Sharpness in Young's Inequality for Convolution Products

1994 ◽  
Vol 46 (06) ◽  
pp. 1287-1298 ◽  
Author(s):  
Ole A. Nielsen
Keyword(s):  
Lie Groups ◽  
Open Subgroup ◽  
Modular Function ◽  
Convolution Product ◽  
Young’S Inequality ◽  
Solvable Lie Groups ◽  
Compact Open Subgroup ◽  
Image Position ◽  
Simply Connected ◽  
Young's Inequality

AbstractSuppose that Gis a locally compact group with modular function Δ and that p, q, r are three numbers in the interval (l,∞) satisfying. If cp,q(G) is the smallest constant c such thatfor all functions f, g ∈ Cc(G) (here the convolution product is with respect to left Haar measure andis the exponent which is conjugate to p) then Young's inequality asserts that cp,q(G) ≤ 1. This paper contains three results about these constants. Firstly, if G contains a compact open subgroup then cp,q(G) = 1 and, as an extension of an earlier result of J. J. F. Fournier, it is shown that there is a constant cp,q< 1 such that if G does not contain a compact open subgroup then c<(G) ≤ c≤. Secondly, Beckner's calculation ofis used to obtain the value of cp,q(G) for all simply-connected solvable Lie groups and all nilpotent Lie groups. And thirdly, it is shown that for a nilpotent Lie group the setis not contained in the union of the spaces Ls(G),.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals Totally disconnected locally compact groups locally of finite rank

2015 ◽  
Vol 158 (3) ◽  
pp. 505-530 ◽  
Author(s):  
PHILLIP WESOLEK
Keyword(s):  
Lie Groups ◽  
Finite Rank ◽  
Open Subgroup ◽  
Simple Groups ◽  
Locally Compact ◽  
Compact Open Subgroup ◽  
Elementary Group ◽  
Finite Set ◽  
Finite Direct Product ◽  
Totally Disconnected

AbstractWe study totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contain a compact open subgroup with finite rank. We show such groups that additionally admit a pro-π compact open subgroup for some finite set of primes π are virtually an extension of a finite direct product of topologically simple groups by an elementary group. This result, in particular, applies to l.c.s.c. p-adic Lie groups. We go on to obtain a decomposition result for all t.d.l.c.s.c. groups containing a compact open subgroup with finite rank. In the course of proving these theorems, we demonstrate independently interesting structure results for t.d.l.c.s.c. groups with a compact open pro-nilpotent subgroup and for topologically simple l.c.s.c. p-adic Lie groups.

Changes of Variables Preserving Fourier-Stieltjes Transforms on Simply Connected Nilpotent Lie Groups

1977 ◽  
Vol 29 (4) ◽  
pp. 744-755
Author(s):  
S. W. Drury
Keyword(s):  
Lie Groups ◽  
Nilpotent Lie Groups ◽  
Continuous Map ◽  
Image Position ◽  
Simply Connected ◽  
Stieltjes Transforms

In [1] Beurling and Helson prove the following theorem.THEOREM. Let φ: R → R be a continuous map such thatThen φ is affine. (Here denotes the space of Fourier-Stieltjes transforms on R.)

Restricting Representations of Completely Solvable Lie Groups

1990 ◽  
Vol 42 (5) ◽  
pp. 790-824 ◽  
Author(s):  
R. L. Lipsman
Keyword(s):  
Lie Groups ◽  
Dual Problem ◽  
Nilpotent Groups ◽  
Solvable Lie Groups ◽  
Direct Integral ◽  
Induced Representations ◽  
Simply Connected ◽  
Direct Integral Decomposition ◽  
Integral Decomposition ◽  
Restriction Problem

We are concerned here with the problem of describing the direct integral decomposition of a unitary representation obtained by restriction from a larger group. This is the dual problem to the more commonly investigated problem of decomposing induced representations. In this paper we work in the context of completely solvable Lie groups—more general than nilpotent, but less general than exponential solvable. Moreover, the groups involved are simply connected. The restriction problem was considered originally in [2] and in [6] for nilpotent groups.

scholarly journals Homogeneous Ricci solitons

2015 ◽  
Vol 2015 (699) ◽  
Author(s):  
Michael Jablonski
Keyword(s):  
Lie Groups ◽  
Ricci Flow ◽  
Isometry Group ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Solvable Lie Groups ◽  
Compact Case ◽  
Group Of Isometries ◽  
Simply Connected ◽  
Special Case

AbstractIn this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat, it is necessarily simply-connected and diffeomorphic to ℝIn the general case, we prove that homogeneous Ricci solitons must be semi-algebraic Ricci solitons in the sense that they evolve under the Ricci flow by dilation and pullback by automorphisms of the isometry group. In the special case that there exists a transitive semi-simple group of isometries on a Ricci soliton, we show that such a space is in fact Einstein. In the compact case, we produce new proof that Ricci solitons are necessarily Einstein.Lastly, we characterize solvable Lie groups which admit Ricci soliton metrics.

scholarly journals TOPOLOGICAL LOOPS HAVING SOLVABLE INDECOMPOSABLE LIE GROUPS AS THEIR MULTIPLICATION GROUPS

2020 ◽  
Author(s):  
Á. FIGULA ◽  
A. AL-ABAYECHI
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Abelian Extension ◽  
Solvable Lie Groups ◽  
Multiplication Group ◽  
Simply Connected

Abstract We prove that the solvability of the multiplication group Mult(L) of a connected simply connected topological loop L of dimension three forces that L is classically solvable. Moreover, L is congruence solvable if and only if either L has a non-discrete centre or L is an abelian extension of a normal subgroup ℝ by the 2-dimensional nonabelian Lie group or by an elementary filiform loop. We determine the structure of indecomposable solvable Lie groups which are multiplication groups of three-dimensional topological loops. We find that among the six-dimensional indecomposable solvable Lie groups having a four-dimensional nilradical there are two one-parameter families and a single Lie group which consist of the multiplication groups of the loops L. We prove that the corresponding loops are centrally nilpotent of class 2.

scholarly journals Three-dimensional solvsolitons and the minimality of the corresponding submanifolds

2017 ◽  
Vol 28 (06) ◽  
pp. 1750048 ◽  
Author(s):  
Takahiro Hashinaga ◽  
Hiroshi Tamaru
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Solvable Lie Groups ◽  
Left Invariant Riemannian Metric ◽  
Invariant Riemannian Metrics ◽  
Simply Connected

In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.

scholarly journals Topological Loops with Decomposable Solvable Multiplication Groups

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Ameer Al-Abayechi ◽  
Ágota Figula
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Solvable Lie Groups ◽  
3 Dimensional ◽  
Multiplication Group ◽  
Simply Connected ◽  
Solvable Lie Group

AbstractIn this paper we deal with the class $$\mathcal {C}$$ C of decomposable solvable Lie groups having dimension six. We determine those Lie groups in $$\mathcal {C}$$ C and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.

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