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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.14496 |
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| _version_ | 1866908968028209152 |
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| author | González-Cervantes, J. O. González-Campos, D. Bory-Reyes, J. |
| author_facet | González-Cervantes, J. O. González-Campos, D. Bory-Reyes, J. |
| contents | The theory of the operator $$G(x) = |\underline{x}|^2 \frac{\partial }{\partial x_0} + \underline{x} \sum_{j=1}^n x_j \frac{\partial }{\partial x_j} $$
is deeply associated with the slice monogenic function theory and has grown in recent years. In particular, for $n=3$ the quaternionic version of $G$ has been recently used to study the quaternionic slice regular function theory. This work extends the study of the $G$ operator in two senses: a) Clifford's analysis structure. The function theory induced by the operator \begin{align*}\mathcal H_a (x) = {\underline a} ( {x}) \frac{\partial }{\partial x_0} - \sum_{i=1}^n \left( \sum_{j=1}^n a_j ( {x}) \frac{\partial (a^{-1})_i}{\partial y_j}\circ a ( {x}) \right) \frac{\partial}{\partial x_i}, \end{align*} where $a$ is a function with certain properties with domain in $\mathbb R^{n+1}$ is presented extending the already known results of the $G$. Also some properties of the material derivative are presented as consequences of function theory induced by $\mathcal H_a$.
b) Structure of quaternionic analysis. In particular, the case $n=3$ is approached from the point of view of quaternionic analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_14496 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some global operators and the material derivative González-Cervantes, J. O. González-Campos, D. Bory-Reyes, J. Complex Variables Primary 47S05. Secondary 46S05, 30G35 The theory of the operator $$G(x) = |\underline{x}|^2 \frac{\partial }{\partial x_0} + \underline{x} \sum_{j=1}^n x_j \frac{\partial }{\partial x_j} $$ is deeply associated with the slice monogenic function theory and has grown in recent years. In particular, for $n=3$ the quaternionic version of $G$ has been recently used to study the quaternionic slice regular function theory. This work extends the study of the $G$ operator in two senses: a) Clifford's analysis structure. The function theory induced by the operator \begin{align*}\mathcal H_a (x) = {\underline a} ( {x}) \frac{\partial }{\partial x_0} - \sum_{i=1}^n \left( \sum_{j=1}^n a_j ( {x}) \frac{\partial (a^{-1})_i}{\partial y_j}\circ a ( {x}) \right) \frac{\partial}{\partial x_i}, \end{align*} where $a$ is a function with certain properties with domain in $\mathbb R^{n+1}$ is presented extending the already known results of the $G$. Also some properties of the material derivative are presented as consequences of function theory induced by $\mathcal H_a$. b) Structure of quaternionic analysis. In particular, the case $n=3$ is approached from the point of view of quaternionic analysis. |
| title | Some global operators and the material derivative |
| topic | Complex Variables Primary 47S05. Secondary 46S05, 30G35 |
| url | https://arxiv.org/abs/2604.14496 |