[Paper Review] A four dimensional Jensen formula

Alessandro Perotti|arXiv (Cornell University)|Feb 18, 2019

Algebraic and Geometric Analysis10 references4 citations

TL;DR

This paper establishes a four-dimensional Jensen formula for slice-regular functions of one quaternionic variable, relating the function's value and first two derivatives at a point to its integral mean over a 3-sphere and the distribution of its zeros and poles. The formula extends previous results to non-slice-preserving functions and incorporates both isolated and spherical zeros and poles via a generalized Blaschke-type factorization.

ABSTRACT

We prove a Jensen formula for slice-regular functions of one quaternionic variable. The formula relates the value of the function and of its first two derivatives at a point with its integral mean on a three dimensional sphere centred at that point and with the disposition of its zeros. The formula can be extended to semiregular slice functions.

Motivation & Objective

• To generalize the Jensen formula for slice-regular functions to non-slice-preserving functions in the quaternionic setting.
• To extend the classical Jensen formula to four real dimensions using the structure of slice-regular functions on the quaternions.
• To incorporate both isolated and spherical zeros and poles into a unified integral formula via Blaschke-type factors.
• To provide a formula valid for semiregular functions, including those with poles, by leveraging the slice product and lifting to complex quaternions.
• To establish a framework for computing logarithmic integrals and derivatives at the origin using spherical symmetry and trace operations.

Proposed method

• Lift slice-regular functions to holomorphic functions on complexified quaternions via the stem function construction in $\mathbb{H} \otimes_{\mathbb{R}} \mathbb{C}$.
• Use the slice product and reciprocal Blaschke factors to construct a slice-preserving multiplier $g$ such that $gf$ extends to a slice-regular function $h$.
• Define $g$ using $r$-Blaschke factors for real poles $p_k$ and normal functions for spherical poles $b_i$, ensuring $|g|=1$ on $\partial\mathbb{B}_r$.
• Apply the known Jensen formula for $h$ and decompose the logarithmic terms to recover the formula for $f$.
• Use the spherical operator and Laplacian $\Delta_4$ to relate the second-order derivatives at the origin to the logarithmic integral of $|N(f)|$.
• Account for non-constant pole orders by introducing isolated multiplicities $i_f(z_j)$ and matching contributions via a derived identity.

Experimental results

Research questions

• RQ1 How can the classical Jensen formula be extended to slice-regular functions of one quaternionic variable in four real dimensions?
• RQ2 What is the role of spherical and isolated zeros and poles in a generalized Jensen-type formula for quaternionic functions?
• RQ3 How do the first and second derivatives of a slice-regular function at the origin relate to its integral mean on a 3-sphere?
• RQ4 Can a unified formula be derived for semiregular functions that includes both zeros and poles with variable orders?
• RQ5 What is the contribution of spherical poles with non-constant order to the logarithmic integral in the Jensen formula?

Key findings

• The formula expresses $\log|f(0)|$ and its first two derivatives in terms of the integral mean of $\log|f|$ and $\log|f \circ S_f|$ over the 3-sphere $\partial\mathbb{B}_r$.
• The formula includes correction terms for isolated zeros $a_i$, isolated poles $p_k$, and spherical poles $\SS_{b_i}$, with explicit dependence on $|a_i|$, $|p_k|$, and $|b_i|$.
• For spherical poles with non-constant order, the formula accounts for isolated multiplicities $i_f(z_j)$, and their contribution matches a sum over $a_i$ with $q' = q$.
• The term $\frac{r^2}{4}\operatorname{Re}\left(\left(f(0)^{-1}\overline{\frac{\partial f}{\partial x}(0)}\right)^2\right)$ captures the influence of the first derivative on the logarithmic mean.
• The term $-\frac{r^2}{4}\operatorname{Re}\left(f(0)^{-1}\frac{\partial^2 f}{\partial x^2}(0)\right)$ accounts for the second derivative's contribution at the origin.
• The formula reduces correctly to the slice-preserving case when all poles and zeros are real or spherical with constant order, and matches known results from [1] and [10].