In this article, we are concerned with the equations of Krylov type on compact Hermitian manifolds, which are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix. Under the assumption of the 𝒞-subsolution, we obtain a priori estimates in cone. By using the method of continuity, we prove an existence theorem, which generalizes the relevant results. As an application, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.
Let be a compact Kähler manifold of complex dimension . In 1978, Yau  proved the famous Calabi-Yau conjecture by solving the following complex Monge-Ampère equation on
with positive function . There have been many generalizations of Yau’s work. One extension of Yau’s Theorem is to the case of Hermitian manifolds, which is initiated by Cherrier  in 1987. The Monge-Ampère equation on compact Hermitian manifolds was solved by Tosatti and Weinkove , building on several earlier works. See [2,4,5, 6,7] and the references therein.
The complex Hessian equation can be expressed as follows:
On compact Kähler manifolds , Hou  proved the existence of a smooth admissible solution of the complex Hessian equation by assuming the nonnegativity of the holomorphic bisectional curvature. Later, Hou et al.  obtained the second-order estimate without any curvature assumption. Using Hou et al.’s estimate, Dinew and Kolodziej  applied a blow-up argument to prove the gradient estimate and solved the complex Hessian equation on compact Kähler manifolds. The corresponding problem on Hermitian manifolds was solved by Zhang  and Székelyhidi  independently.
The complex Hessian quotient equations include the complex Monge-Ampère equation and the complex Hessian equation. Let be a real (1, 1) form, the complex Hessian quotient equations can be expressed as follows:
When is constant, one special case is the so-called Donaldson equation :
In this article, we are concerned with Hessian equations of Krylov type in the form of the linear combinations of the Hessian, which can be written as follows:
The Dirichlet problem of (1.1) on -convex domain in was first studied by Krylov  about 20 years ago. He observed that if for , the natural admissible cone to make (1.1) elliptic is also the -cone, which is the same as the -Hessian equation case, where
Guan and Zhang  solved the equation of Krylov type on the problem of prescribing convex combination of area measures. Pingali  proved a priori estimates to the following equation in Kähler case:
where are smooth real functions such that either or , and . Recently, Phong and Tô  solved Hessian equations of Krylov type on compact Kähler manifolds, where are non-negative constants for . When are non-negative smooth functions for , analogous results on compact Kähler manifolds are obtained by Chen  and Zhou  independently.
Naturally, we want to extend this result to Hermitian manifolds. On the other hand, Zhou  believed that the condition on is not necessary. In fact, Guan and Zhang  considered equation (1.1) without the sign requirement for the coefficient function .
In this article, we mainly concern equation (1.1) on Hermitian manifold without any sign requirement for . Let be the set of all the real (1, 1) forms, eigenvalues of which belong to . To ensure the ellipticity and non degeneracy of the equation in , we require smooth real functions to satisfy the conditions: for , either or , and . Let and . To state our main results, we need also the following condition of -subsolution, which is similar to -subsolution introduced by Székelyhidi .
A smooth real function is a -subsolution to (1.1), if and at each point , the set
Let be a compact Hermitian manifold, a real (1, 1) form on . Suppose that is a -subsolution of equation (1.1) and at each point ,
Then there exists a smooth real function on and a unique constant solving
with and .
Let be a compact Kähler manifold, a closed -form. Suppose that is a -subsolution of equation (1.1) and
where , . Then there exists a smooth real function u on M and a unique constant b solving
with and .
Lately, Pingali  proved an existence result of the deformed Hermitian Yang-Mills equation with phase angle on compact Kähler threefold. Let be a closed form, . From , the deformed Hermitian Yang-Mills equation on compact Kähler threefold can be written as follows:
As an application of Corollary 1.3, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.
Let be a compact Kähler threefold, constant phase angle , and a positive definite closed form, satisfying the following conditions:
Then there exists a smooth solution to equation (1.6) with and .
The rest of this article is organized as follows. In Section 2, we set up some notations and provide some preliminary results. In Section 3, we give the estimate by the Alexandroff-Bakelman-Pucci maximum principle. In Section 4, we establish the estimate for equation (1.1) by the method of Hou et al.  and the -subsolution condition. In Section 5, we give the gradient estimate. In Section 6, we give the proof of Theorem 1.2, Corollaries 1.3, and 1.4 by the method of continuity. Although the method is very standard in the study of elliptic PDEs, it is not easy to carry out this method on a compact Hermitian manifold. Since the essential -subsolution condition depends on , we have to find a uniform -subsolution condition for the solution flow of the continuity method.
In this section, we set up the notation and establish some lemmas. Let denote the th elementary symmetric function
For completeness, we define and . Let denote the symmetric function with and the symmetric function with . Also denote by the symmetric function with deleting the th row and th column, and the symmetric function with deleting the th, th rows and th, th columns, for all . In local coordinates,
Define as the eigenvalue set of with respect to . In local coordinates, equation (1.1) can be written in the following form:
Equivalently, we can rewrite equation (2.1) as follows:
The following lemma is similar to Lemma 2.3 in , but we need to discuss it more widely, that is, instead of .
If is a solution of (2.2), and , , then
If , then we obtain from equation (2.2)
If , i.e., , we see from Lemma 2.1 that
for , which completes the proof of Lemma 2.2.□
For any point , choose a local frame such that . For the convenience of notations, we will write equation (2.2) as follows:
where and . Let
then at , we have
 If and , , then the operator F is elliptic and concave in .
If is a solution of (2.4), , then at ,
Under assumptions of Theorem 1.2, there is a constant such that
Without loss of generality, we may assume that . Thus,
Since is a -subsolution, if is small enough, still satisfies Definition 1.1. Since is compact, there are uniform constants and such that
Direct calculation yields
Since is concave in , from (2.8), we obtain
In this section, we obtain the estimate by using the Alexandroff-Bakelman-Pucci maximum principle and prove the following Proposition 3.1, which is similar to the approach of Székelyhidi .
Let for and be a smooth real form on . Assume that and are solution and -subsolution to (1.1) with , , respectively. We normalize u such that . There is a constant C depending on the given data, such that
To simplify notation, we can assume , otherwise we modify the background form . Therefore, . The following goal is to prove that has a uniform lower bound. Notice that , so , that is,
Let be the Green’s function of a Gauduchon metric conformal to . From , there is a uniform constant such that
Since , there is a point such that . Hence,
Next, we choose local coordinates at the minimum point of and . Let and for a small . From the Alexandroff-Bakelman-Pucci maximum principle, we obtain
is a subsolution, which means that is bounded. Since is compact, there is uniform constant such that satisfies Definition 1.1. Since is a contact set, we have , for , which implies . Choosing such that , on , we have
we obtain that is bounded, which yields . Then
From this and (3.2), we obtain
On the other hand, we have for
Since is uniformly bounded, this inequality contradicts (3.3) if is very large.□
In this section, we establish the estimate to equation (1.1). Our calculation is similar to that in , but on Hermitian manifolds, equation (1.1) are much more difficult to treat due to the torsion terms.
4.1 Notations and lemma
In local coordinates , the Chern connection and torsion are given, respectively, by
while the curvature tensor by
For , we denote
Let , and . For a fixed point , choose a local coordinates such that . Since need not be distinct at , we will perturb slightly such that become smooth functions near . Let be a diagonal matrix such that and are small, satisfying . Define the matrix . At , has eigenvalues
Commuting derivative of gives
implies that . If the matrix is sufficiently small, then , which means that
We are trying to bound from mentioned earlier, so we can assume . Hence,
From here on, will always denote such a constant, which depends on the given data and may vary from line to line. Using
From (4.2), we obtain
From this, we have
Let for and be a smooth real form on . Assume that and are solution and -subsolution to (1.1) with , , respectively. Then there is an estimate as follows:
where C is a uniform constant.
We assume that the subsolution , since otherwise we modify the background form . We normalize so that . Consider the function
and is to be determined later. Direct calculation gives
Since is compact, attains its maximum at some point . From now on, all the calculations will be carried out at the point and the Einstein summation convention will be used. Calculating covariant derivatives, we obtain
Multiplying (4.13) by and summing it over index yield
Direct calculation deduces that
It is shown by Krylov in  that the is concave in for , which means that
Direct computation gives
where the last inequality is given by Lemma 2.2. Noting that
we estimate the second term in (4.14)