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A class of solvable non-homogeneous differential operators on the Heisenberg group

Detlef Müller, Zhenqiu Zhang (2001)

Studia Mathematica

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

Rudra P. Sarkar (2002)

Colloquium Mathematicae

A result by G. H. Hardy ([11]) says that if f and its Fourier transform f̂ are O ( | x | m e - α x ² ) and O ( | x | e - x ² / ( 4 α ) ) respectively for some m,n ≥ 0 and α > 0, then f and f̂ are P ( x ) e - α x ² and P ' ( x ) e - x ² / ( 4 α ) respectively for some polynomials P and P’. If in particular f is as above, but f̂ is o ( e - x ² / ( 4 α ) ) , 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 geometric classification of Lie groups.

Nicholas T. Varopoulos (2000)

Revista Matemática Iberoamericana

This paper is part of a general program that was originally designed to study the Heat diffusion kernel on Lie groups.

A maximal function on harmonic extensions of H -type groups

Maria Vallarino (2006)

Annales mathématiques Blaise Pascal

Let N be an H -type group and S N × + be its harmonic extension. We study a left invariant Hardy–Littlewood maximal operator M ρ on S , 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 M ρ is of weak type ( 1 , 1 ) .

A multiplier theorem for H-type groups

Rita Pini (1991)

Studia Mathematica

We prove an L p -boundedness result for a convolution operator with rough kernel supported on a hyperplane of a group of Heisenberg type.

A nilpotent Lie algebra and eigenvalue estimates

Jacek Dziubański, Andrzej Hulanicki, Joe Jenkins (1995)

Colloquium Mathematicae

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 n 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.

Andrea Bonfiglioli, Francesco Uguzzoni (2005)

Revista Matemática Iberoamericana

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 Paley-Wiener theorem for step two nilpotent Lie groups.

Sundaram Thangavelu (1994)

Revista Matemática Iberoamericana

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

Francesca Astengo, Bianca di Blasio (1999)

Colloquium Mathematicae

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 restriction theorem for the Heisenberg motion

P. Ratnakumar, Rama Rawat, S. Thangavelu (1997)

Studia Mathematica

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.

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