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M. E. Egwe

Publications and source records attributed to M. E. Egwe.

18 recordsLinked to original sources
arXiv2026

Haar Measure on Fuzzy Lie Group

Let $\mathfrak{G}$ be a locally compact Lie group and \(dμ\) a Haar measure defined on $\mathfrak{G}$. We consider the existence of a fuzzy analogue of $\mathfrak{G}$ denoted by $\mathfrak{G_f}$ called a fuzzy Lie group and develop a fuzzy Haar measure \(μ_f\) on $\mathfrak{G_f}$. Also we construct a fuzzy Haar integral with respect to the corresponding fuzzy Haar measure on $\mathfrak{G_f}$. And finally we show that there exists a fuzzy Haar integral that is unique up to a multiplicative constant.

math.GM↗
arXiv2025

On Dirichlet Spaces of Homogeneous Type Via Heat Kernel

This paper considers the properties of Dirichlet Spaces of Homogeneous type which consist of band limited functions that are nearly exponential localizations on $\mathbb{R}^k.$ This is a powerful tool in harmonic analysis and it makes various spaces of functions and distributions more approachable, utilizable and providing non-zero representation of natural function spaces, such as Besov space, on $\mathbb{R}^k$. Spheres and homogeneous spaces can also admit such frames on the intervals and balls. Here, we present mainly the band limited frames that are well-localized in the general setting of Dirichlet spaces of Homogeneous type which have doubling measure and a local scale-invariant Poincare inequality which generates heat kernels through the Gaussian bounds and H$\ddot{o}$lder's continuity. As an application of this build-up, band limited frames are generated in the context of Lie groups which are homogeneous in nature with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and admits the volume doubling property, together with other settings. In this general setting, decomposition of Besov spaces was done with the new frames.

math.FA↗
arXiv2025

Partial Dynamical Systems of $L^p$-Spaces and their Stability Spaces

Using the convolution product and weak derivatives, we consider the partial dynamical systems of the locally convex $L^p(Ω)$ spaces defined by the action of the smooth algebra $\mathscr{K}(Ω)$ through its nets. Slice analysis is then employed to show that the Sobolev spaces $W^{k,p}(Ω)$ are the stable states or space of these partial dynamical systems as limit spaces of the convolution actions of the smooth algebra $K(Ω)$ on the Banach spaces $L^p(Ω)$. Thus, the Sobolev spaces $W^{k,p}(Ω)$ are closed subspaces of the $Lp(Ω)$-spaces under convolution product and weak derivatives, with the weak derivative operators acting as equivariant maps of the slice spaces.

math.FA↗
arXiv2025

Grushin Operator on Infinite Dimensional Homogeneous Lie Groups

A collection of infinite dimensional complete vector fields $\left\{V_i\right\}_{i=1}^{\infty}$ acting on a locally convex manifolds $M$ on which a smooth positive measure $μ$ is defined was considered. It was assumed that the vector fields generates an infinite dimensional Lie algebra $\mathfrak{g}$ and satisfies H$\ddot{o}$rmander's condition. The sum of squares of Grushin operators related to the vector fields was examined and the operator is then considered as the generalized Grushin operator. The paramount proofs were Poincar$\acute{e}$ inequality, Gaussian two-bounded estimate for the related heat kernels and the doubling condition for the metric defined by the underlying vector fields.

math.FA↗
arXiv2025

Counter Examples in Non-Archimedean Locally Convex Spaces

In this paper, we shall consider some counter examples in non-archimedean locally convex spaces with special closed subspaces and Schauder basis in non-archimedean Fréchet spaces as well as closed subspaces \emph{without} Schauder basis in non-archimedean Fréchet spaces.

math.FA↗
arXiv2025

Quaternionic Spherical Fourier Multipliers for Gelfand Pairs

Let $\mathbb{H}_q$ denote the quaternionic Heisenberg group with $\mathbb{R}^4\times\mathbb{R}^3$ stratification and $K$ a compact subgroup of Automorphism. We construct the spherical Fourier multiplier related to a Gelfand Pairs on the Quaternionic Heisenberg group $\mathbb{H}_q$ via the motion group $G=K\ltimes \mathbb{H}_q$ of $\mathbb{H}_q$, where $K\ltimes \mathbb{H}_q$ is the semidirect product of $\mathbb{H}_q$ and its compact subgroup of automorphism $K\in Aut(\mathbb{H}_q)$.

math.RT↗
arXiv2025

A New Spherical Harmonics on the Heisenberg Group

Let $\h_n$ be the $(2n+1)$-dimensional Heisenberg group. and let ${\cal L}_α$ be the sublaplacian of the Lie algebra of $\h_n$ A new spherical harmonics with its orthogonal polynomial properties is presented for the group.

math.RT↗
arXiv2025

Equivalence of Quaternionic Heisenberg Homogeneous Quasi-norms

Let $\mathbb{H}_q$ denote the quaternionic Heisenberg group of dimension $(4n+3)$ with $\mathbb{R}^4\times\mathbb{R}^3$ stratification. We identify certain homogeneous norms on the group and show that any two quasi-norms on $\mathbb{H}_q$ are equivalent for $n<\infty$.

math.FA↗
arXiv2025

Groupoid Characterization of Partial Algebras on Sobolev Spaces

The $L^p$-spaces, with $p \not = \infty$, form a partial algebra $(L^p(Ω), Γ, \cdot)$ with pointwise multiplication of functions. The Sobolev spaces $W^{k,p}(Ω)$, delineated by weak derivatives as subspaces of $L^p$-spaces is shown to contain the partial algebra $(L^p(Ω), Γ, \cdot)$ generalized by the partial action of the smooth algebra $\mathscr{K}(Ω)$ by convolution on the Banach spaces $L^p(Ω)$. We characterised the Sobolev space $W^{k,p}(Ω)$, invariant under $\mathscr{K}(Ω)$ partial action, using Lie groupoid framework, and study the partial algebra as defining the partial dynamical systems on the $L^p$-space associated with the weak differential operators. The locally convex partial $^*$-algebra $(L^p(Ω), Γ, \cdot,^*)$ defines the stable local flows coinciding with local bisections of the Lie groupoid. The unitary representation of resulting Lie groupoid $\mathscr{W} \rightrightarrows W^{k,p}(Ω)$ on the associated Hilbert bundle demonstrates the simplification achieved by the characterisation.

math.FA↗
arXiv2025

On Tempered Ultradistributions in Classical Sobolev Spaces

We construct and investigate the properties of tempered ultradistribution spaces in Sobolev spaces. A new Sobolev space preserving the original properties and condition whose derivatives are linear continuous operators embedding in $L^p$ for $1\leq p\leq \infty$ is characterized. Moreover, we also consider some Sobolev embedding theorems involving rapidly decreasing functions, and finally, we prove the extension of Rellich's compactness theorem.

math.FA↗
arXiv2022

Spherical Functions on Fuzzy Lie group

Let $G$ be a locally compact Lie group and $\mathfrak{g}$ its Lie algebra. We consider a fuzzy analogue of $G,$ denoted by $\mathfrak{G_f}$ called a fuzzy Lie group. Spherical functions on $\mathfrak{G_f}$ are constructed and a version of the existence result of the Helgason-spherical function on $G$ is then established on $\mathfrak{G_f}.$

math.GM↗
arXiv2021

Groupoid approach to the dynamical system of commutative von Neumann Algebras

The automorphism group $Aut(X,μ)$ of a compact, complete metric space $X$ with a Radon measure $μ$ is a subgroup of $\mathcal{U}(L^2(X,μ))$-the unitary group of operators on $L^2(X,μ)$. The $Aut(X,μ)$-action on the generalized space $\mathcal{M}(X)$ is a proper action. Hence, there exists a slice at each point of the generalized space $\mathcal{M}(X)$. Measure Groupoid (virtual group) is subsequently employed to analyze the resulting dynamical system as that of the ergodic action of the commutative algebra (a lattice) $C(X)$ on the generalized space $\mathcal{M}(X)$ which is represented on a commutative von Neumann algebra.

math.DS↗
arXiv2021

Groupoid Characterization of Locally Convex Partial $^*$-Algebras

Given a locally convex space $(\mathcal{A},τ)$ with a Hausdorff locally convex topology $τ$ such that the following maps are continuous; $u \mapsto u^*$ for all $u \in \mathcal{A}$, $x \mapsto x\cdot y$ and $x \mapsto z\cdot x$ for every left and right multipliers of $\mathcal{A}$. In this paper we re-characterized the locally convex partial $*$-algebra $(\mathcal{A}, Γ,\cdot,*,τ)$ arising from these continuous maps in terms of convolution algebra of a Lie groupoid $Γ\rightrightarrows \mathcal{A}$. This is advantageous because the pathologies of the underlying spaces owing to their quantum mechanical nature are easily resolved in groupoid terms.

math.OA↗
arXiv2021

A Fixed Point Theorem On Fuzzy Locally Convex Spaces

Let $X$ be a linear space over a field $\mathbb{K}$ and $(X, ρ, *)$ a fuzzy seminorm space where $(ρ, *)$ a fuzzy seminorm with $*$ a continuous $t$-norm. We give a fixed point theorem for Fuzzy Locally Convex Space.

math.GM↗
arXiv2021

Cohomogeneity One Groupoid Analysis of the Dynamical System of Rings of Continuous Functions

Using the group $G(1)$ of invertible elements and the maximal ideals $\mathfrak{m}_x$ of the commutative algebra $C(X)$ of real-valued functions on a compact regular space $X$, we define a Borel action of the algebra on the measure space $(X,μ)$ with $μ$ a Radon measure. The zero sets $Z(X)$ of the algebra $C(X)$ is used to study the ergodicity of the $G(1)$-action via its action on the maximal ideals $\mathfrak{m}_x$ which defines an action groupoid $\mathcal{G} = \mathfrak{m}_x \ltimes G(1)$ trivialized on $X$. The resulting measure groupoid $(\mathcal{G},\mathcal{C})$ is used to define a proper action on the generalized space $\mathcal{M}(X)$. The existence of slice at each point of $\mathcal{M}(X)$ present it as a cohomogeneity-one $\mathcal{G}$-space. The dynamical system of the algebra $C(X)$ is defined by the action of the measure groupoid $(\mathcal{G},\mathcal{C}) \times \mathcal{M}(X) \to \mathcal{M}(X)$.

math.FA↗