International Seminar Series on Advances in Operator Theory and Inequalities #2

Let $U,V\subset \mathbb R^{n }$ be two domains and $(s,t) \subset \mathbb R$ be an open interval such that there exists a $C^1$ diffeomorphism $\underline{a} \in C^1( U, V)$. Define $a=I\times \underline{a} : (s,t)\times U \to (s,t)\times V$ given by $y= a(x)$ where $x_0 =(a(x))_0 = y_0$ and $\underline{y}= \underline{ a}( \underline{ x})$. In addition, denote $a(x) =x_0 + \sum_{i=1}^n a_{i}(\underline{x}) e_i$, $a^{-1}(y)=y_{0}+\sum_{j=1}^{n}(a^{-1})_{j}(\underline{y})e_{j}$ and
$$ H_a (x)[f] = {\underline a} (\underline{x}) \frac{\partial f }{\partial x_0} - \sum_{i=1}^n \left( \sum_{j=1}^n a_j (\underline{x}) \frac{\partial (a^{-1})_i}{\partial y_j}\circ a (\underline{x}) \right) \frac{\partial f}{\partial x_i},$$ for all $f\in C^{1} (U,R^{n})$, where $a$ is a function with certain properties with domain in $\mathbb{R}^{n+1}$. The operator $ H_{a}$ is related to the global operators $$G(x) = |\underline{x}|^2 \frac{\partial }{\partial x_0} + \underline{x} \sum_{j=1}^3 x_j \frac{\partial }{\partial x_j},$$ $$G(x) = |\underline{x}|^2 \frac{\partial }{\partial x_0} + \underline{x} \sum_{j=1}^n x_j \frac{\partial }{\partial x_j},$$ associated with the slice regular function and monogenic function theories, respectively. The operator global $G$ is associated with slice regular functions (monogenic functions) when the domain of the functions is axially symmetric, case $n=3$ ($n\neq 3$). Nowadays, thanks to the operator $G$ we have global integral theorems for slice regular and monogenic functions, a Splitting Lemma, and a Representation theorem. We note that if $a=I$ is the identity function, we have that $G=\underline{x}H_{I}$. In addition, the material derivative $$D_u=\frac{\partial}{\partial x_0}+\sum_{j=1}^{n} \frac{\partial}{\partial x_j}$$ where $u$ is a certain $\mathbb{R}^{n}$-valued function with domains in $\mathbb{R}^{n}$ or $\mathbb{R}^{n+1}$ and has good relationship with the global operator $ H_a$. For example, the 4-dimensional incompressible Navier-Stokes equation and the material derivative and Euler equation of gas-dynamics can be written in terms of the global operator $H_a$. For the previous reasons, the function theory induced by $H_a$ is presented, extending the already known results of the operator $G$, such as the Splitting Lemma, Representation Theorem, Cauchy formula, a characterization of the zero sets of $H_a$, and a development in power series. The principal results, about the global operator $H_{a}$ and the multidimensional material derivative, were proved in https://doi.org/10.48550/arXiv.2604.14496.