Some Differential Operators Related to the Heisenberg Sub - Laplacian
نویسنده
چکیده
Let Lα = − 12 ∑n j=1 ( ZjZj + ZjZj ) + iαT be the sub-Laplacian on the nonisotropic Heisenberg group Hn where Zj , Z̄j for j = 1, 2, · · · , n and T are a basis of the Lie algebra hn. We apply the Laguerre calculus to obtain the fundamental solution of the heat kernel exp{−sLα}, the Schrödinger operator exp{−isLα} and the operator ∆λ,α = − 12 ∑n j=1 λj(ZjZ̄j + Z̄jZj) + iαT. We also discuss some basic properties of the wave operator.
منابع مشابه
Heat kernel asymptotic expansions for the Heisenberg sub-Laplacian and the Grushin operator.
The sub-Laplacian on the Heisenberg group and the Grushin operator are typical examples of sub-elliptic operators. Their heat kernels are both given in the form of Laplace-type integrals. By using Laplace's method, the method of stationary phase and the method of steepest descent, we derive the small-time asymptotic expansions for these heat kernels, which are related to the geodesic structure ...
متن کاملWeighted Rellich Inequality on H-Type Groups and Nonisotropic Heisenberg Groups
The study of partial differential operators constructed from noncommutative vector fields satisfying the Hörmander condition 1 has hadmuch development. We refer to 2, 3 and the references therein for a systematic account of the study. Recently there have been considerable interests in studying the sub-Laplacians as square sums of vector fields that are not invariant or do not satisfy the Hörman...
متن کاملAdams-spanne Type Estimates for Certain Sublinear Operators and Their Commutators Generated by Fractional Integrals in Generalized Morrey Spaces on Heisenberg Groups and Some Applications
In this paper we consider the Spanne type boundedness of sublinear operators and prove the Adams type boundedness theorems for these operators and also give BMO (bounded mean oscillation space) estimates for their commutators in generalized Morrey spaces on Heisenberg groups. The boundedness conditions are formulated in terms of Zygmund type integral inequalities. Based on the properties of the...
متن کاملOn certain conformally invariant systems of differential equations
Several systems of differential operators are constructed and their study is commenced. These systems are generalizations, in a reasonable sense, of the Heisenberg Laplacian operators introduced by Folland and Stein. In particular, they admit large groups of conformal symmetries; various real form of the special linear groups, even special orthogonal groups, and the exceptional group of type E6...
متن کاملEstimates for Spectral Projection Operators of the Sub-Laplacian on the Heisenberg Group
In this paper, we use Laguerre calculus to find the Lp spectrum (λ, μ) of the pair (L, iT). Here L = −12 ∑n j=1(ZjZj+ZjZj) andT = ∂ ∂t with {Z1, . . . ,Zn,Z1, , . . . ,Zn,T} a basis for the left-invariant vector fields on the Heisenberg group. We find kernels for the spectral projection operators on the ray λ > 0 in the Heisenberg brush and show that they are Calderón-Zygmund-Mikhlin operators....
متن کامل