### Maximal functions related to subelliptic operators with polynomially growing coefficients.

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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...

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).

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

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 ${\mu}_{t}$ generated by a second order subelliptic left-invariant operator ${\sum}_{j=0}^{m}{Y}_{j}+Y$ is considered. Under natural conditions there is a $\mu {\u030c}_{t}$-invariant measure m on N, i.e. $\mu {\u030c}_{t}*m=m$. Precise asymptotics of m at infinity is given for a large class of operators with Y₀,...,Yₘ generating the Lie algebra of S.

Let N be a simply connected nilpotent Lie group and let $S=N\u22ca{\left(\mathbb{R}\u207a\right)}^{d}$ be a semidirect product, ${\left(\mathbb{R}\u207a\right)}^{d}$ 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 $li{m}_{t\to \infty}{t}^{\chi \u2080}\nu x:\left|x\right|>t=C>0$. In particular, this applies to classical Poisson kernels on symmetric spaces,...

We study a natural system of second order differential operators on a symmetric Siegel domain $\mathcal{D}$ that is invariant under the action of biholomorphic transformations. If $\mathcal{D}$ 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.

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.

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 $\mathbb{E}[log{A}_{1}]=0$ , 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...

In this paper we treat noncoercive operators on simply connected manifolds of negative curvature.

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