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Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group

Ewa DamekAndrzej Hulanicki — 1991

Studia Mathematica

On the domain S_a = {(x,e^b): x ∈ N, b ∈ ℝ, b > a} where N is a simply connected nilpotent Lie group, a certain N-left-invariant, second order, degenerate elliptic operator L is considered. N × {e^a} is the Poisson boundary for L-harmonic functions F, i.e. F is the Poisson integral F(xe^b) = ʃ_N f(xy)dμ^b_a(x), for an f in L^∞(N). The main theorem of the paper asserts that the maximal function M^a f(x) = sup{|ʃf(xy)dμ_a^b(y)| : b > a} is of weak type (1,1).

Asymptotic behavior of the invariant measure for a diffusion related to an NA group

Ewa DamekAndrzej Hulanicki — 2006

Colloquium Mathematicae

On a Lie group NA that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, the heat semigroup μ t generated by a second order subelliptic left-invariant operator j = 0 m Y j + Y is considered. Under natural conditions there is a μ ̌ t -invariant measure m on N, i.e. μ ̌ t * m = m . Precise asymptotics of m at infinity is given for a large class of operators with Y₀,...,Yₘ generating the Lie algebra of S.

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