On algebraic and uniqueness properties of 3d harmonic quaternion fields

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 inj…

On algebras of harmonic quaternion fields in ${\mathbb R}^3$

Let ${\mathscr A}(D)$ be an algebra of functions continuous in the disk $D=\{z\in{\mathbb C}\,|\,\,\,|z|\leqslant 1\}$ and {\it holomorphic} into $D$. The well-known fact is that the set ${\mathscr M}$ of its characters (homomorphisms ${\mathscr A}(D)\to\mathbb C$) is exhausted by the Dirac measures $\{δ_{z_0}\,|\,\,z_0\in D\}$ and a homeomorphism ${\mathscr M}\cong D$ holds. We present a 3d analog of this classical result as follows. Let $B=\{x\in{\mathbb R}^3\,|\,\,|x|\leqslant 1\}$. A quaternion field is a pair $p=\{α,u\}$ of a function $α$ and vector field $u$ in the ball $B$. A field $p$ …

Uniqueness theorems for (sub-)harmonic functions with applications to operator theory

We obtain uniqueness theorems for harmonic and subharmonic functions of a new type. They lead to new analytic extension criteria and new conditions for stability of operator semigroups in Banach spaces with Fourier type.

Self-mappings of the quaternionic unit ball: multiplier properties, Schwarz-Pick inequality, and Nevanlinna--Pick interpolation problem

We study several aspects concerning slice regular functions mapping the quaternionic open unit ball into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space. In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.

Boundary uniqueness of harmonic functions and spectral subspaces of operator groups

We obtain new uniqueness theorems for harmonic functions defined on the unit disc or in the half plane. These results are applied to obtain new resolvent descriptions of spectral subspaces of polynomially bounded groups of operators on Banach spaces.

球上$H$-调和莫比乌斯不变内积的唯一性

Uniqueness of $H$-harmonic Moebius invariant inner products on the ball

该研究证明了实n维空间单位球上双曲调和函数的莫比乌斯不变半内积具有唯一性，明确了被双曲拉普拉斯算子零化的函数空间上该内积的唯一特性。