On octonionic Monge-Ampère equation and pluripotential theory associated to octonionic plurisubharmonic functions of two variables
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912820231143424 |
|---|---|
| author | Wang, Wei |
| author_facet | Wang, Wei |
| contents | Several aspects of pluripotential theory are generalized to octonionic plurisubharmonic (OPSH) functions of two variables. We prove the comparison principle for continuous OPSH functions and the quasicontinuity of locally bounded ones. An important tool is a formula of integration by parts for mixed octonionic Monge-Ampère operator. Various useful properties of octonionic relative extremal functions and octonionic capacity are established. The main difficulty is the non-associativity of octonions. However, some weak form of associativity can be used to covercome this difficulty. Another important ingredient in pluripotential theory is the solution to the Dirichlet problem for the homogeneous octonionic Monge-Ampère equation on the unit ball, for which we show the $C_{loc}^{1,1}$-regularity by applying Bedford-Taylor's method. The obstacle to do so is that an OPSH function is usually not OPSH under automorphisms of the unit ball. This issue can be solved by finding a weighted transformation formula of OPSH functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08437 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On octonionic Monge-Ampère equation and pluripotential theory associated to octonionic plurisubharmonic functions of two variables Wang, Wei Complex Variables Analysis of PDEs 32W20, 32U15 Several aspects of pluripotential theory are generalized to octonionic plurisubharmonic (OPSH) functions of two variables. We prove the comparison principle for continuous OPSH functions and the quasicontinuity of locally bounded ones. An important tool is a formula of integration by parts for mixed octonionic Monge-Ampère operator. Various useful properties of octonionic relative extremal functions and octonionic capacity are established. The main difficulty is the non-associativity of octonions. However, some weak form of associativity can be used to covercome this difficulty. Another important ingredient in pluripotential theory is the solution to the Dirichlet problem for the homogeneous octonionic Monge-Ampère equation on the unit ball, for which we show the $C_{loc}^{1,1}$-regularity by applying Bedford-Taylor's method. The obstacle to do so is that an OPSH function is usually not OPSH under automorphisms of the unit ball. This issue can be solved by finding a weighted transformation formula of OPSH functions. |
| title | On octonionic Monge-Ampère equation and pluripotential theory associated to octonionic plurisubharmonic functions of two variables |
| topic | Complex Variables Analysis of PDEs 32W20, 32U15 |
| url | https://arxiv.org/abs/2601.08437 |