathbb{H}) $ and the weak Morrey space $ WL^{\gamma, \theta}_{1, \kappa}(\mathbb{H}) $, respectively.&lt;/p&gt;&lt;/abstract&gt;

Characterizations of Lipschitz functions via the commutators of maximal function in Orlicz spaces on stratified Lie groups

Characterizations of Lipschitz functions via the commutators of maximal function in Orlicz spaces on stratified Lie groups

On criticality coupled sub-Laplacian systems with Hardy type potentials on Stratified Lie groups

&lt;abstract&gt;&lt;p&gt;In this work, our main concern is to study the existence and multiplicity of solutions for the following sub-elliptic system with Hardy type potentials and multiple critical exponents on Carnot group&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation*} \left\{\begin{aligned} &amp;amp;-\Delta_{\mathbb{G}}u = \frac{\psi^{\alpha}|u|^{2^*(\alpha)-2}u}{d(z)^{\alpha}}+ \frac{p_{1}}{2^*(\gamma)}\frac{\psi^{\gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{d(z, z_{0})^{\gamma}} +\lambda h(z)\frac{\psi^{\sigma}|u|^{q-2}u}{d(z)^{\sigma}} \, \, &amp;amp; \text{in } \, \, \Omega, \\ &amp;amp;-\Delta_{\mathbb{G}}v = \frac{\psi^{\beta}|v|^{2^*(\beta)-2}v}{d(z)^{\beta}}+ \frac{p_{2}}{2^*(\gamma)}\frac{\psi^{\gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{d(z, z_{0})^{\gamma}} +\lambda h(z)\frac{\psi^{\sigma}|v|^{q-2}v}{d(z)^{\sigma}}\, \, &amp;amp;\text{in } \, \, \Omega, \\ &amp;amp;\quad u = v = 0\, \, &amp;amp;\text{on } \, \, \partial\Omega, \end{aligned}\right. \end{equation*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ -\Delta_{\mathbb{G}} $ is a sub-Laplacian on Carnot group $ \mathbb{G} $, $ \alpha, \beta, \gamma, \sigma\in [0, 2) $, $ d $ is the $ \Delta_{\mathbb{G}} $-natural gauge, $ \psi = |\nabla_{\mathbb{G}}d| $ and $ \nabla_{\mathbb{G}} $ is the horizontal gradient associated to $ \Delta_{\mathbb{G}} $. The positive parameters $ \lambda $, $ q $ satisfy $ 0 &amp;lt; \lambda &amp;lt; \infty $, $ 1 &amp;lt; q &amp;lt; 2 $, and $ p_{1} $, $ p_{2} &amp;gt; 1 $ with $ p_{1}+p_{2} = 2^*(\gamma) $, here $ 2^*(\alpha): = \frac{2(Q-\alpha)}{Q-2} $, $ 2^*(\beta): = \frac{2(Q-\beta)}{Q-2} $ and $ 2^*(\gamma) = \frac{2(Q-\gamma)}{Q-2} $ are the critical Hardy-Sobolev exponents, $ Q $ is the homogeneous dimension of the space $ \mathbb{G} $. By means of variational methods and the mountain-pass theorem of Ambrosetti and Rabonowitz, we study the existence of multiple solutions to the sub-elliptic system.&lt;/p&gt;&lt;/abstract&gt;

Critical problems with Hardy potential on Stratified Lie Groups

We prove existence and nonexistence results for positive solutions to the subelliptic Brezis-Nirenberg type problem with Hardy potential $$- \Delta_{\mathbb G} u -\mu \frac{\psi^2}{d^2} u =u^{2^*-1}+\lambda u\,\,\, \mbox{in}\,\,\, \Omega,\quad u=0\,\,\, \mbox{on}\,\, \partial \Omega, $$ extending to the Stratified setting well-known Euclidean results by Jannelli [J. Diff. Equ. 156, 1999]. Here, $\Delta_{\mathbb G}$ is a sub-Laplacian on an arbitrary Carnot group ${\mathbb G}$, $\Omega$ is a bounded domain of ${\mathbb G}$, $0\in \Omega$, $d$ is the $\Delta_{\mathbb G}$-gauge, $\psi:=|\Delta_{\mathbb G} d|$, where $\Delta_{\mathbb G}$ is the horizontal gradient associated to $\Delta_{\mathbb G}$, $0 \leq \mu < \bar \mu$, where $\bar \mu =\left ( \frac{Q-2}{2} \right )^2$ is the best Hardy constant on ${\mathbb G}$ and $\lambda\in {\mathbb R}$. The main difficulty in this abstract framework is the lack of knowledge of the ground state solutions to the limit problem $$- \Delta_{\mathbb G} u -\mu \dfrac{\psi^2}{d^2} u =u^{2^*-1}\,\, \mbox{on}\,\, {\mathbb G}, $$ whose explicit expression is not known, except for the case when $\mu=0$ and ${\mathbb G}$ is a group of Iwasawa-type. So, a preliminary refined analysis of qualitative properties of solutions to the above problem on the whole space is required, which has independent interest. In particular, Lorentz regularity and a priori decay estimates are obtained.