Onofri Trace Inequality in Conformal and Matrix Analysis

• Onofri trace inequality is a family of results linking the integrability of exponential-type functionals with energy and trace expressions in conformal geometry and PDEs.
• It serves as a critical endpoint for Sobolev-type embeddings and logarithmic inequalities, with extensions to higher dimensions and matrix settings.
• Its applications span uniqueness, rigidity, and stability analyses, influencing areas from iterative optimization in signal processing to quantum information theory.

The Onofri trace inequality comprises a family of analytical results that link integrability properties of exponential-type functionals with energy or trace expressions, often arising in conformal geometry, nonlinear analysis, and the theory of partial differential equations. Initially formulated on the two-dimensional sphere and Euclidean plane, the Onofri inequality was extended as a trace inequality for matrices and, more recently, for functions on higher-dimensional domains, manifolds, and spaces with boundary. The Onofri trace inequality is closely related to sharp Sobolev-type embeddings, logarithmic Sobolev inequalities, and plays a central role in problems concerning uniqueness, rigidity, and stability.

1. Classical Formulation and Extensions

The original Onofri inequality, established for conformal factors on the two-sphere $\mathbb{S}^2$, asserts that

$$\log \int_{\mathbb{S}^2} e^{2u} d\sigma \le \frac{1}{4} \| u \|^2_{L^2(\mathbb{S}^2)} + C$$

for $u$ with zero mean. Stereographic projection yields a Euclidean form: $$\log\left(\int_{\mathbb{R}^2} e^u d\mu_2\right) - \int_{\mathbb{R}^2} u d\mu_2 \le \frac{1}{16\pi} \int_{\mathbb{R}^2} |\nabla u|^2 dx,\quad d\mu_2(x) = \frac{1}{\pi(1+|x|^2)^2} dx.$$ The inequality is recognized as an endpoint of Gagliardo–Nirenberg interpolation inequalities and functions as a logarithmic Sobolev inequality in the conformally invariant setting (Pino et al., 2012).

Trace inequalities in matrix analysis generalize the Onofri form: $$\operatorname{Tr} \{ (A - B)[B^{-1} - A^{-1}] \} \geq 0$$ for positive definite matrices $A, B$. The extension to arbitrary sums with cumulative matrices and differences weighted by inverses yields

$$\operatorname{Tr} \left\{ \sum_{k=1}^K (A_k - B_k) \left[ \left(\sum_{\ell=1}^k B_\ell\right)^{-1} - \left(\sum_{\ell=1}^k A_\ell\right)^{-1} \right] \right\} \geq 0,$$

where $A_1, B_1$ are positive definite and $A_k, B_k$ are positive semidefinite for $k\geq 2$ (Belmega et al., 2010).

2. Generalizations to Higher Dimensions and Weighted Spaces

Del Pino and Dolbeault established a higher-dimensional Euclidean Onofri inequality: $$\ln\left(\int_{\mathbb{R}^d} e^u d\mu_d\right) - \int_{\mathbb{R}^d} u d\mu_d \leq \alpha_d \mathcal{H}_d(x, \nabla u)$$ with the probability measure

$$d\mu_d(x) = \frac{d}{|\mathbb{S}^{d-1}|} \frac{dx}{(1 + |x|^{d/(d-1)})^d}$$

and the nonlinear Sobolev–Orlicz norm involving

$$\mathcal{R}_d(X, Y) = |X+Y|^d - |X|^d - d|X|^{d-2}(X \cdot Y)$$

(Pino et al., 2012).

Extensions to weighted Sobolev spaces (e.g., $WAN(\mathbb{R}^N)$ with norms involving $|\nabla u|^2 |V U_N|^{N-2}$) allow the inequality to hold outside the sphere context and in higher dimensions (Borgia et al., 8 Aug 2024). Importantly, optimal constants are established, coinciding with those of the sharp logarithmic Moser–Trudinger inequalities on balls (Borgia et al., 8 Aug 2024).

3. Matrix Trace Inequalities and Applications

In the context of positive definite and semidefinite matrices, the generalized trace inequality provides recursive and cumulative control:

• For $K$ terms, cumulative matrices $X_k = \sum_{i=1}^k A_i$, $Y_k = \sum_{i=1}^k B_i$ remain positive definite.
• The trace inequality holds recursively: $T_K$ can be written in terms of $T_{K-1}$ plus an additional nonnegative contribution.

Applications include:

• Uniqueness conditions for Nash equilibria in MIMO game-theoretic models, where the trace inequality ensures diagonally strict concavity (Belmega et al., 2010).
• Matrix stability analysis in control theory and signal processing.
• Derivation of bounds in iterative optimization and cumulative perturbation models.

4. Rigidity, Optimizers, and Extremal Functions

Rigidity results for Onofri-type inequalities identify cases where equality is achieved only for highly symmetric or trivial functions:

• For Onofri trace inequalities on spheres, optimizers are conformal maps or their linearizations (Dolbeault et al., 2014, Ma et al., 20 Aug 2025).
• In higher dimensions (e.g., 4-manifolds), extremals are classified through invariant tensor methods and second-order derivative estimates, showing that equality arises only for specific conformal factors connected to the underlying geometry (Ma et al., 20 Aug 2025).
• On domains with boundary (e.g., half-spaces), extremals for the sharp Onofri trace inequality are explicit "bubble" functions parameterized by translation and scaling, classified through analysis of quasi-linear Liouville equations with Neumann boundary conditions and integral identities (Serrin–Zou and Pohozaev) (Dou et al., 21 Sep 2025).

5. Improvements under Constraints and Interplay with Moment Conditions

The coefficient in the Moser–Trudinger–Onofri inequality can be improved by imposing vanishing moments for $e^{2u}$:

• For functions with zero first-order moment, the coefficient drops from $1/4$ to $1/8$; with vanishing moments up to degree $m$, to $1/(4N_m)$, where $N_m$ comes from cubature formulas (Chang et al., 2019).
• The analogous improvement exists for Lebedev–Milin-type inequalities for functions on the disk and circle, connecting to the spectral properties of the Laplacian.

Fine constraints (e.g., control of second-order moment deviation via a matrix $A(u)$) interpolate between full and partial symmetry constraints, yielding nearly optimal constants and serving as uniqueness criteria for related mean field equations—especially for the sphere (Chen et al., 2023).

6. Noncommutative and Quantum Information Theory Contexts

Extensions to operator and matrix settings in quantum information theory are significant:

• Trace inequalities involving CPTP maps and recovery maps ($R_{\varrho,{\cal N}}(T)$) establish monotonicity and contractivity of quantum relative entropy beyond the strictly positive case (now including nonnegative matrices) and provide explicit bounds: $$\operatorname{Tr} R_{\varrho, {\cal N}}(T) < \operatorname{Tr} T$$ (Sharma, 2015).
• Multivariate trace inequalities (Araki–Lieb–Thirring and Golden–Thompson extensions) yield explicit remainder terms in entropy inequalities and recovery measures via measured relative entropy and Rényi divergences, informed by complex interpolation and spectral pinching (Sutter et al., 2016).

7. Functional and Geometric Analytical Perspectives

The Onofri trace inequality underlies geometric flows, spectral analysis, and critical embedding theorems:

• Appears as a limiting case of families of functional or interpolation inequalities (Gagliardo–Nirenberg, Beckner).
• Related to duality formulas in mass transport and nonlinear flows.
• Sharp constants are intimately tied to concentration–compactness principles and quantization phenomena in mean field equations and critical nonlinearity PDEs (Chen et al., 2 Jun 2024).
• Trace and Poincaré inequalities for Sobolev spaces on intervals and domains illustrate boundary value control as an extension of Onofri-type principles (Martínez et al., 2021).

Table: Selected Key Formulations

Inequality type	Formal statement	Reference
Matrix trace (K terms)	$$\operatorname{Tr}\left\{\sum_{k=1}^K (A_k - B_k) [\sum_{\ell=1}^k B_\ell^{-1} -\sum_{\ell=1}^k A_\ell^{-1}]\right\} \geq 0$$	(Belmega et al., 2010)
Euclidean Onofri (d-dim)	$$\ln \left( \int_{\mathbb{R}^d} e^u d\mu_d \right) - \int_{\mathbb{R}^d} u\,d\mu_d \leq \alpha_d \mathcal{H}_d(x,\nabla u)$$	(Pino et al., 2012)
4-sphere, 4th-order	$$\log \int_{\mathbb{S}^4} e^f - \int_{\mathbb{S}^4} f \le \frac{\lambda}{\kappa^2} \int_{\mathbb{S}^4} |\Delta f|^2 + \frac{1}{\kappa}(\frac{1}{8} - 4\lambda)\int_{\mathbb{S}^4} |\nabla f|^2$$	(Ma et al., 20 Aug 2025)
Upper half space trace	$$\log\left(\int_{\partial\mathbb{R}_+^n} e^w\,d\mu_n\right)- \int_{\partial\mathbb{R}_+^n} w d\mu_n \le \alpha_n \int_{\mathbb{R}_+^n} K_n(x,\nabla w)dx$$	(Dou et al., 21 Sep 2025)

These inequalities and their variants characterize critical embedding phenomena, stability and uniqueness in nonlinear PDEs, optimization in matrix analysis, and monotonicity in quantum channel entropy. The systematic extension of the Onofri trace inequality to higher dimensions, weighted spaces, and operator frameworks reveals deep connections to geometric analysis, quantum information, and the theory of concentration and rigidity.