Sharp $\mathrm{L}^\infty$ estimates for fully non-linear elliptic equations on compact complex manifolds
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913584320086016 |
|---|---|
| author | Qiao, Yuxiang |
| author_facet | Qiao, Yuxiang |
| contents | We study the sharp $\mathrm{L}^\infty$ estimates for fully non-linear elliptic equations on compact complex manifolds. For the case of Kähler manifolds, we prove that the oscillation of any admissible solution to a degenerate fully non-linear elliptic equation satisfying several structural conditions can be controlled by the $\mathrm{L}^1(\log\mathrm{L})^n(\log\log\mathrm{L})^r(r>n)$ norm of the right-hand function (in a regularized form). This result improves that of Guo-Phong-Tong. In addition to their method of comparison with auxiliary complex Monge-Ampère equations, our proof relies on an inequality of Hölder-Young type and an iteration lemma of De Giorgi type. For the case of Hermitian manifolds with non-degenerate background metrics, we prove a similar $\mathrm{L}^\infty$ estimate which improves that of Guo-Phong. An explicit example is constucted to show that the $\mathrm{L}^\infty$ estimates given here may fail when $r\leqslant n-1$. The construction relies on a gluing lemma of smooth, radial, strictly plurisubharmonic functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05157 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp $\mathrm{L}^\infty$ estimates for fully non-linear elliptic equations on compact complex manifolds Qiao, Yuxiang Analysis of PDEs Differential Geometry 32Q99, 35J60 (Primary) 32Q15, 35J70 (Secondary) We study the sharp $\mathrm{L}^\infty$ estimates for fully non-linear elliptic equations on compact complex manifolds. For the case of Kähler manifolds, we prove that the oscillation of any admissible solution to a degenerate fully non-linear elliptic equation satisfying several structural conditions can be controlled by the $\mathrm{L}^1(\log\mathrm{L})^n(\log\log\mathrm{L})^r(r>n)$ norm of the right-hand function (in a regularized form). This result improves that of Guo-Phong-Tong. In addition to their method of comparison with auxiliary complex Monge-Ampère equations, our proof relies on an inequality of Hölder-Young type and an iteration lemma of De Giorgi type. For the case of Hermitian manifolds with non-degenerate background metrics, we prove a similar $\mathrm{L}^\infty$ estimate which improves that of Guo-Phong. An explicit example is constucted to show that the $\mathrm{L}^\infty$ estimates given here may fail when $r\leqslant n-1$. The construction relies on a gluing lemma of smooth, radial, strictly plurisubharmonic functions. |
| title | Sharp $\mathrm{L}^\infty$ estimates for fully non-linear elliptic equations on compact complex manifolds |
| topic | Analysis of PDEs Differential Geometry 32Q99, 35J60 (Primary) 32Q15, 35J70 (Secondary) |
| url | https://arxiv.org/abs/2409.05157 |