Equivalence of Sobolev norms in Lebesgue spaces for Hardy operators in a half-space
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908971332272128 |
|---|---|
| author | Bui, The Anh Merz, Konstantin |
| author_facet | Bui, The Anh Merz, Konstantin |
| contents | We consider Hardy operators, i.e., homogeneous Schrödinger operators consisting of the ordinary or fractional Laplacian in a half-space plus a potential, which only depends on the appropriate power of the distance to the boundary of the half-space. We compare the scales of homogeneous $L^p$-Sobolev spaces generated by these Hardy operators with and without potential with each other. To that end, we prove and use new square function estimates for operators with slowly decaying heat kernels. Our results hold for all admissible coupling constants in the local case and for repulsive potentials in the fractional case, and extend those obtained recently in $L^2$. They also cover attractive potentials in the fractional case, once expected heat kernel estimates are available. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02928 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Equivalence of Sobolev norms in Lebesgue spaces for Hardy operators in a half-space Bui, The Anh Merz, Konstantin Analysis of PDEs Classical Analysis and ODEs Functional Analysis We consider Hardy operators, i.e., homogeneous Schrödinger operators consisting of the ordinary or fractional Laplacian in a half-space plus a potential, which only depends on the appropriate power of the distance to the boundary of the half-space. We compare the scales of homogeneous $L^p$-Sobolev spaces generated by these Hardy operators with and without potential with each other. To that end, we prove and use new square function estimates for operators with slowly decaying heat kernels. Our results hold for all admissible coupling constants in the local case and for repulsive potentials in the fractional case, and extend those obtained recently in $L^2$. They also cover attractive potentials in the fractional case, once expected heat kernel estimates are available. |
| title | Equivalence of Sobolev norms in Lebesgue spaces for Hardy operators in a half-space |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2309.02928 |