Edward McDonald, Fedor Sukochev, Dmitriy Zanin
We study Cwikel-type estimates for the singular values and Schatten $\mathcal{L}_p$-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.
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An endpoint estimate for product singular integral operators on stratified Lie groups
本文在两个分层李群的乘积上证明面积函数、平方函数、极大算子及 Calderón–Zygmund 算子具有超弱有界性，给出了乘积奇异积分算子的端点估计。
In this paper, the main aim is to consider the boundedness of the Hardy-Littlewood maximal commutator $M_{b}$ and the nonlinear commutator $[b, M]$ on the Lebesgue spaces and Morrey spaces over some stratified Lie group $\mathbb{G}$ when $b$ belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces on Lie group are given.
In this short paper, we establish a range of Caffarelli-Kohn-Nirenberg and weighted $L^{p}$-Sobolev type inequalities on stratified Lie groups. All inequalities are obtained with sharp constants. Moreover, the equivalence of the Sobolev type inequality and Hardy inequality is shown in the $L^{2}$-case.
Calderón's commutator on Stratified Lie groups
本文受紧Heisenberg型流形上Calderón交换子研究启发，建立了分层李群上Calderón交换子的$L^p$有界性刻画，并利用新颖的双重算子积分技术研究了两步分层李群上二阶交换子的弱Schatten类估计。
Some estimates for commutators of the fractional maximal function on stratified Lie groups
本文研究了分层李群上分数次极大交换子M_{α,b}与非线性交换子[b, M_α]在Lebesgue空间的有界性，并由此给出了Lipschitz空间的新刻画。
We give necessary and sufficient conditions for the boundedness of the maximal commutators $M_{b}$, the commutators of the maximal operator $[b, M]$ and the commutators of the sharp maximal operator $[b, M^{\sharp}]$ in Orlicz spaces $L^Φ(\mathbb{G})$ on any stratified Lie group $\mathbb{G}$ when $b$ belongs to Lipschitz spaces $\dotΛ_β(\mathbb{G})$. We obtain some new characterizations for certain subclasses of Lipschitz spaces $\dotΛ_β(\mathbb{G})$.
The study of Sobolev inequalities can be divided in two cases: p = 1 and 1 < p < +$\infty$. In the case p = 1 we study here a relaxed version of refined Sobolev inequalities. When p > 1, using as base space classical Lorentz spaces associated to a weight from the Arino-Muckenhoupt class Bp, we will study Gagliardo-Nirenberg inequalities. As a by-product we will also consider Morrey-Sobolev inequalities. These arguments can be generalized to many different frameworks, in particular the proofs are given in the setting of stratified Lie groups.
We study in this article the Improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the Lq norm of a function by the geometric mean of two norms corresponding to Sobolev spaces W(s;p) and Besov spaces B(-b, infty, infty). When the value p which characterizes Sobolev space is strictly larger than 1, the required result is well known in R^n and is classically obtained by a Littlewood-Paley dyadic blocks manipulation. For these inequalities we will develop here another totally different technique. When p = 1, thes…
We prove various Hardy-type and uncertainty inequalities on a stratified Lie group $G$. In particular, we show that the operators $T_α: f \mapsto |.|^{-α} L^{-α/2} f$, where $|.|$ is a homogeneous norm, $0 < α< Q/p$, and $L$ is the sub-Laplacian, are bounded on the Lebesgue space $L^p(G)$. As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg-Pauli-Weyl inequality, relating the $L^p$ norm of a function $f$ to the $L^q$ norm of $|.|^βf$ and th…
We introduce a general definition of homogeneous Besov spaces on a stratified Lie group $G$, based on a Littlewood-Paley-type decomposition of Schwartz functions with all moments vanishing. We show that under mild and intuitive conditions the spaces thus defined are independent of the decomposition employed. A corollary of this is that previously constructed versions of homogeneous Besov spaces on $G$, relying on the spectral calculus of a sub-Laplacian of the group, are consistent, i.e., independent of the choice of sub-Laplacian. We further prove characterizations of homogeneous Besov spaces…
In this paper, generalised weighted $L^p$-Hardy,$ L^p$-Caffarelli-Kohn-Nirenberg, and $L^p$-Rellich inequalities with boundary terms are obtained on stratified Lie groups. As consequences, most of the Hardy type inequalities and Heisenberg- Pauli-Weyl type uncertainty principles on stratified groups are recovered. Moreover, a weighted $L^2$-Rellich type inequality with the boundary term is obtained.
Let G be a stratified Lie group and L be the sub-Laplacian on G. Let 0 \neq f\in S(R^+). We show that Lf(L)δ, the distribution kernel of the operator Lf(L), is an admissible function on G. We also show that, if ξf(ξ) satisfies Daubechies' criterion, then L f(L)δgenerates a frame for any sufficiently fine lattice subgroup of G.
Restriction type estimates on general two-step stratified Lie groups
本文在一般两步分层李群上证明次拉普拉斯算子的限制型估计，核心方法是通过谱簇估计控制各向异性扭曲拉普拉斯算子族的特征值分布。
On a stratified Lie group $G$ equipped with hypoelliptic heat kernel measure, we study the behavior of the dilation semigroup on $L^p$ spaces of log-subharmonic functions. We consider a notion of strong hypercontractivity and a strong logarithmic Sobolev inequality, and show that these properties are equivalent for any group $G$. Moreover, if $G$ satisfies a classical logarithmic Sobolev inequality, then both properties hold. This extends similar results obtained by Graczyk, Kemp and Loeb in the Euclidean setting.
Optimal subelliptic super-Poincaré and isoperimetric inequalities on stratified Lie groups
本文在分层李群及 Grushin、Heisenberg-Greiner 等次椭圆情形中，对指数幂型概率测度证明了 $q\in[1,2]$ 的超庞加莱不等式，并得到一般最优等周不等式。
On subelliptic equations on stratified Lie groups driven by singular nonlinearity and weak $L^1$ data
该研究针对分层李群上含奇异非线性与弱 $L^1$ 数据的次椭圆方程，借助对称山路定理证明在适当参数下存在无穷多解。
We show that for a hypoelliptic Dirichlet form operator A on a stratified complex Lie group, if the logarithmic Sobolev inequality holds, then a holomorphic projection of A is strongly hypercontractive in the sense of Janson. This extends previous results of Gross to a setting in which the operator A is not holomorphic.
Some super-Poincaré inequalities for gaussian-like measures on stratified Lie groups
本文延续$U$-界计划，利用Hardy型不等式证明了Métivier群上次椭圆概率测度的超庞加莱不等式，并从概率论角度推广了已有结果。
We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov space $\dot{B}_{p,q}^s$ in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield th…
We define and investigate $α$-modulation spaces $M_{p,q}^{s,α}(G)$ associated to a step two stratified Lie group $G$ with rational structure constants. This is an extension of the Euclidean $α$-modulation spaces $M_{p,q}^{s,α}(\mathbb{R}^n)$ that act as intermediate spaces between the modulation spaces ($α= 0$) in time-frequency analysis and the Besov spaces ($α= 1$) in harmonic analysis. We will illustrate that the the group structure and dilation structure on $G$ affect the boundary cases $α= 0,1$ where the spaces $M_{p,q}^{s}(G)$ and $\mathcal{B}_{p,q}^{s}(G)$ have non-standard translation …
Schatten Properties of Calderón--Zygmund Singular Integral Commutator on stratified Lie groups
本文完整刻画了分层李群上Calderón--Zygmund奇异积分交换子的Schatten性质，证明了在临界指标处范数等价性，并提出了可推广的新方法。
Characterizations of product Hardy spaces on stratified groups by singular integrals and maximal functions
本文通过发展基于 Heisenberg 群思想的新技术，在分层 Lie 群乘积上利用 Christ--Geller 型奇异积分和垂直或非切向极大函数刻画了 Hardy 空间，填补了经典理论中的空白。
Given a graded group $G$ and commuting, formally self-adjoint, left-invariant, homogeneous differential operators $\mathcal{L}_1,\dots, \mathcal{L}_n$ on $G$, one of which is Rockland, we study the convolution operators $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ and their convolution kernels, with particular reference to the case in which $G$ is abelian and $n=1$, and the case in which $G$ is a $2$-step stratified group which satisfies a slight strengthening of the Moore-Wolf condition and $\mathcal{L}_1,\dots,\mathcal{L}_n$ are either sub-Laplacians or central elements of the Lie algebra of $G$.…
In this paper we investigate critical Gagliardo-Nirenberg, Trudinger-type and Brezis-Gallouet-Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which includes the cases of $\mathbb R^n$, Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo-Nirenberg inequality, the existence of least energy solutions of nonlinear Schrödinger type equations is obtained. We also express the best constant in the critical Gagliardo-Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state s…
In this paper, we obtain asymptotic formulae on nilmanifolds $Γ\backslash G$, wher $G$ is any stratified (or even graded) nilpotent Lie group equipped with a co-compact discrete subgroup $Γ$. We study especially the asymptotics related to the sub-Laplacians naturally coming from the stratified structure of the group $G$ (and more generally any positive Rockland operators when $G$ is graded). We show that the short-time asymptotic on the diagonal of the kernels of spectral multipliers contains only a single non-trivial term. We also study the associated zeta functions.
We prove a general multiplier theorem for symmetric left-invariant sub-Laplacians with drift on non-compact Lie groups. This considerably improves and extends a result by Hebisch, Mauceri, and Meda. Applications include groups of polynomial growth and solvable extensions of stratified groups.
Function spaces are central topic in analysis. Often those spaces and related analysis involves symmetries in form of an action of a Lie group. Coorbit theory as introduced by Feichtinger and Gröchenig and then later extended in [3] gives a unified method to construct Banach spaces of functions based on representations of Lie groups. In this article we identify the homogeneous Besov spaces on stratified Lie groups introduced in [13] as coorbit spaces in the sense of [3] and use this to derive atomic decompositions for the Besov spaces.
From a theorem of Christ and Mauceri and Meda it follows that, for a homogeneous sublaplacian $L$ on a $2$-step stratified group $G$ with Lie algebra $\mathfrak{g}$, an operator of the form $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for $1 < p < \infty$ if the spectral multiplier $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$, where $Q = \dim \mathfrak{g} + \dim[\mathfrak{g},\mathfrak{g}]$ is the homogeneous dimension of $G$. Here we show that the condition can be pushed down to $s > d/2$, where $d = \dim \mathfrak{g}$ is the topological dimension of $G$, p…
In this paper we present a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups. Moreover, using the comparison principle we obtain blow-up type results and global in $t$-boundedness of solutions of nonlinear equations for the heat $p$-sub-Laplacian on stratified Lie groups. In particular, this paper generalises and extends previous results obtained by the first author and Suragan in [RS18].
On a general Lie group $G$ endowed with a sub-Riemannian structure and of local dimension $d$, we characterize the pointwise multipliers of Triebel--Lizorkin spaces $F^{p,q}_α$ for $p,q\in (1,\infty)$ and $α>d/p$, and those of Besov spaces $B^{p,q}_α$ for $q\in [1,\infty]$, $p>d$ and $d/p< α<1$. When $G$ is stratified, we extend the latter characterization to all $p,q\in [1,\infty]$ and $α>d/p$.
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