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We will present an up-to-date account of the recent advances made in the study of Poincaré inequalities for differential forms and related operators.

The Poincaré inequalities have been playing an important role in analysis and related fields during the last several decades. The study and applications of Poincaré inequalities are now ubiquitous in different areas, including PDEs and potential analysis. Some versions of the Poincaré inequality with different conditions for various families of functions or differential forms have been developed in recent years. For example, in 1989, Staples in [

Throughout this paper, we assume that

Let

We begin the discussion with the following definitions and weak reverse Hölder inequality in [

A weight

A pair of weights

Let

We first discuss the Poincaré inequality for some differential forms. These forms are not necessary to be the solutions of any version of the

We call a proper subdomain

In 1993, the following Poincaré-Sobolev inequality was proved in [

Let

From Corollary 4.1 in [

Let

The above Poincaré-Sobolev inequalities are about differential forms. We know that the

Let

Note that (

Let

Note that (

Next, we will prove the following global weighted Poincaré-Sobolev inequality in

Let

Clearly, we can write (

In [

Let

Let

Let

Let

In recent years, several versions of the two weight Poincaré inequalities have been developed; see [

Let

Let

We remark that the exponents

Let

Clearly, in this result if

Let

(1) Theorems

Next, we discuss the following version of two-weight Poincaré inequality for differential forms.

Let

If we choose

Let

Selecting

Let

When

Let

Normally, most of these inequalities are developed with the Lebesgue measure. It is noticeable that the following results from [

We first introduce the following lemmas that will be used to prove the local Poincaré inequality with the Radon measure.

Let

Now, we prove the following local Poincaré inequality with the Radon measure which will be used to establish the global inequality.

Let

Assume that

Let

Let

Let

Let

Then, we will prove the global Poincaré inequalities with the Radon measures in the following statement. We firstly introduce the definition of John domains and the Lemma.

A proper subdomain

Each

Let

We may assume

Similarly, choose

Let

Using (

Assume that all conditions in Theorem

In this section, we establish the local Poincaré inequalities for the differential forms in any bounded domain. A continuously increasing function

We say a Young function

From [

The following results were proved in [

Let

From (3.5) in [

Since each of

Let

From (

Similar to (

Based on the above discussion, we extend the local Poincaré inequalities into the global cases in the following

Let

From above definition, we see that

Let

Similar to the local case, the following global Poincaré inequality with the Luxemburg norm

Let

Note that (

It has been proved that any John domain is a special

Let

Choosing

Let

Note that (

Let

In this section, we say that

We will need the following lemma about

Let

The following results about the composition of the Laplace-Beltrami operator and Green’s operator were proved in [

Let

Let

Using Minkowski’s inequality and combining Theorems

Let

Let

Let

For

We have made necessary preparation in the previous section to prove the following Poincaré-type inequality for Green’s operator.

Let

As an application of Theorem

Let

Since

Since Green’s operators can commute with

We notice that all the results developed so far in this section are about differential forms. We do not require that differential form

Let

Similarly, we can extend inequalities (

Let

In [

Let

From Theorem

The first sum can be estimated by (

Using Theorem

Let

The authors declare that there is no conflict of interests regarding the publication of this paper.