arXiv subjects

Arvish Dabra

Publications and source records attributed to Arvish Dabra.

9 recordsLinked to original sources

arXiv2026

Uncertainty Principles for the Short-Time Fourier Transform on the Heisenberg Group

We develop a systematic theory of uncertainty principles for the short-time Fourier transform (STFT) on the Heisenberg group. Building on recent developments in modulation spaces and time-frequency analysis on the Heisenberg group introduced by Fischer et al. and later by Biswas-Thangavelu, we establish noncommutative analogues of several fundamental uncertainty principles in time-frequency analysis, including Benedicks' theorem, the Donoho-Stark uncertainty principle, and Lieb's inequality. As a consequence of Lieb's inequality, we derive an entropy-based uncertainty principle of Hirschman type. We further establish a Heisenberg-type uncertainty inequality and local uncertainty principles in the spirit of Price. In addition, we prove a Beurling-Hardy-type theorem that captures the interplay between decay and phase-space localization. Finally, we investigate decay properties of the STFT and their implications for time-frequency concentration. These results extend a broad spectrum of classical uncertainty phenomena from the Euclidean setting to the Heisenberg group, highlighting the role of noncommutative harmonic analysis in the study of phase-space localization.

math.FA↗

arXiv2025

Uncertainty Principles for the Strichartz Fourier transform on the Heisenberg Group

In this article, we establish several fundamental uncertainty principles for the Strichartz Fourier transform on the Heisenberg group, including Benedicks' theorem, the Donoho-Stark principle, the local uncertainty principle of Price, and a weak form of Beurling's theorem. The Strichartz Fourier transform, introduced by Thangavelu (2023), provides a scalar-valued analogue of the classical operator-valued Fourier transform on the Heisenberg group. We first prove an analogue of Benedicks' theorem asserting that a nonzero function and its Strichartz Fourier transform cannot both be supported on sets of finite measure. As a consequence, we obtain Nazarov's uncertainty inequality. We then establish the Donoho-Stark principle, providing quantitative bounds on simultaneous concentration in space and frequency, and extend the local uncertainty principle of Price to this framework. Finally, we present a weak form of Beurling's theorem for radial functions on the Heisenberg group.

math.FA↗

arXiv2025

Topological Center of the Double Dual of the Orlicz Figà-Talamanca Herz Algebra

Let $G$ a locally compact group and $(Φ,Ψ)$ be a complementary pair of Young functions. Let $A_Φ(G)$ be the Orlicz analogue of the classical Figà-Talamanca Herz algebra $A_p(G).$ In this article, we establish a necessary and sufficient condition for the equality $Λ(A_Φ(G)^{\ast\ast}) = A_Φ(G)$ to hold, where $Λ(A_Φ(G)^{\ast\ast})$ denotes the topological center of the double dual of $A_Φ(G)$ when equipped with the first Arens product. Furthermore, we prove several results concerning the semi-simplicity of the Banach algebras $A_Φ(G)^{\ast\ast}$ and $UCB_Ψ(\widehat{G})^\ast.$

math.FA↗

arXiv2025

$L^p$-Heisenberg--Pauli--Weyl Uncertainty Inequalities on the Laguerre Hypergroup

In this paper, we establish the first $L^p$-Heisenberg--Pauli--Weyl uncertainty inequalities on the Laguerre hypergroup for the full range $1\le p\le2$. These results extend Xiao's Euclidean $L^p$ theory to the setting of the Laguerre hypergroup, which is the fundamental manifold of the radial function space for the Heisenberg group. The analysis is carried out through the Fourier--Laguerre transform and exploits the mixed discrete--continuous spectral structure of the Laguerre hypergroup, requiring estimates adapted to its Plancherel measure and dilation structure. As a consequence, in the endpoint case $p=2$, we obtain a refined $L^2$-Heisenberg--Pauli--Weyl uncertainty inequality valid for all positive exponents $a,b>0$, thereby improving the earlier result of Atef (2013), where the assumptions $a,b\ge1$ arose from the heat kernel methods. Our proofs rely on the Fourier--Laguerre transform, dilation and scaling invariance, the Hausdorff--Young inequality and the Plancherel identity, completely avoiding heat kernel techniques. These results provide a unified Fourier-analytic framework for Heisenberg--Pauli--Weyl uncertainty inequalities on the Laguerre hypergroup and further strengthen the connections between Euclidean, Heisenberg and hypergroup harmonic analysis.

math.FA↗

arXiv2025

Isomorphism Theorems for the Algebras of $Φ-$Pseudofunctions and $Φ-$Pseudomeasures

In this article, we study the isomorphism problem for the algebras of $Φ-$Pseudofunctions and $Φ-$Pseudomeasures, denoted by $PF_Φ(G)$ and $PM_Φ(G),$ respectively. More precisely, for a certain class of Young functions $Φ,$ we prove that if there exists an isometric isomorphism between $PF_Φ(G_1)$ and $PF_Φ(G_2),$ or between $PM_Φ(G_1)$ and $PM_Φ(G_2),$ then $G_1$ and $G_2$ are isomorphic as topological groups. In addition, we present an Orlicz version of Parrott's theorem.

math.FA↗

arXiv2025

Amenability and Invariant subspaces of the algebra of pseudomeasures

Let $G$ be a locally compact group and $(Φ,Ψ)$ a complimentary pair of Young functions. In this article, we consider the Banach algebra of $Ψ$-pseudomeasures $PM_Ψ(G)$ and the Orlicz Figà-Talamanca Herz algebra $A_Φ(G).$ We prove sufficient conditions for a group $G$ to be amenable in terms of the norm closed topologically invariant subspaces of $PM_Ψ(G).$ Further, for an amenable group $G$ with the Young function $Φ$ satisfying the MA condition, we establish a one-to-one correspondence between certain topologically invariant subalgebras of $PM_Ψ(G)$ and the class of closed subgroups of $G.$ Moreover, we prove a similar result for the predual $A_Φ(G)$ and derive a bijection between certain topologically invariant subalgebras of $A_Φ(G)$ and the set of compact subgroups of $G.$

math.FA↗

arXiv2024

Ideals in the dual of introverted subspaces of $Ψ$-pseudomeasures

Let $G$ be a locally compact group and $(Φ,Ψ)$ a complementary pair of Young functions satisfying the $Δ_2$-condition. Let $A_Φ(G)$ be the Orlicz analogue of the Figà-Talamanca Herz algebra $A_p(G).$ The dual of the algebra $A_Φ(G)$ is the space of $Ψ$-pseudomeasures, denoted by $PM_Ψ(G).$ For certain topologically introverted subspaces $\mathcal{A}$ of $PM_Ψ(G)$ and the Banach algebras $W_Φ(G)$ or $B_Φ(G),$ denoted by $\mathcal{B},$ we characterise the maximal regular left/right/two-sided ideals of the Banach algebras $\mathcal{A}^{'}$ and $\mathcal{B}^{''}$ considered with the Arens product. We further characterise the minimal left ideals of $\mathcal{A}^{'}$ and prove the necessary and sufficient conditions for the existence of minimal ideals in the algebras $A_Φ(G)$ and $\mathcal{B}.$

math.FA↗

arXiv2024

Restriction theorems for the $p$-analog of the Fourier-Stieltjes algebra

For a locally compact group $G$ and $1 < p < \infty,$ let $B_p(G)$ denote the $p$-analog of the Fourier-Stieltjes algebra $B(G) \, (\text{or} \, B_2(G))$. Let $r: B_p(G) \to B_p(H)$ be the restriction map given by $r(u) = u|_H$ for any closed subgroup $H$ of $G.$ In this article, we prove that the restriction map $r$ is a surjective isometry for any open subgroup $H$ of $G.$ Further, we show that the range of the map $r$ is dense in $B_p(H)$ when $H$ is either a compact normal subgroup of $G$ or compact subgroup of an [SIN]$_H$-group.

math.FA↗

arXiv2024

Arens regularity of the Orlicz Figà-Talamanca Herz Algebra

Let G be a locally compact group and let $A_Φ(G)$ be the Orlicz-version of the Figà-Talamanca Herz algebra of G associated with a Young function $Φ.$ We show that if $A_Φ(G)$ is Arens regular, then $G$ is discrete. We further explore the Arens regularity of $A_Φ(G)$ when the underlying group $G$ is discrete. In the running, we also show that $A_Φ(G)$ is finite-dimensional if and only if $G$ is finite. Further, for amenable groups, we show that $A_Φ(G)$ is reflexive if and only if $G$ is finite, under the assumption that the associated Young function $Φ$ satisfies the MA-condition.

math.FA↗