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Higher order logarithms of Bessel operators and an extension problem

This paper investigates the Bessel operator $B_\lambda$ by establishing pointwise representations and asymptotic Taylor expansions for its fractional powers and higher-order logarithms, while also characterizing the logarithm $\log B_\lambda$ as the solution to an extension problem.

Original authors: Jorge J. Betancor, Estefanía. Dalmasso, Juan C. Fariña, Pablo Quijano

Published 2026-08-21

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CC BY 4.0

Original authors: Jorge J. Betancor, Estefanía. Dalmasso, Juan C. Fariña, Pablo Quijano

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

In the vast landscape of mathematics, there are tools designed to measure change, much like a ruler measures length or a scale measures weight. One of the most fundamental of these tools is the derivative, which tells us how quickly a quantity is changing at any given moment. When we look at waves, heat spreading through a metal rod, or the vibration of a drumhead, these changes are often described by equations involving a specific type of operator, a mathematical machine that takes a function and returns its rate of change. In the flat, familiar world of a straight line or a flat plane, this machine is well understood. However, the universe is not always flat or simple. In many physical situations, such as when dealing with circular or spherical shapes, the rules of change become more complex. Mathematicians use a specialized version of the derivative, known as the Bessel operator, to handle these curved geometries. While the standard derivative is like a straight edge, the Bessel operator accounts for the way space stretches and shrinks as you move away from a center point, making it essential for understanding phenomena in physics and engineering that involve radial symmetry.

For decades, mathematicians have been able to raise these operators to fractional powers, essentially asking what it means to take "half" of a derivative or "one-third" of a rate of change. This allows for a more nuanced description of physical processes that evolve in a non-standard way. However, a more subtle question has remained: what happens when we look at the logarithm of these operators? In the world of numbers, the logarithm tells us the power to which a base must be raised to produce a given number. For operators, the logarithm is a much more abstract concept, representing a kind of "infinitesimal" change that sits at the very heart of the operator's behavior. Recently, researchers have begun to map out this territory for the Bessel operator, but a complete picture of its higher-order logarithms and how they relate to the fractional powers was still missing. This gap in understanding is significant because these logarithmic operators act as the building blocks for understanding the fine structure of complex systems, and without them, our mathematical models of curved spaces remain incomplete.

In a recent study, a team of mathematicians set out to fill this gap by defining and analyzing these higher-order logarithmic operators for the Bessel operator. They did not simply guess at the answers; they constructed a rigorous framework that allows these abstract operators to be written down as concrete, point-by-point calculations. The researchers showed that these logarithmic operators can be represented as specific integrals, which are essentially sums of weighted values over a range. By breaking down the problem into these manageable pieces, they were able to prove that these operators behave exactly as one would expect them to when they are very close to zero. Specifically, they demonstrated that the fractional powers of the Bessel operator can be expanded into a series, much like a Taylor series in calculus, where the terms of the series are built from these newly defined logarithmic operators. This means that if you know the logarithmic behavior of the operator, you can reconstruct its fractional power behavior with high precision.

The team also tackled a different, yet related, challenge: finding a way to visualize these logarithmic operators through a physical extension problem. In mathematics, an extension problem involves taking a function defined on a line and extending it into a higher dimension, such as a half-plane, to solve a difficult equation. The researchers proved that the logarithm of the Bessel operator can be recovered by looking at the behavior of a specific solution to an extension equation as it approaches the boundary. They constructed a function that lives in a two-dimensional space and showed that by observing how this function changes as it gets closer to the original line, one can extract the logarithm of the operator. This method is powerful because it transforms a difficult, abstract calculation into a problem of studying the limits of a physical-looking process. The solution to this extension problem involves a specific correction factor that depends on the position, a detail that arises because the Bessel operator does not behave exactly like the standard derivative in flat space.

The results of this work provide a clear and complete description of how these higher-order logarithms work. The researchers established that for a wide class of functions, these operators are well-defined and can be calculated directly. They also proved that the fractional powers of the Bessel operator can be approximated by a sum of these logarithmic terms, with the error becoming vanishingly small as the power approaches zero. This confirms that the logarithmic operators are indeed the fundamental components that generate the fractional powers. Furthermore, the extension problem they solved offers a new way to compute these values, linking the abstract algebraic properties of the operator to the concrete behavior of a function in a higher-dimensional space. The study does not claim to solve every mystery in this field, but it provides a solid foundation, proving that these complex operators can be understood through explicit formulas and physical analogies.

By connecting the abstract world of operator theory with concrete integral representations and extension problems, this research deepens our understanding of how mathematical tools behave in curved spaces. The findings suggest that the logarithmic operators are not just theoretical curiosities but are essential keys to unlocking the behavior of fractional powers in non-Euclidean settings. The work stands as a testament to the power of breaking down complex mathematical structures into their fundamental parts, showing that even the most abstract concepts can be brought into focus through careful analysis and the construction of precise mathematical models. The path forward is now clearer, with these new definitions and representations ready to be applied to other problems in analysis and mathematical physics where the geometry of space plays a crucial role.

Technical Summary: Higher Order Logarithms of Bessel Operators and an Extension Problem

Problem Statement
The paper investigates the Bessel operator $B_\lambda = -\frac{d^2}{dx^2} + \frac{\lambda^2 - 1/4}{x^2}$ defined on $(0, \infty)$ with $\lambda > -1$ . While the fractional powers $B_\lambda^s$ and the logarithm $\log B_\lambda$ have been studied previously (notably by Fu, Li, and Zhang [15]), this work aims to:

Define and analyze higher-order logarithmic operators $\log^m B_\lambda$ for $m \in \mathbb{N}$ .
Derive pointwise integral representations for these operators and for fractional powers $B_\lambda^s$ ( $s \in (-1, 1), s \neq 0$ ).
Establish asymptotic Taylor expansions of $B_\lambda^s$ and $B_\lambda^{-s}$ as $s \to 0^+$ in terms of the logarithmic operators $\log^m B_\lambda$ .
Characterize $\log B_\lambda$ as the solution to a specific extension problem, analogous to the Caffarelli-Silvestre extension for the fractional Laplacian.

Methodology
The authors employ the Hankel transform $h_\lambda$ , which serves as the spectral tool for the Bessel setting analogous to the Fourier transform in the Euclidean setting. The operator $B_\lambda$ is defined spectrally via $h_\lambda(B_\lambda f)(x) = x^2 h_\lambda(f)(x)$ .

Key methodological tools include:

Semigroup Approach: The Bessel heat semigroup $\{W^\lambda_t\}_{t>0}$ , generated by $-B_\lambda$ , is utilized. The kernel $W^\lambda_t(x, y)$ involves the modified Bessel function of the first kind, $I_\lambda$ .
Functional Calculus: Operators are defined via spectral multipliers. For a function $\Phi$ , $\Phi(B_\lambda)f = h_\lambda(\Phi(x^2)h_\lambda f)$ .
Integral Representations: The fractional powers $B_\lambda^s$ are expressed using Bochner integrals involving the semigroup difference $W^\lambda_t f - f$ (for $s>0$ ) or the semigroup itself (for $s<0$ ). These integrals are decomposed to isolate singularities and ensure absolute convergence.
Extension Problem: A function $u^\lambda_f(x, t)$ is constructed on the upper half-plane $(0, \infty) \times (0, \infty)$ using a subordination formula involving the heat kernel and the semigroup. The behavior of this extension as $t \to 0^+$ is analyzed to recover the logarithmic operator.
Asymptotic Analysis: Taylor expansions are derived by differentiating the fractional power operators with respect to the parameter $s$ at $s=0$ , utilizing the dominated convergence theorem and properties of the Gamma function.

Key Contributions and Results

Pointwise Representations (Theorem 1.1):
The authors establish explicit pointwise formulas for $B_\lambda^s f(x)$ for $s \in (0, 1)$ and $s \in (-1, 0)$ . These formulas involve integrals of the heat kernel $W^\lambda_t(x, y)$ against the function $f$ and its local averages. Crucially, the representation includes a "corrector" term involving the difference between the Bessel heat kernel and the Euclidean heat kernel $W_t(x-y)$ , necessitated by the non-Markovian nature of the Bessel semigroup (i.e., $W^\lambda_t(1) \neq 1$ ).
Higher-Order Logarithms (Theorem 1.2):
The paper defines the higher-order logarithmic operators $\log^m B_\lambda$ for $m \in \mathbb{N}$ . It provides a pointwise representation for $(\log^m B_\lambda)f(x)$ involving integrals weighted by powers of $\log t$ . The authors note that their proof is shorter than the existing proof for the first-order case in [15] and extends the domain of definition to functions in $L^1((0, \infty), \frac{dy}{1+y}) \cap \text{Lip}^{\theta}_{\text{loc}, x}(0, \infty)$ .
Asymptotic Expansions (Theorem 1.3):
The study derives asymptotic Taylor expansions for the fractional powers as $s \to 0^+$ . Specifically, it proves that:
$\lim_{s \to 0^+} \frac{(m+1)!}{s^{m+1}} \left( B_\lambda^s f - \left( f + \sum_{j=1}^m \frac{s^j}{j!} (\log^j B_\lambda)f \right) \right) = (\log^{m+1} B_\lambda)f$
in $L^p(0, \infty)$ for $1 < p \leq \infty$ . A similar expansion is provided for $B_\lambda^{-s}$ . This generalizes results by Chen [8] for the Euclidean Laplacian to the Bessel setting.
Extension Problem Characterization (Theorem 1.4):
The authors define an extension $u^\lambda_f(x, t)$ and prove that it satisfies the degenerate elliptic equation $(\partial_t^2 + \frac{1}{t}\partial_t)u^\lambda_f = B_{\lambda, x} u^\lambda_f$ . The main result characterizes the logarithm of the Bessel operator as the boundary limit of this extension:
$(\log B_\lambda)f(x) = -\lim_{t \to 0^+} (u^\lambda_f(x, t) + f(x)\log t) + f(x)K(x)$
where $K(x)$ is a specific function involving Euler's constant $\gamma$ , logarithmic terms, and integrals of the heat kernels. This result parallels the extension theorem for $\log(-\Delta)$ established by Chen, Hauer, and Weth [9].

Significance and Claims
The paper claims to provide a comprehensive framework for higher-order logarithmic operators in the Bessel context. The authors emphasize that their approach yields shorter proofs for known results (specifically the representation of $\log B_\lambda$ ) and extends these results to higher orders and broader function spaces.

The significance lies in:

Generalization: Extending the theory of logarithmic operators from the Euclidean Laplacian to the Bessel operator, which is fundamental in harmonic analysis on the half-line.
Unification: Connecting fractional powers, logarithmic operators, and extension problems within a single spectral framework using the Hankel transform.
Technical Precision: Providing rigorous pointwise representations that account for the specific singularities and non-Markovian properties of the Bessel heat semigroup, which differ from the Euclidean case.

The authors do not propose new applications or experimental setups but focus on the theoretical development of operator theory and harmonic analysis related to the Bessel operator. The work builds upon and refines existing literature, particularly the works of Fu, Li, and Zhang [15] and Chen [8, 9].

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