arXiv subjects

Rupak Kumar Dalai

Publications and source records attributed to Rupak Kumar Dalai.

3 recordsLinked to original sources

arXiv2021

Quaternion Weyl Transform and some uniqueness results

In this article, we study the boundedness and several properties of the quaternion Wigner transform. Using the quaternion Wigner transform as a tool, we define the quaternion Weyl transform (QWT) and prove that the QWT is compact for a certain class of symbols in $L^{r}\left(\mathbb{R}^{4}, \mathbb{Q}\right)$ with $1 \leq r \leq 2.$ Moreover, it can not be extended as a bounded operator for symbols in $L^{r}\left(\mathbb{R}^{4},\mathbb{Q}\right)$ for $2<r<\infty.$ In addition, we prove a rank analogue of the Benedicks-Amrein-Berthier theorem for the QWT. Further, we remark about the set of injectivity and Helgason's support theorem for the quaternion twisted spherical means.

math.FA↗

arXiv2021

Injectivity of spherical mean on Métivier Group

In this article, we study the injectivity of the spherical mean for continuous functions on the Métivier group. The spherical mean is injective for $f(z, .)\in L^p(\mathbb{R}^m),~1\leq p \leq 2$ with tempered growth in $z$ variable. This result is also true for a class of functions in $L^p(\mathbb{C}^n),\,1\leq p\leq\infty$ without tempered growth. Further, we obtain a two-radii theorem for functions on the Métivier group, which are tempered in $z$ variable and periodic in the centre variable.

math.FA↗

arXiv2021

Spherical means on Métivier groups and support theorem

Let $Z_{r, R}$ be the space of continuous functions on the annulus $B_{r, R}$ in $\mathbb C^n$ whose $λ$-twisted spherical mean, in the set up of the Métivier group, vanishes over the spheres $S_s(z)\subset B_{r, R} $ with ball $B_r(0)\subseteq B_s(z).$ We characterize the spherical harmonic coefficients of functions in $Z_{r, R},$ eventually, in terms of polynomial growth, by which we infer support theorem. Further, we prove that non-harmonic complex cone and the boundary of a bounded domain are sets of injectivity for the $λ$-twisted spherical means.

math.FA↗