Spectral asymptotic formula of Bessel--Riesz commutator
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917845182447616 |
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| author | Fan, Zhijie Li, Ji Sukochev, Fedor Zanin, Dmitriy |
| author_facet | Fan, Zhijie Li, Ji Sukochev, Fedor Zanin, Dmitriy |
| contents | Let $R_{λ,j}$ be the $j$-th Bessel--Riesz transform, where $n\geq 1$, $λ>0$, and $j=1,\ldots,n+1$. In this article, we establish a Weyl type asymptotic for $[M_f,R_{λ,j}]$, the commutator of $R_{λ,j}$ with multiplication operator $M_f$, based on building a preliminary result that the endpoint weak Schatten norm of $[M_f,R_{λ,j}]$ can be characterised via homogeneous Sobolev norm $\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})$ of the symbol $f$. Specifically, the asymptotic coefficient is equivalent to $\|f\|_{\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})}.$ Our main strategy is to relate Bessel--Riesz commutator to classical Riesz commutator via Schur multipliers, and then to establish the boundedness of Schur multipliers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14928 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral asymptotic formula of Bessel--Riesz commutator Fan, Zhijie Li, Ji Sukochev, Fedor Zanin, Dmitriy Functional Analysis 47B10, 42B20, 43A85 Let $R_{λ,j}$ be the $j$-th Bessel--Riesz transform, where $n\geq 1$, $λ>0$, and $j=1,\ldots,n+1$. In this article, we establish a Weyl type asymptotic for $[M_f,R_{λ,j}]$, the commutator of $R_{λ,j}$ with multiplication operator $M_f$, based on building a preliminary result that the endpoint weak Schatten norm of $[M_f,R_{λ,j}]$ can be characterised via homogeneous Sobolev norm $\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})$ of the symbol $f$. Specifically, the asymptotic coefficient is equivalent to $\|f\|_{\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})}.$ Our main strategy is to relate Bessel--Riesz commutator to classical Riesz commutator via Schur multipliers, and then to establish the boundedness of Schur multipliers. |
| title | Spectral asymptotic formula of Bessel--Riesz commutator |
| topic | Functional Analysis 47B10, 42B20, 43A85 |
| url | https://arxiv.org/abs/2411.14928 |