A calculus of symbols and convolution semigroups on the Heisenberg group
A class of solvable non-homogeneous differential operators on the Heisenberg group
In [8], we studied the problem of local solvability of complex coefficient second order left-invariant differential operators on the Heisenberg group ℍₙ, whose principal parts are "positive combinations of generalized and degenerate generalized sub-Laplacians", and which are homogeneous under the Heisenberg dilations. In this note, we shall consider the same class of operators, but in the presence of left invariant lower order terms, and shall discuss local solvability for these operators in a complete...
A complete analogue of Hardy's theorem on semisimple Lie groups
A result by G. H. Hardy ([11]) says that if f and its Fourier transform f̂ are and respectively for some m,n ≥ 0 and α > 0, then f and f̂ are and respectively for some polynomials P and P’. If in particular f is as above, but f̂ is , then f = 0. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.
A functional calculus for Rockland operators on nilpotent Lie groups
A geometric classification of Lie groups.
This paper is part of a general program that was originally designed to study the Heat diffusion kernel on Lie groups.
A Hardy inequality with remainder terms in the Heisenberg group and the weighted eigenvalue problem.
A Heat Semigroup Version of Bernstein's Theorem on Lie Groups.
A maximal function on harmonic extensions of -type groups
Let be an -type group and be its harmonic extension. We study a left invariant Hardy–Littlewood maximal operator on , obtained by taking maximal averages with respect to the right Haar measure over left-translates of a family of neighbourhoods of the identity. We prove that the maximal operator is of weak type .
A multiplier theorem for H-type groups
We prove an -boundedness result for a convolution operator with rough kernel supported on a hyperplane of a group of Heisenberg type.
A new criterion for local non-solvability of homogeneous left invariant differential operators on nilpotent Lie groups.
A nilpotent Lie algebra and eigenvalue estimates
The aim of this paper is to demonstrate how a fairly simple nilpotent Lie algebra can be used as a tool to study differential operators on with polynomial coefficients, especially when the property studied depends only on the degree of the polynomials involved and/or the number of variables.
A note on lifting of Carnot groups.
We prove that every homogeneous Carnot group can be lifted to a free homogeneous Carnot group. Though following the ideas of Rothschild and Stein, we give simple and self-contained arguments, providing a constructive proof, as shown in the examples.
A note on Schrödinger operators with polynomial potentials
A Note on the Amenable Subgroups of PSL (2, R).
A Paley-Wiener theorem for step two nilpotent Lie groups.
It is an interesting open problem to establish Paley-Wiener theorems for general nilpotent Lie groups. The aim of this paper is to prove one such theorem for step two nilpotent Lie groups which is analogous to the Paley-Wiener theorem for the Heisenberg group proved in [4].
A Paley-Wiener theorem on NA harmonic spaces
Let N be an H-type group and consider its one-dimensional solvable extension NA, equipped with a suitable left-invariant Riemannian metric. We prove a Paley-Wiener theorem for nonradial functions on NA supported in a set whose boundary is a horocycle of the form Na, a ∈ A.
A Plancherel measure for the discrete Heisenberg group
A remark on a Marcinkiewicz-Hörmander multiplier theorem for some non-differential convolution operators
A Remark on Continuity of Translation Invariant Functionals on L... (G) for Compact Lie Groups G.
A restriction theorem for the Heisenberg motion
We prove a restriction theorem for the class-1 representations of the Heisenberg motion group. This is done using an improvement of the restriction theorem for the special Hermite projection operators proved in [13]. We also prove a restriction theorem for the Heisenberg group.