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The purpose of this survey paper is to present an up-to-date account of the recent advances made in the study of

The homotopy operator has been playing an important role in the study of

Let

The nonlinear partial differential equation

Let

A continuously increasing function

We say a Young function

From [

Let

Suppose that

The following Hölder inequality will be used in this paper.

Let

The following

Let

Using (

The basic

We say the weight

A weight

Let

If

It is well known that

Let

It is clear that

The following result shows that

If we put

The definitions of the following several weight classes can be found in [

We say that the weight

A pair of weights

A pair of weights

A pair of weights

Using the basic Poincaré-type estimate for the homotopy operator

Let

The above

Also, using the procedure developed to extend the local inequalities into the John domains, we have the following global Poincaré-type inequality.

Let

By the same method used to prove the imbedding inequalities, we can prove the following local and global imbedding inequalities, Theorems

Let

Let

So far, we have presented the

Let

Note that inequality (

Let

The above inequalities have integral representations; for example, inequality (

Let

In Theorem

Let

Note that inequality

Let

If we choose

Let

Choosing

Let

Letting

Let

Finally, when

Let

The following local Poincaré-type inequality with the

Let

From (

Since each of

Let

From (

Similar to (

In this section, we will present Lipschitz and BMO norm inequalities for the homotopy operator. All results presented in this section and next section can be found in [

Let

The following Theorem

Let

From Theorem

Using the similar method involved in the proof of Theorem

Let

We have discussed some estimates for the Lipschitz norm

If a differential form

Using Theorems

Let

Since inequality (

As in the proof of Theorem

Let

In this section, we present the weighted Lipschitz and

Let

First, we note that

Next, we present the

Let

Let

Replacing

Substituting (

In this section, we discuss the global inequalities in the following

Let

From the above definition, we see that

Let

From Definition

Similar to the local case, the following global inequality with the Orlicz norm

Let

Note that (

Let

Choosing

Let

Note that (

Let

In this section, we present the norm estimates for the composition of the homotopy operator and projection operator. The results presented in this section can be found in [

Let

Let

Each

Let

Let

Let

Let

Let

In applications, such as in calculating electric or magnetic fields, we often face the fact that the integrand contains a singular factor. So, the above result was extended into the following singular weighted case.

Let

Let

(1) Replacing

The following definition of

Let

Let