\(L^{\varphi}\)-Embedding inequalities for some operators on differential forms
- Guannan Shi^{1},
- Shusen Ding^{2} and
- Yuming Xing^{1}Email author
https://doi.org/10.1186/s13660-015-0887-x
© Shi et al. 2015
Received: 14 August 2015
Accepted: 6 November 2015
Published: 10 December 2015
Abstract
In this paper, the local Poincaré inequality and embedding inequality are proved first. Then the global embedding inequality of composite operators for differential forms on \(L^{\varphi}\)-averaging domains with \(L^{\varphi}\)-norm is established. Some examples are also given to illustrate applications.
Keywords
1 Introduction
- (1)
\(\Omega=\bigcup^{\infty}_{k=1}\Omega_{k}\);
- (2)
\(\Omega_{i}^{0}\cap\Omega_{j}^{0}=\emptyset\), if \(j\neq i\);
- (3)there exist two constants \(C_{1}>0\) and \(C_{2}>0\), such that$$ C_{1}\operatorname{diam}(\Omega_{k})\leq \operatorname{distance}(\Omega_{k},F) \leq C_{2} \operatorname{diam}(\Omega_{k}). $$
In addition, for proving the global embedding inequality on the \(L^{\varphi}\)-averaging domain, we also need the following definitions and notations.
A function \(\varphi(x)\) is called an Orlicz function, if \(\varphi(x)\) satisfies: (1) \(\varphi(x)\) is continuously increasing; (2) \(\varphi(0)=0\). Furthermore, if the Orlicz function \(\varphi(x)\) is a convex function, \(\varphi(x)\) is named a Young function. Therefore, based on this type of functions, the Orlicz norm for differential forms can be denoted as follows.
It is easy to prove that \(L^{\varphi}_{E}\) is a Banach space. Actually, provided that \(\varphi(x)\) is taken as \(\varphi(x)=x^{s}\) (\(s>0\)), then \(\varphi(x)\) is an Orlicz function, and it trivially corresponds to a \(L^{s}(\mu)\) space. As a result of that, we can say that \(L^{\varphi}_{E}\) is the generalization of \(L^{s}_{E}\).
Definition 1.1
[13]
Using the same analysis method as of the \(L^{\varphi}_{E}\)-norm, we can conclude that \(L^{\varphi}\)-averaging domains are the generalization of \(L^{s}\)-averaging domains.
2 Main results
- (1)
\(1/C \leq{\varphi(t^{1/p})}/{f(t)}\leq C\);
- (2)
\(1/C\leq{\varphi(t^{1/q})}/{g(t)}\leq C\),
Now, we establish four important theorems based on the above-mentioned conditions.
Theorem 2.1
Theorem 2.2
Based on the above theorem, we cannot only establish the following global embedding inequality on \(L^{\varphi}\)-averaging domains, but we also get the global Poincaré-type inequality fortunately.
Theorem 2.3
Theorem 2.4
3 Preliminary results
For proving the theorems in Section 2, we shall show and demonstrate some lemmas in this section.
Lemma 3.1
[1]
The inequality in Lemma 3.1 is actually the generalized Hölder inequality. Specifically, if \(t=1\) and \(1< p,q<\infty\), the inequality above is the classical Hölder inequality.
Lemma 3.2
[1]
In fact, we can get a valuable result; if u satisfies the inequality in Lemma 3.2, we say u belongs to the WRH-class.
Lemma 3.3
[2]
Lemma 3.4
[10]
According to the above results, we can prove a very useful lemma as follows.
Lemma 3.5
Proof
Note
Lemma 3.6
Proof
This is the end of the proof of Lemma 3.6. □
Lemma 3.7
Proof
Lemma 3.8
(Covering lemma) [1]
4 Demonstration of main results
According to the above definitions and lemmas, we will prove four theorems in detail. First of all, let us prove Theorem 2.1.
Proof of Theorem 2.1
Remark
Proof for Theorem 2.2
This is the end of the proof of Theorem 2.2. □
Proof for Theorem 2.3
Next, we will prove Theorem 2.4 by using Definition 1.1 and Theorem 2.1.
Proof of Theorem 2.4
In addition, we can obtain a global estimate about the composite operators using the same method as of Theorem 2.1.
Corollary 4.1
Remark
5 Applications
In this section, we will discuss the applications of the results obtained.
Example 5.1
As the application of the obtained theorems, our goal is to get the upper bound of \(TDG(u)\) satisfying the above conditions. Normally, we would consider calculating the integral of \(TDG(u)\), however, one will see that \(TDG(u)\) with the \(L^{\varphi}\)-norm is very complicated. In this case, we can use Theorem 2.3 to get the estimate of \(TDG(u)\) with the \(L^{\varphi}\)-norm in \(W^{1,\varphi}_{\Omega}\).
To end this section, we take a 3-form in \(\mathbb{R}^{3}\) as an example.
Example 5.2
Declarations
Acknowledgements
The authors express their deep appreciation to the referees’ efforts and suggestions, which highly improved the quality of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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