A slice–Bochner criterion for joint logarithmic potentials

Definitions and mathematical statement

The missing display in the source is recovered from companion records as the real logarithmic determinant identity for a bounded commuting tuple: \[ \log\Delta\left(1-\sum_{i=1}^n\alpha_iT_i\right) =\int_{{\mathbb{C}}^n}\log\left|1-\sum_{i=1}^n\alpha_i\zeta_i\right| \,\mu({\,\mathrm{d}}\zeta). \] The absolute-value bars are essential; a single branch of the complex logarithm cannot replace this real extended-valued expression globally. The question asks which functions $F$ on ${\mathbb{C}}^n$ have such an integral representation. The aid treats compactly supported probability measures and gives the following precise slice-compatibility criterion.

Let $F:{\mathbb{C}}^n\to[-\infty,\infty)$, require $F(0)=0$, and use the complex bilinear pairing $a\cdot\zeta=\sum_i a_i\zeta_i$. For each $a\in{\mathbb{C}}^n$, define on $\lambda\ne0$ \[ U_a(\lambda)=\log|\lambda|+F(a/\lambda). \] Require that $U_a$ extend to a nontrivial subharmonic function on ${\mathbb{C}}$, with normalized behavior $U_a(\lambda)-\log|\lambda|\to0$ at infinity, whose Riesz measure \[ \nu_a=(2\pi)^{-1}\Delta_{{\mathbb{R}}^2}U_a \] is a compactly supported probability measure. These conditions identify $U_a$ with its logarithmic potential $\int\log|\lambda-w|\,\nu_a({\,\mathrm{d}} w)$. At $a=0$, $U_0=\log|\lambda|$ and $\nu_0=\delta_0$.

Impose two additional compatibility conditions. First, for one finite $R$, uniformly over all $a$, \[ \operatorname{supp}\nu_a\subseteq\{w:|w|\le R\|a\|_2\}. \] Second, the function \[ \Phi(a)=\int_{\mathbb{C}} e^{i\operatorname{Re}w}\,\nu_a({\,\mathrm{d}} w) \] is continuous and positive definite on the real additive group ${\mathbb{C}}^n$: for every finite choice $a_1,\ldots,a_k$ and $z_1,\ldots,z_k\in{\mathbb{C}}$, \[ \sum_{r,s=1}^k z_r\overline{z_s}\Phi(a_r-a_s)\ge0. \] The inequality means that this sum is real and nonnegative. The slice formula automatically gives the scaling consistency $\nu_{ca}=(w\mapsto cw)_*\nu_a$ for $c\in{\mathbb{C}}$, with the stated convention at $c=0$.

The aid reports that these conditions are necessary and sufficient for a unique probability measure supported in the closed Euclidean ball of radius $R$ to satisfy \[ F(\alpha)=\int_{{\mathbb{C}}^n}\log|1-\alpha\cdot\zeta|\,\mu({\,\mathrm{d}}\zeta) \quad(\alpha\in{\mathbb{C}}^n). \] Indeed Bochner's theorem reconstructs a joint measure from $\Phi$ using the real pairing $\operatorname{Re}(a\cdot\zeta)$; scaling consistency supplies all complex frequencies of each slice, so $(a\cdot\zeta)_*\mu=\nu_a$. The uniform all-direction support estimate gives the ball bound, not merely coordinatewise bounds. Conversely a compact joint measure produces exactly those slices and a continuous positive-definite characteristic function.

Every such measure is reported realizable as the joint spectral distribution of a commuting normal tuple in a $\mathrm{II}_1$ factor, by representing its bounded coordinate functions in an abelian subalgebra. For that tuple the determinant identity follows from normal functional calculus. This is a range criterion for the stated compact-probability transform; it is not a characterization of arbitrary multivariable subharmonic functions without joint slice compatibility.

Short English statement

Characterize logarithmic determinant transforms on ${\mathbb{C}}^n$. The aid reconstructs each one-dimensional slice by its Riesz measure and tests whether those slices come from one compact joint probability law through a continuous positive-definite characteristic function and a uniform directional support bound.

Sources

• American Institute of Mathematics, Free Analysis problem list, physical page 6, subsection 0.9, "Focus Group on Invariant Subspaces (day 4)," https://aimath.org/WWN/freeanalysis/freeanalysis.pdf.
• Local UnsolvedMath registry, permanent ID 20002684, source code catalogue entry, canonical location aim-probability-notes.json, notes[125], record dated 14 August 2026. Its original question refers only to the omitted display, and the literature field is empty.
• Companion permanent ID 20002682, AIM-PROBABILITY-0124, notes[123], contains the display. Companion permanent ID 20002683, AIM-PROBABILITY-0125, notes[124], identifies missing modulus signs by reference to Schultz's primary theorem and Charlesworth--Dykema--Sukochev--Zanin; full bibliographic entries are not supplied.
• Supplied research attempt 1: partial result, problem status at run unknown, rigor labelled proved, novelty confidence low. Its slice--Bochner theorem is machine-generated research aid, not peer reviewed; independently verify before citation. Bochner's theorem is invoked without a particular bibliographic edition.