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 $G$, provided that $d \leq 7$ or $\dim [\mathfrak{g},\mathfrak{g}] \leq 2$.
In this article, we address sparse bounds for a class of spectral multipliers that include oscillating multipliers on stratified Lie groups. Our results can be applied to obtain weighted bounds for general Riesz means and for solutions of dispersive equations.
We consider a class of spectral multipliers on stratified Lie groups which generalise the class of Hörmander multipliers and include multipliers with an oscillatory factor. Oscillating multipliers have been examined extensively in the euclidean setting where sharp, endpoint $L^p$ estimates are well known. In the Lie group setting, corresponding $L^p$ bounds for oscillating spectral multipliers have been established by several authors but only in the open range of exponents. In this paper we establish the endpoint $L^p(G)$ bound when $G$ is a stratified Lie group. More importantly we begin to a…
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.
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.
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 $α>d/p$.
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups.
Let $G$ be a $2$-step stratified group of topological dimension $d$ and homogeneous dimension $Q$. Let $L$ be a homogeneous sub-Laplacian on $G$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in (1,\infty)$ whenever the multiplier $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$. It is known that, for several $2$-step groups and sub-Laplacians, the threshold $Q/2$ in the smoothness condition is not sharp and in many cases it is possible to push it down to $d/2$. Here we show …
We propose analogues of Green's and Picone's identities for the $p$-sub-Laplacian on stratified Lie groups. In particular, these imply a generalised Díaz-Saá inequality for the $p$-sub-Laplacian on stratified Lie groups. Using these we derive a comparison principle and uniqueness of a positive solution to nonlinear hypoelliptic equations on general stratified Lie groups extending to the setting of the general stratified Lie groups previously known results on Euclidean and Heisenberg groups.
In this paper we obtain logarithmic Hardy and Rellich inequalities on general Lie groups. In the case of graded groups, we also show their refinements using the homogeneous Sobolev norms. In fact, we derive a family of weighted logarithmic Hardy-Rellich inequalities, for which logarithmic Hardy and Rellich inequalities are special cases. As a consequence of these inequalities, we also derive a Gross type logarithmic Hardy inequality on general stratified groups. An interesting feature of such estimate is that we consider the measure which is Gaussian only on the first stratum of the group. Suc…
We establish the optimal $C_{H}^{1,1}$ interior regularity of solutions to \[ Δ_{H}u=fχ_{\{u\ne0\}}, \] where $Δ_{H}$ denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on $f$, namely $f*Γ$ is $C_{H}^{1,1}$, where $Γ$ is the fundamental solution of $Δ_{H}$. The $C_{H}^{1,1}$ regularity is understood in the sense of Folland and Stein. In the classical Euclidean setting, the first seeds of the above problem are already present in the 1991 paper of Sakai and are also related to quadrature domains. As a special instance of our results, when $u$ is …
We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only …
Weighted $L^2$ theory for the Euclidean Dirac operator in higher dimensions
University of Science and Technology of China（中国科学技术大学）；Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
