Class Of Nilpotent Lie Groups Research Articles

Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms

We introduce a class of nilpotent Lie groups which arise naturally from the notion of composition of quadratic forms, and show that their standard sublaplacians admit fundamental solutions analogous to that known for the Heisenberg group.

A Plancherel Formula for Idyllic Nilpotent Lie Groups

A procedure is developed which can be used to compute the Plancherel measure for a certain class of nilpotent Lie groups, including the Heisenberg groups, free groups, two-and three-step groups, the nilpotent part of an Iwasawa decomposition of the R-split form of the classical simple groups ${A_l},{C_l},{G_2}$. Let

A Plancherel formula for idyllic nilpotent Lie groups

A procedure is developed which can be used to compute the Plancherel measure for a certain class of nilpotent Lie groups, including the Heisenberg groups, free groups, two-and three-step groups, the nilpotent part of an Iwasawa decomposition of the R-split form of the classical simple groups A l , C l , G 2 {A_l},{C_l},{G_2} . Let G be a connected, simply connected nilpotent Lie group. The Plancherel formula for G can be expressed in terms of Plancherel measure of a normal subgroup N and projective Plancherel measures of certain subgroups of G / N G/N . To get an explicit measure for G, we need an explicit formula for (1) the disintegration of Plancherel measure of N under the action of G on N, and (2) projective Plancherel measures of G γ / N {G_\gamma }/N , where G γ {G_\gamma } is the stability subgroup at γ \gamma in N. When both N and G γ / N {G_\gamma }/N are abelian, the measures (1) and (2) are obtained as special cases of more general problems. These measures combine into Plancherel measure for G.

Singular integrals on nilpotent Lie groups

Convolution operators T f ( x ) = ∫ f ( x y − 1 ) K ( y ) d y Tf(x) = \smallint f(x{y^{ - 1}})K(y)\;dy on a class of nilpotent Lie groups are shown to be bounded on L p , 1 > p > ∞ {L^p},\;1 > p > \infty , provided the Euclidean Fourier transform of K K (with respect to a coordinate system in which the group multiplication is in a special form) satisfies familiar “multiplier” conditions.