scholarly journals A generalization of the Kobayashi–Oshima uniformly bounded multiplicity theorem

2021 ◽  
Author(s):  
Taito Tauchi
Keyword(s):  
Parabolic Subgroup ◽  
Closed Subgroup ◽  
Regular Representation ◽  
Borel Subgroup ◽  
Uniform Boundedness ◽  
Representation Formula ◽  
Group Invariant ◽  
Multiplicity Theorem ◽  
Minimal Parabolic Subgroup ◽  
Uniformly Bounded

Let [Formula: see text] be a minimal parabolic subgroup of a real reductive Lie group [Formula: see text] and [Formula: see text] a closed subgroup of [Formula: see text]. Then it is proved by Kobayashi and Oshima that the regular representation [Formula: see text] contains each irreducible representation of [Formula: see text] at most finitely many times if the number of [Formula: see text]-orbits on [Formula: see text] is finite. Moreover, they also proved that the multiplicities are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite, where [Formula: see text] are complexifications of [Formula: see text], respectively, and [Formula: see text] is a Borel subgroup of [Formula: see text]. In this paper, we prove that the multiplicities of the representations of [Formula: see text] induced from a parabolic subgroup [Formula: see text] in the regular representation on [Formula: see text] are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite. For the proof of this claim, we also show the uniform boundedness of the dimensions of the spaces of group invariant hyperfunctions using the theory of holonomic [Formula: see text]-modules.

scholarly journals Uniform boundedness principles for regular Borel vector measures

1980 ◽  
Vol 29 (2) ◽  
pp. 206-218 ◽  
Author(s):  
Joseph Kupka
Keyword(s):  
Banach Space ◽  
Borel Measure ◽  
Borel Subset ◽  
Uniform Boundedness ◽  
Vector Measures ◽  
Boundedness Theorem ◽  
Regular Borel Measure ◽  
Pointwise Limit ◽  
Regular Open ◽  
Uniformly Bounded

The setting is a compact Hausfroff space ω. The notion of a Walls class of subsets of Ω is defined via strange axioms—axioms whose justification rests with examples such as the collection of regular open sets or the range of a strong lifting. Avarient of Rosenthal' famous lwmma which applies directly to Banach space-valued measures is esablished, and it is used to obtain, in elementary fashion, the following two uniform boundedness principles: (1)The Nikodym Boundedness Theorem. If K is a family of regular Borel vector measures on Ω which is point-wise bounded on every set of a fixed Wells class, then K is uniformly bounded. (2)The Nikodym Covergence Theorem. If {μn} is a sequence of regular Borel vector measures on Ω which is converguent on every set of a fixed Wells class, then the μn are uniformly countably additive, the sequence {μn} is convergent on every Borel subset of Ω and the pointwise limit constitutes a regular Borel measure.

scholarly journals Existence and uniform boundedness of strong solutions of the time-dependent Ginzburg-Landau equations of superconductivity

2005 ◽  
Vol 2005 (8) ◽  
pp. 863-887
Author(s):  
Fouzi Zaouch
Keyword(s):  
Magnetic Field ◽  
Existence And Uniqueness ◽  
Uniform Boundedness ◽  
Strong Solutions ◽  
Time Dependent ◽  
Dynamical Process ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equations ◽  
Uniformly Bounded ◽  
Weak And Strong Solutions

The time-dependent Ginzburg-Landau equations of superconductivity with a time-dependent magnetic fieldHare discussed. We prove existence and uniqueness of weak and strong solutions withH1-initial data. The result is obtained under the “φ=−ω(∇⋅A)” gauge withω>0. These solutions generate a dynamical process and are uniformly bounded in time.

scholarly journals The local structure theorem for real spherical varieties

2015 ◽  
Vol 151 (11) ◽  
pp. 2145-2159 ◽  
Author(s):  
Friedrich Knop ◽  
Bernhard Krötz ◽  
Henrik Schlichtkrull
Keyword(s):  
Homogeneous Space ◽  
Parabolic Subgroup ◽  
Local Structure ◽  
Structure Theorem ◽  
Reductive Group ◽  
Spherical Varieties ◽  
Open Orbit ◽  
Levi Decomposition ◽  
Minimal Parabolic Subgroup

Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$-orbit with a bundle $Q\times _{L}S$. Here $Q$ is a parabolic subgroup with Levi decomposition $L\ltimes U$, and $S$ is a homogeneous space for a quotient $D=L/L_{n}$ of $L$, where $L_{n}\subseteq L$ is normal, such that $D$ is compact modulo center.

scholarly journals Uniform boundedness of Kantorovich operators in variable exponent Lebesgue spaces

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 5755-5765 ◽  
Author(s):  
Rabil Ayazoglu ◽  
Sezgin Akbulut ◽  
Ebubekir Akkoyunlu
Keyword(s):  
Variable Exponent ◽  
Closed Interval ◽  
Uniform Boundedness ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
The Difference ◽  
Upper Estimate ◽  
Kantorovich Operators ◽  
Uniformly Bounded

In this paper, the Kantorovich operators Kn, n ? N are shown to be uniformly bounded in variable exponent Lebesgue spaces on the closed interval [0; 1]. Also an upper estimate is obtained for the difference Kn(f)-f for functions f of regularity of order 1 and 2 measured in variable exponent Lebesgue spaces, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.

scholarly journals Fourier algebras of parabolic subgroups

2017 ◽  
Vol 120 (2) ◽  
pp. 272 ◽  
Author(s):  
Søren Knudby
Keyword(s):  
Regular Representation ◽  
Sufficient Conditions ◽  
Locally Compact Group ◽  
Irreducible Representations ◽  
Fourier Algebra ◽  
Real Rank ◽  
Parabolic Subgroups ◽  
Direct Sum ◽  
Rank One ◽  
Minimal Parabolic Subgroup

We study the following question: given a locally compact group when does its Fourier algebra coincide with the subalgebra of the Fourier-Stieltjes algebra consisting of functions vanishing at infinity? We provide sufficient conditions for this to be the case.As an application, we show that when $P$ is the minimal parabolic subgroup in one of the classical simple Lie groups of real rank one or the exceptional such group, then the Fourier algebra of $P$ coincides with the subalgebra of the Fourier-Stieltjes algebra of $P$ consisting of functions vanishing at infinity. In particular, the regular representation of $P$ decomposes as a direct sum of irreducible representations although $P$ is not compact.

The form of locally defined operators in waterman spaces

2021 ◽  
Vol 71 (6) ◽  
pp. 1529-1544
Author(s):  
Małgorzata Wróbel
Keyword(s):  
Banach Spaces ◽  
Bounded Variation ◽  
Composition Operators ◽  
Representation Formula ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
Spaces Of Continuous Functions ◽  
Local Operators ◽  
Uniformly Bounded

Abstract A representation formula for locally defined operators acting between Banach spaces of continuous functions of bounded variation in the Waterman sense is presented. Moreover, the Nemytskij composition operators will be investigated and some consequences for locally bounded as well as uniformly bounded local operators will be given.

scholarly journals Characteristic subgroup lattices and Hopf–Galois structures

2019 ◽  
Vol 29 (02) ◽  
pp. 391-405
Author(s):  
Timothy Kohl
Keyword(s):  
Hopf Algebras ◽  
Regular Representation ◽  
Additive Group ◽  
Representation Formula ◽  
Characteristic Subgroup ◽  
Abstract Group ◽  
Left Regular Representation ◽  
Galois Correspondence ◽  
Left Regular ◽  
Subgroup Lattices

The Hopf–Galois structures on normal field extensions [Formula: see text] with [Formula: see text] are in one-to-one correspondence with the set of regular subgroups [Formula: see text] of [Formula: see text], the group of permutations of [Formula: see text] as a set, that are normalized by the left regular representation [Formula: see text]. Each such [Formula: see text] corresponds to a Hopf algebra [Formula: see text] that acts on [Formula: see text]. Such regular subgroups need not be isomorphic to [Formula: see text] but must have the same order. One can divide all such [Formula: see text] into collections [Formula: see text], where [Formula: see text] is the set of those regular [Formula: see text] normalized by [Formula: see text] and isomorphic to a given abstract group [Formula: see text], where [Formula: see text]. There exists an injective correspondence between the characteristic subgroups of a given [Formula: see text] and the set of subgroups of [Formula: see text] stemming from the Galois correspondence between sub-Hopf algebras of [Formula: see text] and intermediate fields [Formula: see text], where [Formula: see text]. We utilize this correspondence to show that for certain pairs [Formula: see text], the collection [Formula: see text] must be empty. This also shows that for these [Formula: see text], there do not exist skew braces with additive group isomorphic to [Formula: see text] and circle group isomorphic to [Formula: see text].

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