## Abstract

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

## 1 Introduction

Let

with positive function

The complex Hessian equation can be expressed as follows:

On compact Kähler manifolds

The complex Hessian quotient equations include the complex Monge-Ampère equation and the complex Hessian equation. Let

When

After some progresses made in [14,15, 16,17], Song and Weinokove [18] solved the Donaldson equation on closed Kähler manifolds via

and solved this equation on closed Kähler manifolds by assuming a cone condition. When

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

Guan and Zhang [23] solved the equation of Krylov type on the problem of prescribing convex combination of area measures. Pingali [24] proved *a priori* estimates to the following equation in Kähler case:

where

Naturally, we want to extend this result to Hermitian manifolds. On the other hand, Zhou [27] believed that the condition on

In this article, we mainly concern equation (1.1) on Hermitian manifold without any sign requirement for

## Definition 1.1

A smooth real function

is bounded.

## Theorem 1.2

*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*

## Corollary 1.3

*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 [28] proved an existence result of the deformed Hermitian Yang-Mills equation with phase angle

As an application of Corollary 1.3, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.

## Corollary 1.4

*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

## 2 Preliminaries

In this section, we set up the notation and establish some lemmas. Let

For completeness, we define

Define

where

Equivalently, we can rewrite equation (2.1) as follows:

## Lemma 2.1

[29,30] *For*
*and*
*we have*

The following lemma is similar to Lemma 2.3 in [27], but we need to discuss it more widely, that is,

## Lemma 2.2

*If*
*is a solution of* (2.2),
*and*
*then*

## Proof

If

If

for

For any point

where

then at

Let

## Lemma 2.3

[23] *If*
*and*
*then the operator F is elliptic and concave in*

From Lemma 2.4 in [27] and Lemma 2.2, we have the following lemma.

## Lemma 2.4

*If*
*is a solution of* (2.4),
*then at*

## Lemma 2.5

*Under assumptions of*
Theorem 1.2, *there is a constant*
*such that*

*or*

## Proof

Without loss of generality, we may assume that

Since

where

Direct calculation yields

Since

By substituting (2.10) into (2.9), we obtain

Set

## 3
C
0
estimate

In this section, we obtain the

## Proposition 3.1

*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*

## Proof

To simplify notation, we can assume

Let

Since

which yields

Next, we choose local coordinates at the minimum point of

where

Let

Since

we obtain that

From this and (3.2), we obtain

On the other hand, we have for

so

Then,

Since

## 4
C
2
estimate

In this section, we establish the

### 4.1 Notations and lemma

In local coordinates

while the curvature tensor

For

and

Let

### Lemma 4.1

### Proof

Commuting derivative of

We are trying to bound

From here on,

we have

From (4.2), we obtain

From this, we have

Substituting (4.5), (4.6), and (4.8) into (4.4) gives (4.3).□

### 4.2
C
2
estimate

### Proposition 4.2

*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*.

### Proof

We assume that the

Here,

where

and

and

Since

Multiplying (4.13) by

We will control some terms in (4.14). Covariant differentiating equation (2.4) twice in the

and

Direct calculation deduces that

From Lemmas 4.1 and (4.16), we can estimate the first term in (4.14)

It is shown by Krylov in [22] that the

Direct computation gives

which yields

where the last inequality is given by Lemma 2.2. Noting that

we have

Substituting (4.19) into (4.18) and by Lemma 2.2,

Since

we have

From

we estimate the second term in (4.14)