We establish two results concerning the Quantum Limits (QLs) of some sub-Laplacians. First, under a commutativity assumption on the vector fields involved in the definition of the sub- Laplacian, we prove that it is possible to split any QL into several pieces which can be studied separately, and which come from well-characterized parts of the associated sequence of eigenfunctions. Secondly, building upon this result, we study in detail the QLs of a particular family of sub-Laplacians defined on products of compact quotients of Heisenberg groups. We express the QLs through a disintegration of measure result which follows from a natural spectral decomposition of the sub-Laplacian in which harmonic oscillators appear. Both results are based on the construction of an adequate elliptic operator commuting with the sub-Laplacian, and on the associated joint spectral calculus. They illustrate the fact that, because of the possible high degeneracies in the spectrum, the spectral theory of sub-Laplacians is very rich.
We investigate the Fredholm alternative for the $p$-Laplacian in an exterior domain which is the complement of the closed unit ball in $\mathbb{R}^N$ ($N\geq 2$). By employing techniques of Calculus of Variations we obtain the multiplicity of solutions. The striking difference between our case and the entire space case is also discussed.
In this paper we explore the weak solutions of the Cauchy problem and an inverse source problem for the heat equation in the quantum calculus, formulated in abstract Hilbert spaces. For this we use the Fourier series expansions. Moreover, we prove the existence, uniqueness and stability of the weak solution of the inverse problem with a final determination condition. We give some examples such as the q-Sturm-Liouville problem, the q-Bessel operator, the q-deformed Hamiltonian, the fractional Sturm-Liouville operator, and the restricted fractional Laplacian, covered by our analysis.
The purpose of this paper is to give a new proof of results of Moscovici and Stanton on the orbital integrals associated with eta invariants on compact locally symmetric spaces. Moscovici and Stanton used methods of harmonic analysis on reductive groups. Here, we combine our approach to orbital integrals using the hypoelliptic Laplacian, with the introduction of a rotation on certain Clifford algebras. Probabilistic methods play an important role in establishing key estimates. In particular, we construct the proper It{ô} calculus associated with certain hypoelliptic diffusions.
Let $M$ be a compact smooth manifold equipped with a positive smooth density $μ$ and $H$ be a smooth distribution endowed with a fiberwise inner product $g$. We define the Laplacian $Δ_H$ associated with $(H,μ,g)$ and prove that it gives rise to an unbounded self-adjoint operator in $L^2(M,μ)$. Then, assuming that $H$ generates a singular foliation $\mathcal F$, we prove that, for any function $φ$ from the Schwartz space $\mathcal S(\mathbb R)$, the operator $φ(Δ_H)$ is a smoothing operator in the scale of longitudinal Sobolev spaces associated with $\mathcal F$. The proofs are based on pseudo…
Let $\Box_b$ be the Kohn Laplacian acting on $(0,j)$-forms on the unit sphere in $\mathbb{C}^n$. In a recent paper of Casarino, Cowling, Sikora and the author, a spectral multiplier theorem of Mihlin--Hörmander type for $\Box_b$ is proved in the case $0<j<n-1$. Here we prove an analogous theorem in the exceptional cases $j=0$ and $j=n-1$, including a weak type $(1,1)$ endpoint estimate. We also show that both theorems are sharp. The proof hinges on an abstract multivariate multiplier theorem for systems of commuting operators.
This paper concerns Hodge-Dirac operators D = d + $δ$ acting in L p ($Ω$, λ) where $Ω$ is a bounded open subset of R n satisfying some kind of Lipschitz condition, λ is the exterior algebra of R n , d is the exterior derivative acting on the de Rham complex of differential forms on $Ω$, and $δ$ is the interior derivative with tangential boundary conditions. In L 2 ($Ω$, λ), $δ$ = d * and D is self-adjoint, thus having bounded resolvents (I + itD) --1 t$\in$R as well as a bounded functional calculus in L 2 ($Ω$, λ). We investigate the range of values p H \textless{} p \textless{} p H about p = …
Consider a second order, strongly elliptic negative semidefinite differential operator $L$ (maybe a system) on a compact Riemannian manifold $\overline{M}$ with smooth boundary, where the domain of $L$ is defined by a coercive boundary condition. Classically known results, and also recent work in \cite{DOS} and \cite{DM} establish sufficient conditions for $L^\infty-\text{BMO}_L$ continuity of $φ(\sqrt{A})$, where $φ\in S^0_1(\mathbb{R})$, and $A$ is a suitable elliptic operator. Using a variant of the Cheeger-Gromov-Taylor functional calculus due to \cite{MMV}, and short time bounds on the in…
We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite collection of algebraic rotations of the $2$-sphere. Under mild conditions, such joint eigenfunctions are shown to satisfy for large $p$ the same bounds as those known for Laplace eigenfunctions on a surface of non-positive curvature.
We solve $\square_b$ on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related $\square_b$ operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solve $\square_b$ works whenever $\overline{\partial}_b$ on the compact CR manifold has closed range in $L^2$; in particular, as in Beals and Greiner, it does not require the CR manifold to be the boundary of a strongly pseudoconvex domain in $\mathbb{C}^2$. Our result provides in tu…
