Proves an order-interpolation inequality for squares of Bessel functions of the first and second kinds and applies it to bound optimal constants for Schrödinger smoothing estimates across dimensions.
Identities and inequalities for integral transforms involving squares of the Bessel functions
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abstract
We consider an integral transform given by $T_{\nu} f(s) := \pi \int_0^\infty rs J_{\nu}(r s)^2 f(r) \, dr$, where $J_{\nu}$ denotes the Bessel function of the first kind of order $\nu$. 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 Schr\"{o}dinger 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 $T_{\nu} f$ involving the $d$-dimensional Fourier transform of $x \mapsto f(\lvert x \rvert)$ when $\nu = k + d/2 - 1$ 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.
fields
math.CA 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Establishes optimal constants and extremizers for smoothing estimates of quantum harmonic oscillators as direct analogues of prior free-particle results.
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An order-interpolation inequality for Bessel functions
Proves an order-interpolation inequality for squares of Bessel functions of the first and second kinds and applies it to bound optimal constants for Schrödinger smoothing estimates across dimensions.
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Optimal constants of smoothing estimates for quantum harmonic oscillators
Establishes optimal constants and extremizers for smoothing estimates of quantum harmonic oscillators as direct analogues of prior free-particle results.