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

This paper extends a known identity for an integral transform involving squares of Bessel functions to non-integer indices and derives several inequalities from this generalized expression.

Original authors: Soichiro Suzuki

Published 2026-06-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Soichiro Suzuki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a chef trying to bake the perfect cake, but instead of flour and sugar, your ingredients are mathematical waves called Bessel functions. These waves are tricky; they ripple and oscillate, making it hard to predict how they will behave when mixed together.

This paper, written by Soichiro Suzuki, is essentially a new recipe book for mixing these specific waves. The author isn't just baking a single cake; he is discovering new rules about how these waves interact, which helps mathematicians bake better "mathematical cakes" for solving problems in physics (specifically, how particles move in quantum mechanics).

Here is the breakdown of the paper's journey, using everyday analogies:

1. The Problem: The "Smoothing" Mystery

In the world of quantum physics (specifically the Schrödinger equation), scientists want to know how "smooth" a wave stays as it travels through space. To measure this, they use a special tool called an integral transform (let's call it the "Wave Mixer").

The Old Tool: Previous mathematicians (Walther, Bez, et al.) knew how to use this Wave Mixer, but only for specific, "whole number" settings (like setting a dial to 1, 2, or 3).
The Gap: They didn't have a clear rule for "fractional" settings (like 1.5 or 2.7). It was like having a radio that only worked on whole numbers, leaving a gap of static in between.

2. The Big Discovery: The "Universal Adapter"

Suzuki's main achievement is building a Universal Adapter. He extends the Wave Mixer so it works for any number, not just whole numbers.

The Identity (The Magic Formula): He found a new equation that connects the "Wave Mixer" to another tool called the Hankel Transform (think of this as a "Fourier Transform for circular waves").
The Analogy: Imagine you have a complex machine that takes a raw ingredient (a function) and processes it. Suzuki discovered that you don't need to look at the machine's messy insides. Instead, you can first pass the ingredient through a simple filter (the Hankel Transform), and then use a much simpler, predictable machine (the $U$ operator) to get the result.
The Result: This formula works for integers (the old way) and non-integers (the new way), unifying the whole process.

3. The Rules of the Road: Inequalities

Once the machine was fixed to work for all numbers, Suzuki noticed some fascinating patterns, which he calls Inequalities. Think of these as traffic rules for the waves:

Rule A (The Monotonicity Rule): If the input wave is "positive" (all waves pushing in the same direction), the output of the machine never goes down as you turn the dial up. It either stays flat or goes up. It's like a staircase that only goes up or stays level; it never goes down.
Rule B (The Comparison Rule): If you compare two settings on the dial (say, setting 2 vs. setting 2.5), the machine with the lower setting always produces a "larger" or "equal" result than the higher setting, provided the input is positive.
- Why this matters: This helps prove that in certain physics scenarios, the "smoothness" of a wave in a 3D world is always better than or equal to the smoothness in a 4D world. It puts a cap on how bad things can get.

4. The Special Ingredient: "Completely Monotone" Functions

The paper introduces a special type of ingredient called a Completely Monotone function.

The Metaphor: Imagine a perfectly smooth, decaying curve, like a cup of coffee cooling down. It never wiggles; it just slowly, steadily gets cooler.
The Discovery: If you use this "perfectly smooth" ingredient in the Wave Mixer, the output behaves beautifully. It becomes analytic (mathematically smooth and predictable everywhere) and follows strict rules about increasing or decreasing.
The Payoff: This allows mathematicians to prove that for these specific "perfect" ingredients, there is no single "worst-case scenario" (extremiser) that breaks the rules. The system is too well-behaved to have a single breaking point.

5. The "Side Quests" (Remarks)

The paper also looks at what happens in special edge cases:

The "Negative Half" Case: What happens if the dial is set to -0.5? The author shows that even here, the rules hold, but the machine behaves like a simple switch (on/off) rather than a complex mixer.
The Dirac Equation: The author shows that these same rules apply to a different type of physics equation (the Dirac equation, used for electrons), proving the "Universal Adapter" is robust enough for different types of quantum particles.

Summary

In short, this paper is about unifying and simplifying a complex mathematical tool used in quantum physics.

It fixed the machine: It made the tool work for fractional numbers, not just whole ones.
It mapped the terrain: It discovered that the tool's output always follows a "staircase" pattern (never going down) under certain conditions.
It found the perfect ingredient: It showed that if you use a "perfectly smooth" input, the output is predictable and elegant.

These findings don't just sit on a shelf; they help physicists calculate the "best possible constants" for how waves smooth out, ensuring their models of the universe are as accurate as possible.

Technical Summary: Identities and Inequalities for Integral Transforms Involving Squares of the Bessel Functions

Problem Statement
This paper investigates the integral transform $T_\nu$ defined by
$T_\nu f(s) := \pi \int_0^\infty rs J_\nu(rs)^2 f(r) \, dr,$
where $J_\nu$ is the Bessel function of the first kind of order $\nu \in [-1/2, \infty)$ . This transform is central to the study of optimal constants in Kato–Yajima smoothing estimates for the free Schrödinger equation on $\mathbb{R}^d$ . Specifically, the optimal constant $C_S^{(d)}(w, \psi)$ for the inequality
$\int_{\mathbb{R}^d \times \mathbb{R}} w(|x|) |\psi(|\nabla|) e^{it\Delta} u_0(x)|^2 \, dx \, dt \le C \|u_0\|_{L^2(\mathbb{R}^d)}^2$
is determined by the supremum of $s^{-1}\psi(s)^2 T_{k+d/2-1}w(s)$ over $k \in \mathbb{N}$ and $s > 0$ .

While previous work by Bez, Saito, and Sugimoto established an identity relating $T_{k+d/2-1}$ to the Hankel transform $H_{d/2-1}$ for integer indices $k$ and dimensions $d$ , a gap remained in extending these identities to non-integer indices and deriving general inequalities for $T_\nu$ without restricting to integer orders. The paper aims to extend these identities to arbitrary indices and utilize them to establish monotonicity properties and inequalities for the transform.

Methodology
The author employs a combination of integral transform theory, special function identities, and measure-theoretic arguments.

Extension of Identities: The core methodological innovation is the extension of the identity $T_{k+d/2-1}f = U_{k,d/2-1}H_{d/2-1}f$ (previously known for integer $k$ ) to arbitrary indices. This is achieved by introducing the Gegenbauer function of the second kind, $D_\mu^{(\nu)}$ , to define a generalized integral operator $U_{\mu,\nu}$ with a kernel $K_{\mu,\nu}(r,s)$ that is non-zero for $r > 2s$ when $\mu \notin \mathbb{N}$ .
Hankel Transform Analysis: The proof of the generalized identity relies on calculating the Hankel transform of the kernel function $g_{\mu,\nu,s}(r) = \pi s r^{-2\nu} J_{\mu+\nu}(rs)^2$ . This calculation utilizes Macdonald's formula for integrals involving products of three Bessel functions, expressing the result in terms of Legendre functions and subsequently converting them to Gegenbauer functions.
Parseval's Identity: The validity of the identity $T_{\mu+\nu}f = U_{\mu,\nu}H_\nu f$ is established using Parseval's identity for the Hankel transform, ensuring the necessary integrability conditions for the functions and their transforms.
Monotonicity and Inequalities: By analyzing the kernel $K_{\mu,\nu}(r,s)$ , the author derives pointwise inequalities (e.g., $K_{\mu,\nu} \le K_{0,\nu}$ ) which, when combined with the non-negativity of the Hankel transform $H_\nu f$ , yield inequalities for $T_\nu f$ .
Complete Monotonicity: The paper leverages the Bernstein–Widder and Schoenberg theorems, which characterize functions $f$ such that $r \mapsto f(r^{1/2})$ is completely monotone as superpositions of Gaussians. This allows the extension of results from the Gaussian case to a broader class of functions via integration against a Borel measure.

Key Contributions and Results

Generalized Identity (Theorem 1.6(I)): The paper establishes that for $\mu \in [0, \infty)$ , $\nu \in (-1/2, \infty)$ , and suitable $f$ , the identity $T_{\mu+\nu}f(s) = U_{\mu,\nu}H_\nu f(s)$ holds. This generalizes previous results restricted to integer shifts in the index.
Monotonicity and Inequalities (Theorem 1.6(II)–(V)):
- If $H_\nu f$ is non-negative, $s \mapsto T_\nu f(s)$ is non-decreasing for $\nu \ge 1/2$ .
- The sum $T_\nu f(s) + T_{\nu+1} f(s)$ is non-decreasing for $\nu > -1/2$ .
- For $\mu \in \mathbb{N}_{\ge 1}$ or $\mu \in (0, 1]$ (with $\mu+2\nu \ge 0$ ), if $H_\nu f$ is non-negative, then $T_{\mu+\nu}f(s) \le T_\nu f(s)$ .
Application to Smoothing Estimates (Theorem 1.5): Using the inequality $T_{d/2-1/2}w \le T_{d/2-1}w$ (derived from Theorem 1.6(V) with $\mu=1/2$ ), the author proves that if the Fourier transforms of a spatial weight $w$ are non-negative on $\mathbb{R}^d$ and $\mathbb{R}^{d+1}$ , then the optimal constant for the smoothing estimate satisfies $C_S^{(d+1)}(w, \psi) \le C_S^{(d)}(w, \psi)$ . This provides a sufficient condition for the dimension-decreasing property of optimal constants, covering cases (A), (B), and (C) mentioned in the introduction.
Analyticity and Complete Monotonicity (Theorem 1.8): For functions $f$ $f$ where $r \mapsto f(r^{1/2})$ $r \mapsto f (r^{1/2})$ is completely monotone, the paper proves that:
- $s \mapsto s^{-\nu-1/2}T_\nu f(s^{1/2})$ is completely monotone.
- $s \mapsto T_\nu f(s)$ is strictly increasing for $\nu \ge 1/2$ .
- $s \mapsto T_\nu f(s) + T_{\nu+1} f(s)$ is strictly increasing.
- $\nu \mapsto T_\nu f(s)$ is strictly decreasing for $\nu \ge 0$ .
- $T_{-\nu}f(s) < T_\nu f(s)$ for $\nu \in (-1/2, 0)$ .
  These results imply the analyticity of $T_\nu f$ on $(0, \infty)$ , which has implications for the non-existence of extremizers in smoothing estimates.
Specific Examples: The paper applies these theorems to derive new proofs for known properties of modified Bessel functions $I_\nu$ and $K_\nu$ , such as the complete monotonicity of $s^{-\nu}e^{-s}I_\nu(s)$ and the monotonicity of $s I_\nu(s)K_\nu(s)$ .
Extensions: The results are extended to the case $\nu = -1/2$ (Proposition 4.1) and to Dirac equations (Theorem 4.3), showing analogous dimension-decreasing properties for optimal constants in those contexts.

Significance
The paper claims to provide a unified framework for understanding the behavior of the integral transform $T_\nu$ across non-integer indices. By extending the identity connecting $T_\nu$ to the Hankel transform, the author derives a suite of inequalities that were previously only known for specific integer cases or required different methods.

The primary significance lies in:

Generalization: Extending the identity $T_{k+d/2-1} = U_{k,d/2-1}H_{d/2-1}$ to arbitrary real indices, thereby removing the restriction to integer dimensions or orders in the analysis of smoothing estimates.
Optimal Constants: Providing a rigorous sufficient condition (non-negativity of Fourier transforms) for the monotonicity of optimal smoothing constants with respect to dimension, specifically proving $C_S^{(d+1)} \le C_S^{(d)}$ under broad conditions.
Analyticity: Establishing the analyticity of $T_\nu f$ for a wide class of weights (those related to completely monotone functions), which supports the non-existence of extremizers for the smoothing estimates.
Unification: Demonstrating that various known inequalities and monotonicity properties of Bessel functions (e.g., involving $I_\nu$ and $K_\nu$ ) are direct consequences of the general properties of the transform $T_\nu$ and the complete monotonicity of the input function.

The author notes that while some specific inequalities (like those in Example 3.2) were known, the proofs provided here are new and stem from the generalized identity, offering a more direct and unified approach.