On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1

Jan-Olav Rönning

Displaying similar documents to “On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1”

Mean field limit for the one dimensional Vlasov-Poisson equation

Maxime Hauray (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

Similarity:

We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in [Tro86]. Here we shall give a simpler proof of this result, and explain why it implies the so-called “Propagation of molecular chaos”. More precisely, both...

Harmonic interpolating sequences, $L^{p}$ and BMO

John B. Garnett (1978)

Annales de l'institut Fourier

Similarity:

Let $(z_{ν})$ be a sequence in the upper half plane. If $1 < p \leq \infty$ and if $y_{ν}^{1 / p} f (z_{ν}) = a_{ν}, ν = 1, 2, ... (*)$ has solution $f (z)$ in the class of Poisson integrals of $L^{p}$ functions for any sequence $(a_{ν}) \in ℓ^{p}$ , then we show that $(z_{ν})$ is an interpolating sequence for $H^{\infty}$ . If $f (z_{ν}) = a_{ν}$ , $ν = 1, 2, ...$ has solution in the class of Poisson integrals of BMO functions whenever $(a_{ν}) \in ℓ^{\infty}$ , then $(z_{ν})$ is again an interpolating sequence for $H^{\infty}$ . A somewhat more general theorem is also proved and a counterexample for the case $p \leq 1$ is described.

On the global regularity of subcritical Euler–Poisson equations with pressure

Eitan Tadmor, Dongming Wei (2008)

Journal of the European Mathematical Society

Similarity:

We prove that the one-dimensional Euler–Poisson system driven by the Poisson forcing together with the usual $γ$ -law pressure, $γ \geq 1$ , admits global solutions for a large class of initial data. Thus, the Poisson forcing regularizes the generic finite-time breakdown in the $2 \times 2$ $p$ -system. Global regularity is shown to depend on whether or not the initial configuration of the Riemann invariants and density crosses an intrinsic critical threshold.

Poisson's equation and characterizations of reflexivity of Banach spaces

Vladimir P. Fonf, Michael Lin, Przemysław Wojtaszczyk (2011)

Colloquium Mathematicae

Similarity:

Let X be a Banach space with a basis. We prove that X is reflexive if and only if every power-bounded linear operator T satisfies Browder’s equality $x \in X : s u p_{n} | | \sum_{k = 1}^{n} T^{k} x | | < \infty$ = (I-T)X $.$ We then deduce that X (with a basis) is reflexive if and only if every strongly continuous bounded semigroup $T_{t} : t \geq 0$ with generator A satisfies $A X = x \in X : s u p_{s > 0} | | \int_{0}^{s} T_{t} x d t | | < \infty$ . The range (I-T)X (respectively, AX for continuous time) is the space of x ∈ X for which Poisson’s equation (I-T)y = x (Ay = x in continuous time) has a solution y ∈ X; the above equalities...

Canonical Poisson-Nijenhuis structures on higher order tangent bundles

P. M. Kouotchop Wamba (2014)

Annales Polonici Mathematici

Similarity:

Let M be a smooth manifold of dimension m>0, and denote by $S_{c a n}$ the canonical Nijenhuis tensor on TM. Let Π be a Poisson bivector on M and $Π^{T}$ the complete lift of Π on TM. In a previous paper, we have shown that $(T M, Π^{T}, S_{c a n})$ is a Poisson-Nijenhuis manifold. Recently, the higher order tangent lifts of Poisson manifolds from M to $T^{r} M$ have been studied and some properties were given. Furthermore, the canonical Nijenhuis tensors on $T^{A} M$ are described by A. Cabras and I. Kolář [Arch. Math. (Brno) 38 (2002),...

One-parameter contractions of Lie-Poisson brackets

Oksana Yakimova (2014)

Journal of the European Mathematical Society

Similarity:

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $𝒜 = 𝕂 [𝔸^{n}]$ is said to be of Kostant type, if its centre $Z (𝒜)$ is freely generated by homogeneous polynomials $F_{1}, ..., F_{r}$ such that they give Kostant’s regularity criterion on $𝔸^{n} (d_{x} F_{i}$ are linear independent if and only if the Poisson tensor has the maximal rank at $x$ ). If the initial Poisson algebra is of Kostant type and $F_{i}$ satisfy a certain degree-equality, then the contraction...

Prolongation of Poisson $2$ -form on Weil bundles

Norbert Mahoungou Moukala, Basile Guy Richard Bossoto (2016)

Archivum Mathematicum

Similarity:

In this paper, $M$ denotes a smooth manifold of dimension $n$ , $A$ a Weil algebra and $M^{A}$ the associated Weil bundle. When $(M, ω_{M})$ is a Poisson manifold with $2$ -form $ω_{M}$ , we construct the $2$ -Poisson form $ω_{M^{A}}^{A}$ , prolongation on $M^{A}$ of the $2$ -Poisson form $ω_{M}$ . We give a necessary and sufficient condition for that $M^{A}$ be an $A$ -Poisson manifold.

A Hardy space related to the square root of the Poisson kernel

Jonatan Vasilis (2010)

Studia Mathematica

Similarity:

A real-valued Hardy space $H ¹_{\sqrt} () \subseteq L ¹ ()$ related to the square root of the Poisson kernel in the unit disc is defined. The space is shown to be strictly larger than its classical counterpart H¹(). A decreasing function is in $H ¹_{\sqrt} ()$ if and only if the function is in the Orlicz space LloglogL(). In contrast to the case of H¹(), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L() contains positive functions which do not belong to $H ¹_{\sqrt} ()$ , and no Orlicz...

Cauchy-Poisson transform and polynomial inequalities

Mirosław Baran (2009)

Annales Polonici Mathematici

Similarity:

We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in $ℝ^{N}$ is Hölder continuous then E admits a Szegö type inequality with weight function $d i s t {(x, \partial E)}^{- (1 - κ)}$ with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.

A note on n-ary Poisson brackets

Michor, Peter W., Vaisman, Izu

Similarity:

An $n$ -ary Poisson bracket (or generalized Poisson bracket) on the manifold $M$ is a skew-symmetric $n$ -linear bracket ${, \dots,}$ of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order $n$ , i.e., $\sum_{σ \in S_{2 n - 1}} (sign σ) {{f_{σ_{1}}, \dots, f_{σ_{n}}}, f_{σ_{n + 1}}, \dots, f_{σ_{2 n - 1}}} = 0,$ $S_{2 n - 1}$ being the symmetric group. The notion of generalized Poisson bracket was introduced by et al. in [J. Phys. A, Math. Gen. 29, No. 7, L151–L157 (1996; Zbl 0912.53019) and J. Phys. A, Math. Gen. 30, No. 18, L607–L616 (1997; Zbl 0932.37056)]....

Kontsevich Deformation Quantization on Lie Algebras

Nabiha Ben Amar, Mouna Chaabouni, Mabrouka Hfaiedh (2007)

Bollettino dell'Unione Matematica Italiana

Similarity:

We consider Kontsevich star product on the dual $\mathfrak{g}^{*}$ of a general Lie algebra g equipped with the linear Poisson bracket. We show that this star product provides a deformation quantization by partial embeddings in the direction of the Poisson bracket.

Iterates and the boundary behavior of the Berezin transform

Jonathan Arazy, Miroslav Engliš (2001)

Annales de l’institut Fourier

Similarity:

Let $μ$ be a measure on a domain $Ω$ in $ℂ^{n}$ such that the Bergman space of holomorphic functions in $L^{2} (Ω, μ)$ possesses a reproducing kernel $K (x, y)$ and $K (x, x) > 0$ $\forall x \in Ω$ . The Berezin transform associated to $μ$ is the integral operator $B f (y) = K {(y, y)}^{- 1} \int_{Ω} f (x) {| K (x, y) |}^{2} d μ (x) .$ The number $B f (y)$ can be interpreted as a certain mean value of $f$ around $y$ , and functions satisfying $B f = f$ as functions having a certain mean-value property. In this paper we investigate the boundary behavior of $B f$ , the existence of functions $f$ satisfying...

Involutivity of truncated microsupports

Masaki Kashiwara, Térésa Monteiro Fernandes, Pierre Schapira (2003)

Bulletin de la Société Mathématique de France

Similarity:

Using a result of J.-M. Bony, we prove the weak involutivity of truncated microsupports. More precisely, given a sheaf $F$ on a real manifold and $k \in ℤ$ , if two functions vanish on ${SS}_{k} (F)$ , then so does their Poisson bracket.

The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney, Roman Urban (2013)

Studia Mathematica

Similarity:

Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to $ℝ^{k}$ , k>1. We consider a class of second order left-invariant differential operators on S of the form $ℒ_{α} = L^{a} + Δ_{α}$ , where $α \in ℝ^{k}$ , and for each $a \in ℝ^{k}, L^{a}$ is left-invariant second order differential operator on N and $Δ_{α} = Δ - ⟨ α, \nabla ⟩$ , where Δ is the usual Laplacian on $ℝ^{k}$ . Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively)...

GCD sums from Poisson integrals and systems of dilated functions

Christoph Aistleitner, István Berkes, Kristian Seip (2015)

Journal of the European Mathematical Society

Similarity:

Upper bounds for GCD sums of the form $\sum_{k, ℓ = 1}^{N} \frac{{(\gcd (n_{k}, n_{ℓ}))}^{2 α}}{{(n_{k} n_{ℓ})}^{α}}$ are established, where ${(n_{k})}_{1 \leq k \leq N}$ is any sequence of distinct positive integers and $0 < α \leq 1$ ; the estimate for $α = 1 / 2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $α = 1 / 2$ . The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to...