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Soichiro Suzuki

Publications and source records attributed to Soichiro Suzuki.

9 recordsLinked to original sources
arXiv2026

Optimized bounds for the product and the ratios of modified Bessel functions

New sharp bounds for the product and the ratios of modified Bessel functions are presented. Most bounds for the product are derived as direct consequences of previously established bounds for the ratios of consecutive orders, except for the lower bound $I_ν(x)K_ν(x) > \frac{1}{2}(x^2 + ν^2 + 1/5)^{-1/2}$, which had been conjectured for $x > 0$ and $ν> -1$ and we prove in the present paper, showing that the constant $1/5$ can not be lowered. Moreover, very sharp bounds are obtained for the ratios (and consequently for the product) by asymptotically optimizing certain uniparametric inequalities. These optimized bounds are remarkably accurate: they remain extremely sharp for both small and large $x$ with fixed $ν$, and for large $ν$ with fixed $x$ or fixed $z = x/ν$. As a consequence, they provide precise upper and lower estimates across a wide range of parameters.

math.CA↗
arXiv2026

Identities and inequalities for integral transforms involving squares of the Bessel functions

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 equation 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 $k \in \mathbb{N}$. In this paper, we extend their identity to non-integer indices and derive several inequalities from it. In particular, we give a dimension-comparison result for the smoothing estimates on $\mathbb{R}^d$ and $\mathbb{R}^{d+1}$.

math.CA↗
arXiv2026

Optimal constants of smoothing estimates for the Dirac equation in arbitrary dimensions

We give optimal constants of smoothing estimates for the $d$-dimensional free Dirac equation for any $d \geq 2$. Our main abstract theorem shows that the optimal constant $C$ of the smoothing estimate associated with a spatial weight $w$ and smoothing function $ψ$ is given by $C = \sup_{k \in \mathbb{N}} \sup_{r > 0} \widetildeλ_k(r)$, where $\{ \widetildeλ_k \}$ is a certain sequence of functions defined via integral formulae involving $(w, ψ)$. This is an analogue of a similar result for Schrödinger equations given by Bez, Saito, and Sugimoto (2015), and also extends previous results of Ikoma (2022) and Ikoma and Suzuki (2025) for $d=2, 3$ to arbitrary dimensions $d \geq 2$. In order to prove this, we establish a modified version of the spherical harmonics decomposition of $L^2(\mathbb{S}^{d-1})$, which is well suited to the Dirac operator and allows us to find optimal constants. Furthermore, using our abstract theorem, we give explicit values of optimal constants associated with typical examples of $(w, ψ)$. As it turns out, optimal constants for Dirac equations can be written explicitly in many cases, even when it is impossible for Schrödinger equations. In particular, the classical result of Simon (1992) for Schrödinger equations, which holds when $d \geq 3$ but fails when $d=2$, is true for Dirac equations whenever $d \geq 3$ and remains valid for the massless two-dimensional case.

math.AP↗
arXiv2026

An order-interpolation inequality for Bessel functions

We show that $J_{μ+ ν}(r)^2 < J_{ν-1/2}(r)^2 + J_{ν+1/2}(r)^2$ holds whenever $μ\in (-1/2, 1/2)$, $ν\in [0, \infty)$, and $r \in (0, \infty)$. In fact, we prove a stronger version for any fixed non-trivial linear combination of the Bessel functions of the first and second kinds. This inequality can be regarded as a kind of interpolation with respect to order. As an application, we establish a dimension-comparison result for optimal constants of smoothing estimates for the free Schrödinger equation. Briefly, the optimal constant on $\mathbb{R}^{d+1}$ is at most twice that on $\mathbb{R}^d$ for each $d \geq 2$.

math.CA↗
arXiv2025

Optimal constants of smoothing estimates for the 3D Dirac equation

Recently, Ikoma (2022) considered optimal constants and extremisers for the $2$-dimensional Dirac equation using the spherical harmonics decomposition. Though its argument is valid in any dimensions $d \geq 2$, the case $d \geq 3$ remains open since it leads us to too complicated calculation: determining all eigenvalues and eigenvectors of infinite dimensional matrices. In this paper, we give optimal constants and extremisers of smoothing estimates for the $3$-dimensional Dirac equation. In order to prove this, we construct a certain orthonormal basis of spherical harmonics. With respect to this basis, infinite dimensional matrices actually become block diagonal and so that eigenvalues and eigenvectors can be easily found. As applications, we obtain the equivalence of the smoothing estimate for the Schrödinger equation and the Dirac equation, and improve a result by Ben-Artzi and Umeda (2021).

math.AP↗
arXiv2024

Optimal constants of smoothing estimates for Dirac equations with radial data

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.

math.AP↗
arXiv2023

Equivalence between the energy decay of fractional damped Klein-Gordon equations and geometric conditions for damping coefficients

We consider damped $s$-fractional Klein--Gordon equations on $\mathbb{R}^d$, where $s$ denotes the order of the fractional Laplacian. In the one-dimensional case $d = 1$, Green (2020) established that the exponential decay for $s \geq 2$ and the polynomial decay of order $s/(4-2s)$ hold if and only if the damping coefficient function satisfies the so-called geometric control condition. In this note, we show that the $o(1)$ energy decay is also equivalent to these conditions in the case $d=1$. Furthermore, we extend this result to the higher-dimensional case: the logarithmic decay, the $o(1)$ decay, and the thickness of the damping coefficient are equivalent for $s \geq 2$. In addition, we also prove that the exponential decay holds for $0 < s < 2$ if and only if the damping coefficient function has a positive lower bound, so in particular, we cannot expect the exponential decay under the geometric control condition.

math.AP↗
arXiv2022

The uncertainty principle and energy decay estimates of the fractional Klein-Gordon equation with space-dependent damping

We consider the $s$-fractional Klein-Gordon equation with space-dependent damping on $\mathbb{R}^d$. Recent studies reveal that the so-called geometric control conditions (GCC) are closely related to semigroup estimates of the equation. Particularly, in the case $d = 1$, a necessary and sufficient condition for the exponential stability in terms of GCC is known for any $s > 0$. On the other hand, in the case $d \geq 2$ and $s \geq 2$, Green-Jaye-Mitkovski (2022) proved that an `$1$-GCC' is sufficient for the exponential stability, but also conjectured that it is not necessary if $s$ is sufficiently large. In this paper, we prove the equivalence between the exponential stability and a kind of the uncertainty principle in Fourier analysis. As a consequence of the equivalence, we show that the $1$-GCC is not necessary for the exponential stability in the case $s \geq 4$. Furthermore, we also establish an extrapolation result with respect to $s$. In particular, we can obtain the polynomial stability for the non-fractional case $s = 2$ from the exponential stability for some $s > 2$.

math.AP↗