SOBOLEV SPACES ARISING FROM A GENERALIZED SPHERICAL FOURIER TRANSFORM

2021 ◽  
Vol 10 (7) ◽  
pp. 2947-2955
Author(s):  
Yaogan Mensah
Keyword(s):  
Fourier Transform ◽  
Sobolev Spaces ◽  
Sobolev Embedding ◽  
Embedding Theorems ◽  
Locally Compact ◽  
Unimodular Group ◽  
Spherical Fourier Transform

In this paper, we define Sobolev spaces on a locally compact unimodular group in link with the spherical Fourier transform of type $\delta$. Properties of these spaces are obtained. Analogues of Sobolev embedding theorems are proved.

Weighted Sobolev spaces and exterior problems for the Helmholtz equation

1987 ◽  
Vol 410 (1839) ◽  
pp. 373-383 ◽  
Keyword(s):  
Hilbert Space ◽  
Helmholtz Equation ◽  
Sobolev Spaces ◽  
Existence And Uniqueness ◽  
Weighted Sobolev Spaces ◽  
Radiation Condition ◽  
Sobolev Embedding ◽  
Embedding Theorems ◽  
Uniqueness Of Solutions ◽  
Exterior Problems

Weighted Sobolev spaces are used to settle questions of existence and uniqueness of solutions to exterior problems for the Helmholtz equation. Furthermore, it is shown that this approach can cater for inhomogeneous terms in the problem that are only required to vanish asymptotically at infinity. In contrast to the Rellich–Sommerfeld radiation condition which, in a Hilbert space setting, requires that all radiating solutions of the Helmholtz equation should satisfy a condition of the form ( ∂ / ∂ r − i k ) u ∈ L 2 ( Ω ) , r = | x | ∈ Ω ⊂ R n , it is shown here that radiating solutions satisfy a condition of the form ( 1 + r ) − 1 2 ( ln ( e + r ) ) − 1 2 δ u ∈ L 2 ( Ω ) , 0 < δ < 1 2 , and, moreover, such solutions satisfy the classical Sommerfeld condition u = O ( r − 1 2 ( n − 1 ) ) , r → ∞ . Furthermore, the approach avoids many of the difficulties usually associated with applications of the Poincaré inequality and the Sobolev embedding theorems.

The Norm of the Lp-Fourier Transform, II

1976 ◽  
Vol 28 (6) ◽  
pp. 1121-1131 ◽  
Author(s):  
Bernard Russo
Keyword(s):  
Fourier Transform ◽  
General Theory ◽  
Measure Space ◽  
Locally Compact ◽  
Unimodular Group ◽  
Image Position

Let G be a locally compact separable unimodular group. The general theory [18] assigns to G a measure space (Λ, μ) whose points ƛ index a family of unitary factor representations of G in such a way that if U ƛ corresponds to ƛ and thenfor all .

SOBOLEV SPACES ON LOCALLY COMPACT ABELIAN GROUPS AND THE BOSONIC STRING EQUATION

2014 ◽  
Vol 98 (1) ◽  
pp. 39-53 ◽  
Author(s):  
PRZEMYSŁAW GÓRKA ◽  
ENRIQUE G. REYES
Keyword(s):  
Sobolev Spaces ◽  
Abelian Groups ◽  
Linear Equations ◽  
Bosonic String ◽  
Sobolev Embedding ◽  
String Equation ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Compactness Theorems

AbstractMotivated by a class of nonlinear nonlocal equations of interest for string theory, we introduce Sobolev spaces on arbitrary locally compact abelian groups and we examine some of their properties. Specifically, we focus on analogs of the Sobolev embedding and Rellich–Kondrachov compactness theorems. As an application, we prove the existence of continuous solutions to a generalized bosonic string equation posed on an arbitrary compact abelian group, and we also remark that our approach allows us to solve very general linear equations in a $p$-adic context.

Sobolev Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Sobolev Spaces ◽  
Embedding Theorems ◽  
Approximation Numbers ◽  
Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

scholarly journals Sobolev spaces on Lie groups: Embedding theorems and algebra properties

2019 ◽  
Vol 276 (10) ◽  
pp. 3014-3050 ◽  
Author(s):  
Tommaso Bruno ◽  
Marco M. Peloso ◽  
Anita Tabacco ◽  
Maria Vallarino
Keyword(s):  
Lie Groups ◽  
Sobolev Spaces ◽  
Embedding Theorems ◽  
And Algebra

scholarly journals DIRECTLY FINITE ALGEBRAS OF PSEUDOFUNCTIONS ON LOCALLY COMPACT GROUPS

2014 ◽  
Vol 57 (3) ◽  
pp. 693-707
Author(s):  
YEMON CHOI
Keyword(s):  
Invertible Element ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Complex Line ◽  
Locally Compact ◽  
The Real ◽  
C Algebra ◽  
Finite Algebras ◽  
Unimodular Group ◽  
Reduced Group

AbstractAn algebraAis said to be directly finite if each left-invertible element in the (conditional) unitization ofAis right invertible. We show that the reduced group C*-algebra of a unimodular group is directly finite, extending known results for the discrete case. We also investigate the corresponding problem for algebras ofp-pseudofunctions, showing that these algebras are directly finite ifGis amenable and unimodular, or unimodular with the Kunze–Stein property. An exposition is also given of how existing results from the literature imply thatL1(G) is not directly finite whenGis the affine group of either the real or complex line.

scholarly journals The algebra of functions with Fourier transforms in a given function space

1973 ◽  
Vol 9 (1) ◽  
pp. 73-82 ◽  
Author(s):  
U.B. Tewari ◽  
A.K. Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Function Space ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Algebra Of Functions

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

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