Inverse Complex Hessian Quotient Operators

• Inverse complex Hessian quotient operators are fully nonlinear elliptic operators constructed from ratios of elementary symmetric polynomials of the complex Hessian eigenvalues.
• They establish a bridge between viscosity and pluripotential frameworks using equivalent formulations and smooth approximation schemes to ensure admissibility and degenerate ellipticity.
• Their Dirichlet theory leverages Gårding cone conditions and concavity, yielding unique solutions under precise structural bounds in both Euclidean domains and compact Hermitian manifolds.

Inverse complex Hessian quotient operators are fully nonlinear complex elliptic operators built from ratios of elementary symmetric polynomials of the eigenvalues of the complex Hessian $Hu=(u_{i\bar j})$, or equivalently from wedge-product quotients of forms involving $dd^c u$. In the literature summarized here, the most explicit model is the reciprocal-type equation

$$\frac{(dd^c u)^n}{(dd^c u)^{\,n-k}\wedge \omega^k}=\psi(z,u),$$

which is, in eigenvalue notation and up to normalization conventions, the quotient

$$\frac{S_n(\lambda(Hu))}{S_{n-k}(\lambda(Hu))}=\psi(z,u).$$

Although the exact phrase “inverse complex Hessian quotient operator” is not the primary terminology of the foundational viscosity paper, that paper directly studies such operators through its treatment of “inverse $\sigma_k$ equations,” while also placing them inside a broader theory of complex Hessian quotient equations of the form $S_k/S_\ell$ (Dinew et al., 2017).

1. Operator class and basic formulas

The local framework begins with equations

$F[u]:=F(z,u,Du,Hu)=0 \qquad \text{in }\Omega,$

where

$$Du=(\partial_{z_1}u,\dots,\partial_{z_n}u),\qquad Hu=(u_{i\bar j})_{i,j=1}^n,$$

and the operator depends only on the eigenvalues $\lambda(Hu)$ of the Hermitian matrix $Hu$ (Dinew et al., 2017). The elementary symmetric polynomials are

$dd^c u$0

with normalized version

$dd^c u$1

For complex Hessian operators,

$dd^c u$2

This provides the standard bridge between the eigenvalue description and the differential-form description. The direct quotient equation treated in the viscosity theory is

$dd^c u$3

while the inverse-type model is

$dd^c u$4

In eigenvalue variables, this is precisely

$dd^c u$5

and equivalently can be rewritten as the reciprocal quotient equation

$dd^c u$6

Accordingly, inverse complex Hessian quotient operators sit naturally inside the general quotient formalism once one allows reciprocal normalization (Dinew et al., 2017).

A later viscosity existence theory on compact Hermitian manifolds adopts the normalized quotient operator

$dd^c u$7

and treats complex Hessian quotient equations in eigenvalue form

$dd^c u$8

That work does not formulate a separate inverse-quotient theorem, but its operator-theoretic discussion is directly relevant after reciprocal reformulation of the right-hand side (Cheng et al., 28 Jan 2025).

2. Admissibility, cones, and ellipticity

The natural admissibility condition is expressed through the Gårding cones

$dd^c u$9

A $$\frac{(dd^c u)^n}{(dd^c u)^{\,n-k}\wedge \omega^k}=\psi(z,u),$$0 function is $$\frac{(dd^c u)^n}{(dd^c u)^{\,n-k}\wedge \omega^k}=\psi(z,u),$$1-admissible if

$$\frac{(dd^c u)^n}{(dd^c u)^{\,n-k}\wedge \omega^k}=\psi(z,u),$$2

The admissible cone $$\frac{(dd^c u)^n}{(dd^c u)^{\,n-k}\wedge \omega^k}=\psi(z,u),$$3 is assumed open, convex, symmetric under permutations, and to contain the positive cone

$$\frac{(dd^c u)^n}{(dd^c u)^{\,n-k}\wedge \omega^k}=\psi(z,u),$$4

For standard Hessian quotient equations the natural cone is $$\frac{(dd^c u)^n}{(dd^c u)^{\,n-k}\wedge \omega^k}=\psi(z,u),$$5, and a viscosity subsolution of

$$\frac{(dd^c u)^n}{(dd^c u)^{\,n-k}\wedge \omega^k}=\psi(z,u),$$6

is $$\frac{(dd^c u)^n}{(dd^c u)^{\,n-k}\wedge \omega^k}=\psi(z,u),$$7-subharmonic (Dinew et al., 2017).

Degenerate ellipticity for a general operator $$\frac{(dd^c u)^n}{(dd^c u)^{\,n-k}\wedge \omega^k}=\psi(z,u),$$8 means

$$\frac{(dd^c u)^n}{(dd^c u)^{\,n-k}\wedge \omega^k}=\psi(z,u),$$9

For eigenvalue-type operators this becomes monotonicity in cone directions, encoded by

$$\frac{S_n(\lambda(Hu))}{S_{n-k}(\lambda(Hu))}=\psi(z,u).$$0

The same viscosity framework assumes

$$\frac{S_n(\lambda(Hu))}{S_{n-k}(\lambda(Hu))}=\psi(z,u).$$1

and, in the Dirichlet problem section, further requires concavity and homogeneity. Homogeneity plus positivity imply

$$\frac{S_n(\lambda(Hu))}{S_{n-k}(\lambda(Hu))}=\psi(z,u).$$2

These are the structural hypotheses that place quotient operators—and inverse quotients after suitable normalization—inside the degenerate elliptic theory (Dinew et al., 2017).

For quotient operators on compact Hermitian manifolds, the asymptotic operator

$$\frac{S_n(\lambda(Hu))}{S_{n-k}(\lambda(Hu))}=\psi(z,u).$$3

plays the role of the cone-at-infinity control. In the quotient case

$$\frac{S_n(\lambda(Hu))}{S_{n-k}(\lambda(Hu))}=\psi(z,u).$$4

the explicit formula is

$$\frac{S_n(\lambda(Hu))}{S_{n-k}(\lambda(Hu))}=\psi(z,u).$$5

with associated cone

$$\frac{S_n(\lambda(Hu))}{S_{n-k}(\lambda(Hu))}=\psi(z,u).$$6

This suggests that reciprocal formulations inherit a nontrivial asymptotic cone geometry rather than a purely formal algebraic inversion (Cheng et al., 28 Jan 2025).

3. Viscosity formulation and automatic admissibility

The Trudinger-style viscosity definition adapted to complex Hessians distinguishes subsolutions and supersolutions in a way compatible with restricted ellipticity cones. An upper semicontinuous $$\frac{S_n(\lambda(Hu))}{S_{n-k}(\lambda(Hu))}=\psi(z,u).$$7 is a viscosity subsolution of

$$\frac{S_n(\lambda(Hu))}{S_{n-k}(\lambda(Hu))}=\psi(z,u).$$8

if every $$\frac{S_n(\lambda(Hu))}{S_{n-k}(\lambda(Hu))}=\psi(z,u).$$9 upper test $\sigma_k$0 touching $\sigma_k$1 from above at $\sigma_k$2 satisfies

$\sigma_k$3

A lower semicontinuous $\sigma_k$4 is a viscosity supersolution if there are no lower test functions $\sigma_k$5 at $\sigma_k$6 such that

$\sigma_k$7

Because the ellipticity region may be confined to a cone, the paper also introduces a cone-restricted supersolution notion requiring the lower test to satisfy

$\sigma_k$8

before checking the sign condition (Dinew et al., 2017).

A central structural fact for quotient and inverse quotient equations is that the viscosity notion itself enforces admissibility. If $\sigma_k$9 is a viscosity subsolution of

$S_k/S_\ell$0

then any upper test $S_k/S_\ell$1 must satisfy

$S_k/S_\ell$2

In particular, any viscosity subsolution of

$S_k/S_\ell$3

is $S_k/S_\ell$4-subharmonic. This identifies the correct admissible class for quotient equations and, by the same structural logic, for reciprocal inverse-quotient formulations (Dinew et al., 2017).

The comparison principle is the uniqueness backbone. For

$S_k/S_\ell$5

with $S_k/S_\ell$6 nondecreasing in $S_k/S_\ell$7, and $S_k/S_\ell$8 satisfying either an ellipticity-growth condition or concavity plus homogeneity, the theory proves

$S_k/S_\ell$9

for subsolutions $F[u]:=F(z,u,Du,Hu)=0 \qquad \text{in }\Omega,$0 and supersolutions $F[u]:=F(z,u,Du,Hu)=0 \qquad \text{in }\Omega,$1. This estimate is the fundamental mechanism behind uniqueness, Perron existence, and stability (Dinew et al., 2017).

On compact Hermitian manifolds, the viscosity definition is similarly stated in terms of upper and lower test functions $F[u]:=F(z,u,Du,Hu)=0 \qquad \text{in }\Omega,$2 for the operator

$F[u]:=F(z,u,Du,Hu)=0 \qquad \text{in }\Omega,$3

with the test Hessian required either to remain inside $F[u]:=F(z,u,Du,Hu)=0 \qquad \text{in }\Omega,$4 or else to violate admissibility on the supersolution side. The theory there is organized around the strict $F[u]:=F(z,u,Du,Hu)=0 \qquad \text{in }\Omega,$5-subsolution condition rather than boundary comparison, reflecting the closed-manifold setting (Cheng et al., 28 Jan 2025).

4. Dirichlet theory for quotient equations and its inverse-quotient implications

The general boundary value problem is

$F[u]:=F(z,u,Du,Hu)=0 \qquad \text{in }\Omega,$6

where $F[u]:=F(z,u,Du,Hu)=0 \qquad \text{in }\Omega,$7 is a bounded $F[u]:=F(z,u,Du,Hu)=0 \qquad \text{in }\Omega,$8 $F[u]:=F(z,u,Du,Hu)=0 \qquad \text{in }\Omega,$9-pseudoconvex domain, $$Du=(\partial_{z_1}u,\dots,\partial_{z_n}u),\qquad Hu=(u_{i\bar j})_{i,j=1}^n,$$0, and $$Du=(\partial_{z_1}u,\dots,\partial_{z_n}u),\qquad Hu=(u_{i\bar j})_{i,j=1}^n,$$1 with $$Du=(\partial_{z_1}u,\dots,\partial_{z_n}u),\qquad Hu=(u_{i\bar j})_{i,j=1}^n,$$2 and $$Du=(\partial_{z_1}u,\dots,\partial_{z_n}u),\qquad Hu=(u_{i\bar j})_{i,j=1}^n,$$3 weakly increasing. Here $$Du=(\partial_{z_1}u,\dots,\partial_{z_n}u),\qquad Hu=(u_{i\bar j})_{i,j=1}^n,$$4-pseudoconvexity means that there exists $$Du=(\partial_{z_1}u,\dots,\partial_{z_n}u),\qquad Hu=(u_{i\bar j})_{i,j=1}^n,$$5 such that

$$Du=(\partial_{z_1}u,\dots,\partial_{z_n}u),\qquad Hu=(u_{i\bar j})_{i,j=1}^n,$$6

is $$Du=(\partial_{z_1}u,\dots,\partial_{z_n}u),\qquad Hu=(u_{i\bar j})_{i,j=1}^n,$$7-subharmonic near $$Du=(\partial_{z_1}u,\dots,\partial_{z_n}u),\qquad Hu=(u_{i\bar j})_{i,j=1}^n,$$8, where $$Du=(\partial_{z_1}u,\dots,\partial_{z_n}u),\qquad Hu=(u_{i\bar j})_{i,j=1}^n,$$9 (Dinew et al., 2017).

Under the structural assumptions of concavity, ellipticity, positivity in $\lambda(Hu)$0, vanishing on $\lambda(Hu)$1, and homogeneity, the Dirichlet problem admits a unique admissible solution

$\lambda(Hu)$2

The proof uses a global admissible subsolution built from a defining function and harmonic extension, a harmonic supersolution, Perron’s method, and the comparison principle. The theory allows degenerate ellipticity in the sense that $\lambda(Hu)$3 may vanish on $\lambda(Hu)$4, so strict ellipticity up to the boundary of the cone is not assumed (Dinew et al., 2017).

Applied to quotient operators, this yields the explicit result that for

$\lambda(Hu)$5

with $\lambda(Hu)$6 a smooth bounded $\lambda(Hu)$7-pseudoconvex domain and $\lambda(Hu)$8, there exists a unique viscosity solution

$\lambda(Hu)$9

for any continuous boundary data $Hu$0. The structural fact used to place the quotient operator inside the theory is that

$Hu$1

is concave and homogeneous (Dinew et al., 2017).

This Dirichlet theorem is stated for direct quotients rather than inverse quotients. A plausible implication is that many inverse quotient equations can be treated by rewriting

$Hu$2

as

$Hu$3

provided the admissible cone remains $Hu$4 and the transformed right-hand side stays positive. That reformulation is explicitly suggested in the compact Hermitian manifold viscosity theory, although it is identified there as an inference from the operator framework rather than a separate theorem (Cheng et al., 28 Jan 2025).

5. The inverse quotient model and the viscosity–pluripotential bridge

The most direct analysis of inverse complex Hessian quotient operators concerns the equation

$Hu$5

together with the associated pluripotential formulation

$Hu$6

This is the distinguished inverse-type model in the complex setting (Dinew et al., 2017).

A major contribution is the equivalence theorem for viscosity and mixed-form formulations. For $Hu$7 and $Hu$8, the following are equivalent:

1.

$Hu$9

1. for all $dd^c u$00,

$dd^c u$01

1. there exist smooth psh approximants $dd^c u$02 satisfying multilinear inequalities;
2. there exist smooth strictly psh approximants $dd^c u$03 with

$dd^c u$04

This equivalence is especially important for inverse quotient operators because it provides several interchangeable analytic languages: viscosity inequalities, mixed-form inequalities, and smooth approximation schemes (Dinew et al., 2017).

The pluripotential consequences are equally sharp. If $dd^c u$05 is a viscosity subsolution of the inverse quotient equation, then

$dd^c u$06

in the pluripotential sense, and also

$dd^c u$07

in the pluripotential sense. The second inequality is singled out in the paper as showing that a viscosity subsolution of the inverse quotient equation automatically satisfies a uniform lower $dd^c u$08-Hessian bound. The authors remark that this implies the natural function space for the inverse equation is much smaller than the whole bounded psh class (Dinew et al., 2017).

There is also a converse criterion: if $dd^c u$09 is locally bounded psh and

$dd^c u$10

pluripotentially, then

$dd^c u$11

in the viscosity sense. For supersolutions, if

$dd^c u$12

is a viscosity supersolution of the inverse quotient equation, then there exist increasing strictly psh approximants $dd^c u$13 converging to $dd^c u$14 in capacity with

$dd^c u$15

pointwise, hence

$dd^c u$16

in the pluripotential sense (Dinew et al., 2017).

The bridge becomes a full equivalence under extra positivity: if

$dd^c u$17

in the viscosity sense and

$dd^c u$18

in the pluripotential sense, and if

$dd^c u$19

then $dd^c u$20 is a viscosity solution of the equality problem. This nondegenerate bridge theorem is one of the clearest structural results for inverse complex Hessian quotient operators in the cited literature (Dinew et al., 2017).

A key algebraic input behind these equivalences is the matrix inequality

$dd^c u$21

used to pass from quotient inequalities to mixed-form inequalities involving $dd^c u$22 (Dinew et al., 2017).

6. Regularity, related developments, and scope

For the general Dirichlet problem

$dd^c u$23

the Perron solution belongs to

$dd^c u$24

and under strict $dd^c u$25-pseudoconvexity, boundary regularity of $dd^c u$26, and $dd^c u$27-type assumptions on $dd^c u$28, the viscosity solution satisfies

$dd^c u$29

The proof uses global lower barriers from $dd^c u$30-subharmonic defining functions, harmonic upper barriers, a Walsh/Bedford–Taylor style translation argument, and homogeneity and concavity of $dd^c u$31. For inverse quotient equations, however, the principal regularity mechanism emphasized in the direct analysis is stability under smooth approximation and convergence in capacity rather than a separate Hölder theorem (Dinew et al., 2017).

Subsequent developments broaden the quotient side of the theory. On compact Hermitian manifolds, existence of viscosity solutions is proved for

$dd^c u$32

under the existence of a strict viscosity $dd^c u$33-subsolution, and the theorem explicitly covers the quotient operators

$dd^c u$34

That work removes the determinant-domination condition required in the authors’ earlier theory and treats precisely the regime where $dd^c u$35 (Cheng et al., 28 Jan 2025).

Several other papers are relevant mainly by analogy rather than by direct treatment of inverse complex quotients. A variational finite-energy framework for twisted complex Hessian equations on bounded $dd^c u$36-hyperconvex domains develops the quotient

$dd^c u$37

and provides tools such as admissible envelopes, capacity domination, and weak stability, but it does not study inverse Hessian quotient equations directly (Badiane et al., 2023). In the quaternionic HKT setting, a $dd^c u$38 estimate is established for the quotient equation

$dd^c u$39

with direct use of a cone condition and concavity of $dd^c u$40, giving a structurally parallel quotient theory outside the complex category (Chen, 2022).

Recent work on degenerate complex $dd^c u$41-Hessian equations isolates concavity mechanisms and sharp degeneracy exponents for $dd^c u$42-type operators, including the quoted concavity of

$dd^c u$43

on $dd^c u$44. That paper does not treat inverse quotient equations, but it strongly suggests that inverse quotient regularity should be approached through a concave transform of the operator rather than the raw reciprocal formulation (Lyu, 28 Feb 2026). By contrast, several recent real-variable papers on $dd^c u$45, $dd^c u$46, and Neumann or Liouville problems are explicitly outside the complex setting, so their significance for inverse complex Hessian quotient operators is structural rather than direct (Lu et al., 16 Feb 2026, Mei et al., 25 Apr 2026, Chen et al., 2020, Gong et al., 10 Jan 2025).

Within this literature, the precise direct takeaway is narrow but substantial. The foundational viscosity paper establishes a general Dirichlet theory for complex Hessian quotient equations and gives a detailed viscosity–pluripotential treatment of the inverse model

$dd^c u$47

including equivalent formulations, approximation theorems, and bridge results between viscosity and pluripotential inequalities (Dinew et al., 2017). The later manifold theory extends existence techniques for quotient operators in the compact Hermitian setting, and the surrounding literature clarifies the structural role of concavity, admissible cones, asymptotic operators, and approximation, but does not yet present a comparably general standalone theory for inverse complex Hessian quotient operators as a separate named class (Cheng et al., 28 Jan 2025).