$L^{p}$ bounds for spectral multipliers on rank one NA-groups with roots not all positive

Emilie David-Guillou

Displaying similar documents to “ $L^{p}$ bounds for spectral multipliers on rank one NA-groups with roots not all positive”

Transferring $L^{p}$ eigenfunction bounds from $S^{2 n + 1}$ to hⁿ

Valentina Casarino, Paolo Ciatti (2009)

Studia Mathematica

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By using the notion of contraction of Lie groups, we transfer $L^{p} - L ²$ estimates for joint spectral projectors from the unit complex sphere $S^{2 n + 1}$ in $ℂ^{n + 1}$ to the reduced Heisenberg group hⁿ. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on hⁿ. As a consequence, we prove, in the spirit of Sogge’s work, a discrete restriction theorem for the sub-Laplacian L on hⁿ.

The 4-string braid group $B_{4}$ has property RD and exponential mesoscopic rank

Sylvain Barré, Mikaël Pichot (2011)

Bulletin de la Société Mathématique de France

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We prove that the braid group $B_{4}$ on 4 strings, its central quotient $B_{4} / 〈 z 〉$ , and the automorphism group $Aut (F_{2})$ of the free group $F_{2}$ on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group $B_{4}$ is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

$D_{p} (3)$ (p ≥ 5) can be characterized by its order components

Huaguo Shi, Zhangjia Han, Guiyun Chen (2012)

Colloquium Mathematicae

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Let G be a finite group, and $M = D_{p} (3)$ (p ≥ 3). It is proved that G ≅ M if G and M have the same order components.

Groups of given intermediate word growth

Laurent Bartholdi, Anna Erschler (2014)

Annales de l’institut Fourier

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We show that there exists a finitely generated group of growth $\sim f$ for all functions $f : ℝ_{+} \to ℝ_{+}$ satisfying $f (2 R) \leq f {(R)}^{2} \leq f (η_{+} R)$ for all $R$ large enough and $η_{+} \approx 2.4675$ the positive root of $X^{3} - X^{2} - 2 X - 4$ . Set $α_{-} = log 2 / log η_{+} \approx 0.7674$ ; then all functions that grow uniformly faster than $exp (R^{α_{-}})$ are realizable as the growth of a group. We also give a family of sum-contracting branched groups of growth $\sim exp (R^{α})$ for a dense set of $α \in [α_{-}, 1]$ .

$L^{p}$ spectral multipliers on the free group $N_{3, 2}$

Alessio Martini, Detlef Müller (2013)

Studia Mathematica

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Let L be a homogeneous sublaplacian on the 6-dimensional free 2-step nilpotent Lie group $N_{3, 2}$ on three generators. We prove a theorem of Mikhlin-Hörmander type for the functional calculus of L, where the order of differentiability s > 6/2 is required on the multiplier.

Some results on $A_{p} (G)$ submodules of $P M_{p} (G)$

Edmond E. Granirer (1987)

Colloquium Mathematicae

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An $M_{q} ()$ -functional calculus for power-bounded operators on certain UMD spaces

Earl Berkson, T. A. Gillespie (2005)

Studia Mathematica

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For 1 ≤ q < ∞, let $_{q} ()$ denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes $_{q} ()$ as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction...

$C^{r}$ -solutions of a system of functional equations

Z. Kominek (1974)

Annales Polonici Mathematici

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On the functional equation $f (x) + \sum_{i = 1}^{n} g_{i} (y_{i}) = h (T (x, y_{1}, y_{2}, . . ., y_{n}))$

C. T. Ng (1973)

Annales Polonici Mathematici

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Completely bounded lacunary sets for compact non-abelian groups

Kathryn Hare, Parasar Mohanty (2015)

Studia Mathematica

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In this paper, we introduce and study the notion of completely bounded $Λ_{p}$ sets ( $Λ_{p}^{c b}$ for short) for compact, non-abelian groups G. We characterize $Λ_{p}^{c b}$ sets in terms of completely bounded $L^{p} (G)$ multipliers. We prove that when G is an infinite product of special unitary groups of arbitrarily large dimension, there are sets consisting of representations of unbounded degree that are $Λ_{p}$ sets for all p < ∞, but are not $Λ_{p}^{c b}$ for any p ≥ 4. This is done by showing that the space of completely bounded $L^{p} (G)$ ...

Algebras whose groups of units are Lie groups

Helge Glöckner (2002)

Studia Mathematica

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Let A be a locally convex, unital topological algebra whose group of units $A^{\times}$ is open and such that inversion $ι : A^{\times} \to A^{\times}$ is continuous. Then inversion is analytic, and thus $A^{\times}$ is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then $A^{\times}$ has a locally diffeomorphic exponential function and multiplication is given locally by the Baker-Campbell-Hausdorff series. In contrast, for suitable non-Mackey complete A, the unit group $A^{\times}$ is an analytic Lie...

Deformation theory and finite simple quotients of triangle groups I

Michael Larsen, Alexander Lubotzky, Claude Marion (2014)

Journal of the European Mathematical Society

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Let $2 \leq a \leq b \leq c \in ℕ$ with $μ = 1 / a + 1 / b + 1 / c < 1$ and let $T = T_{a, b, c} = 〈 x, y, z : x^{a} = y^{b} = z^{c} = x y z = 1 〉$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$ ? (Classically, for $(a, b, c) = (2, 3, 7)$ and more recently also for general $(a, b, c)$ .) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially...

On the Functional Equation $f (λ) + f (ω λ) f (ω^{- 1} λ) = 1$ , $(ω^{5} = 1)$

Yasutaka Sibuya (1984)

Recherche Coopérative sur Programme n°25

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On some symmetrical equations of the form $f_{1} (x_{1} + . . . + x_{n}) = \sum_{i_{1}, . . ., i_{n}}) f_{1} (x_{i} 1) . . . f_{n} (x_{i} n)$

H. Światak (1967)

Annales Polonici Mathematici

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Random walks on finite rank solvable groups

Ch. Pittet, Laurent Saloff-Coste (2003)

Journal of the European Mathematical Society

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We establish the lower bound $p_{2 t} (e, e) ≿ exp (- t^{1 / 3})$ , for the large times asymptotic behaviours of the probabilities $p_{2 t} (e, e)$ of return to the origin at even times $2 t$ , for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer $r$ , such that any of its finitely generated subgroup admits a generating set of cardinality less or equal to $r$ .)

On the solutions of the functional equation $ϕ (x) + ϕ^{2} (x) = F (x)$

M. Malenica (1982)

Matematički Vesnik

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The density of representation degrees

Martin Liebeck, Dan Segal, Aner Shalev (2012)

Journal of the European Mathematical Society

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For a group $G$ and a positive real number $x$ , define $d_{G} (x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$ . We study the asymptotics of $d_{G} (x)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $α > 0$ such that $d_{G} (x) > x^{α}$ for all large $x$ , or $G$ is virtually abelian (in which case $d_{G} (x)$ is bounded). ...

Subalgebra of $L_{1} (G)$ associated with Laplacian on a Lie group

A. Hulanicki (1974)

Colloquium Mathematicae

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On the structural theory of ${II}_{1}$ factors of negatively curved groups

Ionut Chifan, Thomas Sinclair (2013)

Annales scientifiques de l'École Normale Supérieure

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Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor $L Γ$ is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that $L Γ$ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in $Sp (n, 1)$ , $n \geq 2$ , are virtually $W^{*}$ -superrigid.

On a translation property of positive definite functions

Lars Omlor, Michael Leinert (2010)

Banach Center Publications

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If G is a locally compact group with a compact invariant neighbourhood of the identity e, the following property (*) holds: For every continuous positive definite function h≥ 0 with compact support there is a constant $C_{h} > 0$ such that $\int L_{x} h \cdot g \leq C_{h} \int h g$ for every continuous positive definite g≥0, where $L_{x}$ is left translation by x. In [L], property (*) was stated, but the above inequality was proved for special h only. That “for one h” implies “for all h” seemed obvious, but turned out not to be obvious at...

On the Davenport constant and group algebras

Daniel Smertnig (2010)

Colloquium Mathematicae

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For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S = g ₁ \cdot . . . \cdot g_{l}$ over G such that $(X^{g ₁} - a ₁) \cdot . . . \cdot (X^{g_{l}} - a_{l}) \neq 0 \in K [G]$ for all $a ₁, . . ., a_{l} \in K^{\times}$ . If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for...

Coxeter group actions on the complement of hyperplanes and special involutions

Giovanni Felder, A. Veselov (2005)

Journal of the European Mathematical Society

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We consider both standard and twisted actions of a (real) Coxeter group $G$ on the complement $ℳ_{G}$ to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in $G$ and give explicit formulae which describe both actions on the total cohomology $H^{*} (ℳ_{G}, 𝒞)$ in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group $S_{n}$ , the...

A.e. convergence of anisotropic partial Fourier integrals on Euclidean spaces and Heisenberg groups

D. Müller, E. Prestini (2010)

Colloquium Mathematicae

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We define partial spectral integrals $S_{R}$ on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by $l o g (2 + L_{α}) f \in L ²$ , where $L_{α}$ is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that $S_{R} f (x)$ converges a.e. to f(x) as R → ∞.

Limits of relatively hyperbolic groups and Lyndon’s completions

Olga Kharlampovich, Alexei Myasnikov (2012)

Journal of the European Mathematical Society

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We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon’s completion $G^{ℤ [t]}$ of the group $G$ , or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{ℤ [t]}$ containing $G$ is universally equivalent to $G$ . Since finitely...

A Non Solvable Operator Satisfying Condition $(ψ)$

Nicolas Lerner (1992)

Publications mathématiques et informatique de Rennes

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A complete analogue of Hardy's theorem on semisimple Lie groups

Rudra P. Sarkar (2002)

Colloquium Mathematicae

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A result by G. H. Hardy ([11]) says that if f and its Fourier transform f̂ are $O (| x |^{m} e^{- α x ²})$ and $O (| x | ⁿ e^{- x ² / (4 α)})$ respectively for some m,n ≥ 0 and α > 0, then f and f̂ are $P (x) e^{- α x ²}$ and $P^{'} (x) e^{- x ² / (4 α)}$ respectively for some polynomials P and P’. If in particular f is as above, but f̂ is $o (e^{- x ² / (4 α)})$ , then f = 0. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.

Spectral radius of operators associated with dynamical systems in the spaces C(X)

Krzysztof Zajkowski (2005)

Banach Center Publications

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We consider operators acting in the space C(X) (X is a compact topological space) of the form $A u (x) = (\sum_{k = 1}^{N} e^{φ_{k}} T_{α_{k}}) u (x) = \sum_{k = 1}^{N} e^{φ_{k} (x)} u (α_{k} (x))$ , u ∈ C(X), where $φ_{k} \in C (X)$ and $α_{k} : X \to X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $φ = {(φ_{k})}_{k = 1}^{N}$ . We prove that $l n (r (A)) = λ (φ) = m a x_{ν \in M e s} \sum_{k = 1}^{N} \int_{X} φ_{k} d ν_{k} - λ * (ν)$ , where Mes is the set of all probability vectors of measures $ν = {(ν_{k})}_{k = 1}^{N}$ on X × 1,..., N and λ* is some convex lower-semicontinuous functional on...

Solutions for the p-order Feigenbaum’s functional equation $h (g (x)) = g^{p} (h (x))$

Min Zhang, Jianguo Si (2014)

Annales Polonici Mathematici

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This work deals with Feigenbaum’s functional equation ⎧ $h (g (x)) = g^{p} (h (x))$ , ⎨ ⎩ g(0) = 1, -1 ≤ g(x) ≤ 1, x∈[-1,1] where p ≥ 2 is an integer, $g^{p}$ is the p-fold iteration of g, and h is a strictly monotone odd continuous function on [-1,1] with h(0) = 0 and |h(x)| < |x| (x ∈ [-1,1], x ≠ 0). Using a constructive method, we discuss the existence of continuous unimodal even solutions of the above equation.

The functional equation $f^{2} (x) = g (x)$

James C. Lillo (1967)

Annales Polonici Mathematici

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Displaying similar documents to “ L p bounds for spectral multipliers on rank one NA-groups with roots not all positive”

Displaying similar documents to “ $L^{p}$ bounds for spectral multipliers on rank one NA-groups with roots not all positive”