Identities and inequalities for integral transforms involving squares of the Bessel functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911244134383616 |
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| author | Suzuki, Soichiro |
| author_facet | Suzuki, Soichiro |
| contents | We consider an integral transform given by $T_ν f(s) := π\int_0^\infty rs J_ν(r s)^2 f(r) \, dr$, where $J_ν$ denotes the Bessel function of the first kind of order $ν$. As shown by Walther (2002, doi:10.1006/jfan.2001.3863), this transform plays an essential role in the study of optimal constants of smoothing estimates for the free Schrödinger equations on $\mathbb{R}^d$. On the other hand, Bez et al. (2015, doi:10.1016/j.aim.2015.08.025) studied these optimal constants using a different method, and obtained a certain alternative expression for $T_ν f$ involving the $d$-dimensional Fourier transform of $x \mapsto f(\lvert x \rvert)$ when $ν= k + d/2 - 1$ for $k \in \mathbb{N}$. The aims of this paper are to extend their identity for non-integer indices and to derive several inequalities from it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00137 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Identities and inequalities for integral transforms involving squares of the Bessel functions Suzuki, Soichiro Classical Analysis and ODEs Analysis of PDEs 33C10, 35B65, 35Q41, 42A38 We consider an integral transform given by $T_ν f(s) := π\int_0^\infty rs J_ν(r s)^2 f(r) \, dr$, where $J_ν$ denotes the Bessel function of the first kind of order $ν$. As shown by Walther (2002, doi:10.1006/jfan.2001.3863), this transform plays an essential role in the study of optimal constants of smoothing estimates for the free Schrödinger equations on $\mathbb{R}^d$. On the other hand, Bez et al. (2015, doi:10.1016/j.aim.2015.08.025) studied these optimal constants using a different method, and obtained a certain alternative expression for $T_ν f$ involving the $d$-dimensional Fourier transform of $x \mapsto f(\lvert x \rvert)$ when $ν= k + d/2 - 1$ for $k \in \mathbb{N}$. The aims of this paper are to extend their identity for non-integer indices and to derive several inequalities from it. |
| title | Identities and inequalities for integral transforms involving squares of the Bessel functions |
| topic | Classical Analysis and ODEs Analysis of PDEs 33C10, 35B65, 35Q41, 42A38 |
| url | https://arxiv.org/abs/2511.00137 |