arXiv subjects

Prerna Gulia

Publications and source records attributed to Prerna Gulia.

5 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↗

arXiv2026

Boundedness of Fourier Multipliers and Applications to Nonlinear PDEs for the Strichartz Fourier Transform on the Heisenberg Group

We investigate Fourier multipliers associated with the Strichartz Fourier transform on the Heisenberg group. In particular, we establish H\"ormander-type $L^{p}-L^{q}$ boundedness results for the range $1<p\leq 2\leq q<\infty$. The analysis is based on deriving suitable analogues of the Hausdorff-Young and Paley inequalities for the Strichartz Fourier transform, followed by interpolation arguments to obtain the desired multiplier estimates. As an application, we study the local well-posedness of certain nonlinear partial differential equations. Furthermore, we establish an $L^{p}$-boundedness theorem for Fourier multipliers associated with the Strichartz Fourier transform for the full range $1<p<\infty$.

math.FA↗

arXiv2026

Weighted Norm Inequalities for the Strichartz Fourier transform on the Heisenberg Group

In this article, we establish an analogue of Pitt's inequality for the Strichartz Fourier transform on the Heisenberg group $\mathbb{H}^n$. By exploiting the scalar-valued formulation of the transform and the framework of decreasing rearrangements, we derive weighted $L^p$-$L^q$ estimates of Pitt type. In particular, we obtain sufficient conditions for the validity of such inequalities via weighted Hardy inequalities and Calder\'{o}n's interpolation method, and we also prove necessary conditions in the case of radial weights, using structural properties of Laguerre functions and zeros of Bessel function. As an application, we deduce an uncertainty principle of Heisenberg-Pauli-Weyl type in this setting and establish a Paley inequality for the Strichartz Fourier transform. We also derive Pitt's inequality using Hardy's inequality for the case $p=q=2$. These results extend the classical Euclidean theory of Pitt's inequality to the non-commutative, nilpotent setting of $\mathbb{H}^n$ for the sub-Laplacian and conformal Laplacian. Here we highlight the role of Laguerre functions in harmonic analysis on the Heisenberg group.

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

Some Versions of Beurling's Theorem on H-type Groups

We prove an analogue of Beurling's theorem on the H-type groups of certain dimensions after establishing the Gutzmer's formula for the H-type groups. We also obtain some other versions of the theorem using the modified Radon transform.

math.FA↗