Displaying similar documents to “On the Liouville property for sublaplacians”

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

Ewa Damek, Andrzej Hulanicki (1991)

Studia Mathematica

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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).

Note on semigroups generated by positive Rockland operators on graded homogeneous groups

Jacek Dziubański, Waldemar Hebisch, Jacek Zienkiewicz (1994)

Studia Mathematica

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Let L be a positive Rockland operator of homogeneous degree d on a graded homogeneous group G and let p t be the convolution kernels of the semigroup generated by L. We prove that if τ(x) is a Riemannian distance of x from the unit element, then there are constants c>0 and C such that | p 1 ( x ) | ≤ C e x p ( - c τ ( x ) d / ( d - 1 ) ) . Moreover, if G is not stratified, more precise estimates of p 1 at infinity are given.

On the definition of the dual Lie coalgebra of a Lie algebra.

Bertin Diarra (1995)

Publicacions Matemàtiques

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Let L be a Lie algebra over a field K. The dual Lie coalgebra Lº of L has been defined by W. Michaelis to be the sum of all good subspaces V of the dual space L* of L: V is good if m(V) ⊂ V ⊗ V, where m is the multiplication of L. We show that Lº = m(L* ⊗ L*) as in the associative case.