Maximal functions related to subelliptic operators with polynomially growing coefficients.
The geometry of a semi-direct extension of a Heisenberg type nilpotent group
Curvature of a semi-direct extension of a Heisenberg type nilpotent group
Maximal functions related to subelliptic operators invariant under an action of a nilpotent Lie group
Fundamental solutions of differential operators on homogeneous manifolds of negative curvature and related Riesz transforms
Left-invariant degenerate elliptic operators on semidirect extensions of homogeneous groups
Convergence of Poisson integrals on semidirect extensions of homogeneous groups
A Poisson kernel on Heisenberg type nilpotent groups
Master
Pointwise estimates for the Poisson kernel on NA groups by the Ancona method
What can we do to support the applied mathematics in Poland?
In the past two years I talked to a great many people. One of the fundamental questions concerning the further development of applied mathematics in Poland is whether to appoint a new discipline (s) in the field of mathematics or not. Such a possible new discipline could be, for example. Applied statistics and mathematics calculation. Opinions are divided. The basic argument is this: ”If you summon a new discipline, it will give an impulse to the development of this kind of research in Poland. It...
Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group
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).
Boundaries and the Fatou theorem for subelliptic second order operators on solvable Lie groups
Invariant operators and pluriharmonic functions on symmetric irreducible Siegel domains
Let D be a symmetric irreducible Siegel domain. Pluriharmonic functions satisfying a certain rather weak growth condition are characterized by r+2 operators (r+1 in the tube case), r being the rank of the underlying symmetric cone
Asymptotic behavior of the invariant measure for a diffusion related to an NA group
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 generated by a second order subelliptic left-invariant operator is considered. Under natural conditions there is a -invariant measure m on N, i.e. . Precise asymptotics of m at infinity is given for a large class of operators with Y₀,...,Yₘ generating the Lie algebra of S.
Regular behavior at infinity of stationary measures of stochastic recursion on NA groups
Let N be a simply connected nilpotent Lie group and let be a semidirect product, acting on N by diagonal automorphisms. Let (Qₙ,Mₙ) be a sequence of i.i.d. random variables with values in S. Under natural conditions, including contractivity in the mean, there is a unique stationary measure ν on N for the Markov process Xₙ = MₙXn-1 + Qₙ. We prove that for an appropriate homogeneous norm on N there is χ₀ such that . In particular, this applies to classical Poisson kernels on symmetric spaces,...
Hua-harmonic functions on symmetric type two Siegel domains
We study a natural system of second order differential operators on a symmetric Siegel domain that is invariant under the action of biholomorphic transformations. If is of type two, the space of real valued solutions coincides with pluriharmonic functions. We show the main idea of the proof and give a survey of previous results.
Estimates for the Poisson kernels and their derivatives on rank one NA groups
For rank one solvable Lie groups of the type NA estimates for the Poisson kernels and their derivatives are obtained. The results give estimates on the Poisson kernel and its derivatives in a natural parametrization of the Poisson boundary (minus one point) of a general homogeneous, simply connected manifold of negative curvature.
On the invariant measure of the random difference equation Xn = AnXn−1 + Bn in the critical case
We consider the autoregressive model on ℝ defined by the stochastic recursion = −1 + , where {( , )} are i.i.d. random variables valued in ℝ× ℝ+. The critical case, when , was studied by Babillot, Bougerol and Elie, who proved that there exists a unique invariant Radon measure for the Markov chain { }. In the present paper we prove that the weak limit of properly dilated...
Martin boundary for homogeneous riemannian manifolds of negative curvature at the bottom of the spectrum.
In this paper we treat noncoercive operators on simply connected manifolds of negative curvature.
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