Optimal constants of smoothing estimates for Dirac equations with radial data
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917665986052096 |
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| author | Ikoma, Makoto Suzuki, Soichiro |
| author_facet | Ikoma, Makoto Suzuki, Soichiro |
| contents | Kato--Yajima smoothing estimates are one of the fundamental results in study of dispersive equations such as Schrödinger equations and Dirac equations. For $d$-dimensional Schrödinger-type equations ($d \geq 2$), optimal constants of smoothing estimates were obtained by Bez--Saito--Sugimoto (2017) via the so-called Funk--Hecke theorem. Recently Ikoma (2022) considered optimal constants for $d$-dimensional Dirac equations using a similar method, and it was revealed that determining optimal constants for Dirac equations is much harder than the case of Schrödinger-type equations. Indeed, Ikoma obtained the optimal constant in the case $d = 2$, but only upper bounds (which seem not optimal) were given in other dimensions. In this paper, we give optimal constants for $d$-dimensional Schrödinger-type and Dirac equations with radial initial data for any $d \geq 2$. In addition, we also give optimal constants for the one-dimensional Schrödinger-type and Dirac equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08982 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal constants of smoothing estimates for Dirac equations with radial data Ikoma, Makoto Suzuki, Soichiro Analysis of PDEs Classical Analysis and ODEs 33C55, 35B65, 35Q41, 42B10 Kato--Yajima smoothing estimates are one of the fundamental results in study of dispersive equations such as Schrödinger equations and Dirac equations. For $d$-dimensional Schrödinger-type equations ($d \geq 2$), optimal constants of smoothing estimates were obtained by Bez--Saito--Sugimoto (2017) via the so-called Funk--Hecke theorem. Recently Ikoma (2022) considered optimal constants for $d$-dimensional Dirac equations using a similar method, and it was revealed that determining optimal constants for Dirac equations is much harder than the case of Schrödinger-type equations. Indeed, Ikoma obtained the optimal constant in the case $d = 2$, but only upper bounds (which seem not optimal) were given in other dimensions. In this paper, we give optimal constants for $d$-dimensional Schrödinger-type and Dirac equations with radial initial data for any $d \geq 2$. In addition, we also give optimal constants for the one-dimensional Schrödinger-type and Dirac equations. |
| title | Optimal constants of smoothing estimates for Dirac equations with radial data |
| topic | Analysis of PDEs Classical Analysis and ODEs 33C55, 35B65, 35Q41, 42B10 |
| url | https://arxiv.org/abs/2306.08982 |