Injectivity Criteria for Upsilon Transforms

Definitions and mathematical statement

Let $\gamma$ be a $\sigma$-finite Borel measure on $(0,\infty)$, as stated in the source. In the usual $d$-dimensional setting, a Lévy measure $\rho$ on ${\mathbb{R}}^d$ satisfies \[ \rho(\{0\})=0,\qquad \int_{{\mathbb{R}}^d}(1\wedge\|z\|^2)\,\rho(dz)<\infty. \] The dimension and underlying space of $\rho$ are not explicitly supplied, so this standard finite-dimensional convention is made conditional. For a Borel set $A$, put $Ax^{-1}=\{z:xz\in A\}$. The source's transform is the positive scaling mixture \[ \Upsilon_\gamma(\rho)(A)=\int_0^\infty\rho(Ax^{-1})\,\gamma(dx). \] This is inverse scaling of the set, or equivalently forward multiplication of points by $x$, not the reverse convention.

With $\mathfrak L_d$ denoting the Lévy measures above, its natural Lévy domain is \[ \mathcal D_\gamma=\{\rho\in\mathfrak L_d: \Upsilon_\gamma(\rho)\in\mathfrak L_d\}. \] Under this interpretation the integrability requirement can be written \[ \int_{{\mathbb{R}}^d}\int_0^\infty(1\wedge\|xz\|^2)\,\gamma(dx)\rho(dz)<\infty. \] The source asks for general conditions on $\gamma$ ensuring \[ \forall\rho_1,\rho_2\in\mathcal D_\gamma,\qquad \Upsilon_\gamma(\rho_1)=\Upsilon_\gamma(\rho_2) \ \Longrightarrow\ \rho_1=\rho_2. \] Equalities are equalities of measures, not merely of finitely many moments or transform values. The entire intended Lévy domain is the target; no density, moment or support restriction on its elements is added. Whether a criterion is sought for one fixed dimension or uniformly in all dimensions must be specified. A domain with only the zero measure would make injectivity vacuous, and no nontrivial-domain requirement is printed.

The statement assumes only $\sigma$-finiteness of the scaling measure $\gamma$. The literature assessment speaks of a criterion for every Lévy measure $\gamma$, which is a narrower or differently normalized class; no additional Lévy integrability of $\gamma$ is silently imposed. Its triage checked 2026-08-21 reports domain descriptions, injectivity reductions, sufficient criteria and noninjective examples, but no complete general criterion. Those claims, the partially-solved label and notation-restoration verification are retained as dated metadata, not independently checked current results. No Mellin/Fourier nonvanishing condition is guessed from the scaling formula alone.

Short English statement

Find conditions on the scaling measure that make the Upsilon mixture injective on all input Lévy measures for which the output remains a Lévy measure. The inverse-set scaling is explicit. The source allows general $\sigma$-finite scaling measures, despite narrower wording in its assessment, and leaves the ambient dimension unspecified.

Sources

• Local UnsolvedMath registry, permanent ID 30000697, source code catalogue entry, "Injectivity Criteria for Upsilon Transforms," including the full scaling formula, Lévy-domain question and verification metadata.
• Mini-Workshop: Lévy Processes and Related Topics in Modelling (2007), Oberwolfach Reports, pp. 417--446, DOI 10.4171/owr/2007/07: https://doi.org/10.4171/owr/2007/07.
• Supplied reference "general Upsilon transformations": https://alea.impa.br/articles/v4/04-07.pdf.
• Supplied reference "injectivity criteria and counterexample": https://data.math.au.dk/publications/thiele/2008/imf-thiele-2008-02.pdf. These URLs are cited as supplied; no PDFs have been opened.
• Literature triage checked 2026-08-21; record updated 2026-09-16. No external source has been visited or independently checked here.