Work

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

Year: 2026

Type: article

Abstract: We consider an integral transform given by Tνf(s):=π∫0∞rsJν(rs)2f(r) dr, where Jν denotes the Bessel function of the first kind of order ν. As shown by Walther (2002), this transform plays an essential role in the study of optimal constants of smoothing estimates for the free Schrödinger equation on Rd. On the other hand, Bez et al. studied these optimal constants using a different method and obtained a certain alternative expression for Tνf involving the d-dimensional Fourier transform of x↦f(|x|) when ν=k+d/2−1 for k∈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 Rd and Rd+1.

Source: Integral Transforms and Special Functions

DOI: 10.1080/10652469.2026.2732147

Author: Soichiro Suzuki

Institution: Chuo University

Language: English

Topic: Mathematical Analysis and Transform Methods

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Japan Society for the Promotion of Science

Award: G8218740568