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Imbedding inequalities with {L}^{\phi}norms for composite operators
Journal of Inequalities and Applications volume 2013, Article number: 544 (2013)
Abstract
In this paper, we prove imbedding inequalities with {L}^{\phi}norms for the composition of the potential operator and homotopy operator applied to differential forms. We also establish the global imbedding inequality in {L}^{\phi}averaging domains.
MSC:35J60, 35B45, 30C65, 47J05, 46E35.
1 Introduction
The theory about operators applied to functions has been very well developed. However, the study about operators applied to differential forms has just begun. The purpose of this paper is to establish the local and global imbedding inequalities with {L}^{\phi}norms for the composition of the homotopy operator T and the potential operator P applied to differential forms. Specifically, we estimate the upper bound of the OrliczSobolevnorm {\parallel T(P(u)){(T(P(u)))}_{B}\parallel}_{{W}^{1,\phi}(B,{\wedge}^{l})} in terms of the {L}^{\phi}norm {\parallel u\parallel}_{{L}^{\phi}(\sigma B)}, where \sigma >1 is a constant, B is a ball, and u is a differential form satisfying the Aharmonic equation. We also establish the global imbedding theorems in the {L}^{\phi}averaging domains and bounded domains, respectively. Differential forms and operators T and P are widely used not only in analysis and partial differential equations [1–7], but also in physics and potential analysis [8–11]. We all know that any differential form u can be decomposed as u=d(Tu)+T(du), where d is the differential operator, and T is the homotopy operator. In many situations, we need to estimate the composition of the homotopy operator T and the potential operator P. For example, when we consider the decomposition of P(u), we have to study the composition T\circ P of the homotopy operator T and the potential operator P. Our main results are presented and proved in Theorem 2.6, Theorem 3.3 and Theorem 3.6, respectively.
We assume that Ω is a bounded domain in {\mathbb{R}}^{n}, n\ge 2, B and σB are the balls with the same center and diam(\sigma B)=\sigma diam(B) throughout this paper. We do not distinguish the balls from cubes in this paper. We use E to denote the ndimensional Lebesgue measure of a set E\subseteq {\mathbb{R}}^{n}. For a function u, the average of u over B is defined by {u}_{B}=\frac{1}{B}{\int}_{B}u\phantom{\rule{0.2em}{0ex}}dx. All integrals involved in this paper are the Lebesgue integrals. Differential forms are extensions of differentiable functions in {\mathbb{R}}^{n}. For example, the function u({x}_{1},{x}_{2},\dots ,{x}_{n}) is called a 0form. A differential 1form u(x) in {\mathbb{R}}^{n} can be written as u(x)={\sum}_{i=1}^{n}{u}_{i}({x}_{1},{x}_{2},\dots ,{x}_{n})\phantom{\rule{0.2em}{0ex}}d{x}_{i}, where the coefficient functions {u}_{i}({x}_{1},{x}_{2},\dots ,{x}_{n}), i=1,2,\dots ,n, are differentiable. Similarly, a differential kform u(x) can be expressed as
where I=({i}_{1},{i}_{2},\dots ,{i}_{k}), 1\le {i}_{1}<{i}_{2}<\cdots <{i}_{k}\le n. Let {\wedge}^{l}={\wedge}^{l}({\mathbb{R}}^{n}) be the set of all lforms in {\mathbb{R}}^{n}, {D}^{\prime}(\mathrm{\Omega},{\wedge}^{l}) be the space of all differential lforms in Ω, and {L}^{p}(\mathrm{\Omega},{\wedge}^{l}) be the lforms u(x)={\sum}_{I}{u}_{I}(x)\phantom{\rule{0.2em}{0ex}}d{x}_{I} in Ω satisfying {\int}_{\mathrm{\Omega}}{{u}_{I}}^{p}<\mathrm{\infty} for all ordered ltuples I, l=1,2,\dots ,n. We denote the exterior derivative by d and the Hodge star operator by ⋆. The Hodge codifferential operator {d}^{\star} is given by {d}^{\star}={(1)}^{nl+1}\star d\star, l=1,2,\dots ,n. For u\in {D}^{\prime}(\mathrm{\Omega},{\wedge}^{l}) the vectorvalued differential form
consists of differential forms \frac{\partial u}{\partial {x}_{i}}\in {D}^{\prime}(\mathrm{\Omega},{\wedge}^{l}), where the partial differentiation is applied to the coefficients of ω. The nonlinear partial differential equation
is called nonhomogeneous Aharmonic equation, where A:\mathrm{\Omega}\times {\wedge}^{l}({\mathbb{R}}^{n})\to {\wedge}^{l}({\mathbb{R}}^{n}) and B:\mathrm{\Omega}\times {\wedge}^{l}({\mathbb{R}}^{n})\to {\wedge}^{l1}({\mathbb{R}}^{n}) satisfy the conditions
for almost every x\in \mathrm{\Omega} and all \xi \in {\wedge}^{l}({\mathbb{R}}^{n}). Here a,b>0 are constants, and 1<p<\mathrm{\infty} is a fixed exponent associated with (1.1). A solution to (1.1) is an element of the Sobolev space {W}_{\mathrm{loc}}^{1,p}(\mathrm{\Omega},{\wedge}^{l1}) such that
for all \phi \in {W}_{\mathrm{loc}}^{1,p}(\mathrm{\Omega},{\wedge}^{l1}) with compact support. If u is a function (0form) in {\mathbb{R}}^{n}, equation (1.1) reduces to
If the operator B=0, equation (1.1) becomes
which is called the (homogeneous) Aharmonic equation. Let A:\mathrm{\Omega}\times {\wedge}^{l}({\mathbb{R}}^{n})\to {\wedge}^{l}({\mathbb{R}}^{n}) be defined by A(x,\xi )=\xi {\xi }^{p2} with p>1. Then A satisfies the required conditions, and (1.5) becomes the pharmonic equation {d}^{\star}(du{du}^{p2})=0 for differential forms. See [1–3, 12–16] for recent results on the Aharmonic equations and related topics.
Assume that D\subset {\mathbb{R}}^{n} is a bounded, convex domain. The following operator {K}_{y} with the case y=0 was first introduced by Cartan in [8]. Then it was extended to the following general version in [6]. For each y\in D, a linear operator {K}_{y}:{C}^{\mathrm{\infty}}(D,{\mathrm{\Lambda}}^{l})\to {C}^{\mathrm{\infty}}(D,{\mathrm{\Lambda}}^{l1}) defined by ({K}_{y}\omega )(x;{\xi}_{1},\dots ,{\xi}_{l1})={\int}_{0}^{1}{t}^{l1}\omega (tx+yty;xy,{\xi}_{1},\dots ,{\xi}_{l1})\phantom{\rule{0.2em}{0ex}}dt and the decomposition \omega =d({K}_{y}\omega )+{K}_{y}(d\omega ) correspond. A homotopy operator T:{C}^{\mathrm{\infty}}(D,{\mathrm{\Lambda}}^{l})\to {C}^{\mathrm{\infty}}(D,{\mathrm{\Lambda}}^{l1}) is defined by an averaging {K}_{y} over all points y in D
where \phi \in {C}_{0}^{\mathrm{\infty}}(D) is normalized by {\int}_{D}\phi (y)\phantom{\rule{0.2em}{0ex}}dy=1. For simplicity purpose, we write \xi =({\xi}_{1},\dots ,{\xi}_{l1}). Then T\omega (x;\xi )={\int}_{0}^{1}{t}^{l1}{\int}_{D}\phi (y)\omega (tx+yty;xy,\xi )\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt. By substituting z=tx+yty and t=s/(1+s), we have
where the vector function \zeta :D\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n} is given by \zeta (z,h)=h{\int}_{0}^{\mathrm{\infty}}{s}^{l1}{(1+s)}^{n1}\phi (zsh)\phantom{\rule{0.2em}{0ex}}ds. The integral (1.7) defines a bounded operator T:{L}^{s}(D,{\mathrm{\Lambda}}^{l})\to {W}^{1,s}(D,{\mathrm{\Lambda}}^{l1}), l=1,2,\dots ,n, and the decomposition
holds for any differential form u. The lform {\omega}_{D}\in {D}^{\prime}(D,{\mathrm{\Lambda}}^{l}) is defined by
for all \omega \in {L}^{p}(D,{\mathrm{\Lambda}}^{l}), 1\le p<\mathrm{\infty}. Also, for any differential form u, we have
From [[17], p.16], we know that any open subset Ω in {\mathbb{R}}^{n} is the union of a sequence of cubes {Q}_{k}, whose sides are parallel to the axes, whose interiors are mutually disjoint, and whose diameters are approximately proportional to their distances from F. Specifically,

(i)
\mathrm{\Omega}={\bigcup}_{k=1}^{\mathrm{\infty}}{Q}_{k},

(ii)
{Q}_{j}^{0}\cap {Q}_{k}^{0}=\varphi if j\ne k,

(iii)
there exist two constants {c}_{1},{c}_{2}>0 (we can take {c}_{1}=1, and {c}_{2}=4), so that
{c}_{1}diam({Q}_{k})\le distance({Q}_{k},F)\le {c}_{2}diam({Q}_{k}).(1.11)
Thus, the definition of the homotopy operator T can be generalized to any domain Ω in {\mathbb{R}}^{n}: For any x\in \mathrm{\Omega}, x\in {Q}_{k} for some k. Let {T}_{{Q}_{k}} be the homotopy operator defined on {Q}_{k} (each cube is bounded and convex). Thus, we can define the homotopy operator {T}_{\mathrm{\Omega}} on any domain Ω by
Recently, Hui Bi extended the definition of the potential operator to the case of differential forms, see [3]. For any differential lform u(x), the potential operator P is defined by
where the kernel K(x,y) is a nonnegative measurable function defined for x\ne y, and the summation is over all ordered ltuples I. The l=0 case reduces to the usual potential operator,
where f(x) is a function defined on E\subset {R}^{n}. See [3] and [9] for more results about the potential operator. We say a kernel K on {\mathbb{R}}^{n}\times {\mathbb{R}}^{n} satisfies the standard estimates if there exist δ, 0<\delta \le 1 and a constant C such that for all distinct points x and y in {\mathbb{R}}^{n}, and all z with xz<\frac{1}{2}xy, the kernel K satisfies

(i)
K(x,y)\le C{xy}^{n};

(ii)
K(x,y)K(z,y)\le C{xz}^{\delta}{xy}^{n\delta};

(iii)
K(y,x)K(y,z)\le C{xz}^{\delta}{xy}^{n\delta}.
In this paper, we always assume that P is the potential operator defined in (1.13) with the kernel K(x,y) satisfying condition (i) of the standard estimates. Recently, Hui Bi in [3] proved the following inequality for the potential operator.
where u\in {D}^{\prime}(E,{\wedge}^{l}), l=0,1,\dots ,n1, is a differential form defined in a bounded and convex domain E, and p>1 is a constant.
2 Local imbedding inequalities
In this section, we prove the local {L}^{\phi} imbedding inequalities for T\circ P applied to solutions of the nonhomogeneous Aharmonic equation in a bounded domain. We will need the following definitions and a notation. A continuously increasing function \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with \phi (0)=0, is called an Orlicz function. The Orlicz space {L}^{\phi}(\mathrm{\Omega}) consists of all measurable functions f on Ω such that {\int}_{\mathrm{\Omega}}\phi (\frac{f}{\lambda})\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty} for some \lambda =\lambda (f)>0. {L}^{\phi}(\mathrm{\Omega}) is equipped with the nonlinear Luxemburg functional
A convex Orlicz function φ is often called a Young function. If φ is a Young function, then {\parallel \cdot \parallel}_{{L}^{\phi}(\mathrm{\Omega})} defines a norm in {L}^{\phi}(\mathrm{\Omega}), which is called the Luxemburg norm or Orlicz norm. For any subset E\subset {\mathbb{R}}^{n}, we use {W}^{1,\phi}(E,{\wedge}^{l}) to denote the OrliczSobolev space of lforms, which equals {L}^{\phi}(E,{\wedge}^{l})\cap {L}_{1}^{\phi}(E,{\wedge}^{l}) with the norm
If we choose \phi (t)={t}^{p}, p>1 in (2.1), we obtain the usual {L}^{p} norm for {W}^{1,p}(E,{\wedge}^{l})
Definition 2.1 [18]
We say a Young function φ lies in the class G(p,q,C), 1\le p<q<\mathrm{\infty}, C\ge 1, if (i) 1/C\le \phi ({t}^{1/p})/g(t)\le C and (ii) 1/C\le \phi ({t}^{1/q})/h(t)\le C for all t>0, where g is a convex increasing function, and h is a concave increasing function on [0,\mathrm{\infty}).
From [18], each of φ, g and h in above definition is doubling in the sense that its values at t and 2t are uniformly comparable for all t>0, and the consequent fact that
where {C}_{1} and {C}_{2} are constants. Also, for all 1\le {p}_{1}<p<{p}_{2} and \alpha \in \mathbb{R}, the function \phi (t)={t}^{p}{log}_{+}^{\alpha}t belongs to G({p}_{1},{p}_{2},C) for some constant C=C(p,\alpha ,{p}_{1},{p}_{2}). Here {log}_{+}(t) is defined by {log}_{+}(t)=1 for t\le e; and {log}_{+}(t)=log(t) for t>e. Particularly, if \alpha =0, we see that \phi (t)={t}^{p} lies in G({p}_{1},{p}_{2},C), 1\le {p}_{1}<p<{p}_{2}. We will need the following reverse Hölder inequality.
Lemma 2.2 [19]
Let u be a solution of the nonhomogeneous Aharmonic equation (1.1) in a domain Ω and 0<s,t<\mathrm{\infty}. Then there exists a constant C, independent of u, such that
for all balls B with \sigma B\subset \mathrm{\Omega} for some \sigma >1.
We first prove the following local inequality for the composition T\circ P with the {L}^{\phi}norm.
Theorem 2.3 Let φ be a Young function in the class G(p,q,C), 1\le p<q<\mathrm{\infty}, C\ge 1, let Ω be a bounded and convex domain, let T:{C}^{\mathrm{\infty}}(\mathrm{\Omega},{\wedge}^{l})\to {C}^{\mathrm{\infty}}(\mathrm{\Omega},{\wedge}^{l1}), l=1,2,\dots ,n, be the homotopy operator defined in (1.6), and let P be the potential operator defined in (1.13) with the kernel K(x,y) satisfying condition (i) of the standard estimates. Assume that \phi (u)\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega}), and u is a solution of the nonhomogeneous Aharmonic equation (1.1) in Ω. Then there exists a constant C, independent of u, such that
for all balls B with \sigma B\subset \mathrm{\Omega}.
Proof Since u=d(Tu)+T(du) and d(Tu)={u}_{B} hold for any differential form u, we have
Using (1.15) and noticing {\parallel {u}_{B}\parallel}_{q,B}\le {C}_{1}{\parallel u\parallel}_{q,B} for any differential form, it follows that
for q>1, Replacing u by T(P(u)) in (2.5) and using (1.10) and (2.6), we obtain
for any differential form u and all balls B with B\subset \mathrm{\Omega}. From Lemma 2.2, for any positive numbers p and q, it follows that
where σ is a constant \sigma >1. Using Jensen’s inequality for {h}^{1}, (2.2), (2.7), (2.8), (i) in Definition 2.1, and noticing the fact that φ and h are doubling, and φ is an increasing function, we obtain
Since p\ge 1, then 1+\frac{1}{n}+\frac{pq}{pq}>\frac{1}{n}. Hence, we have {B}^{1+\frac{1}{n}+\frac{pq}{pq}}\le {C}_{13}{B}^{\frac{1}{n}}. Note that φ is doubling, we obtain
Combining (2.9) and (2.10) and using {B}^{\frac{1}{n}}={C}_{15}diam(B) yields
Since each of φ, g and h in Definition 2.1 is doubling, from (2.11), we have
for all balls B with \sigma B\subset \mathrm{\Omega} and any constant \lambda >0. From (2.1) and the last inequality, we have the following inequality with the Luxemburg norm
The proof of Theorem 2.3 has been completed. □
In order to prove our main local imbedding theorem, we will need the following Theorems 2.4 and 2.5.
Theorem 2.4 Let φ be a Young function in the class G(p,q,C), 1\le p<q<\mathrm{\infty}, C\ge 1, let Ω be a bounded and convex domain, let T be the homotopy operator defined in (1.6), and let P be the potential operator defined in (1.13) with the kernel K(x,y) satisfying condition (i) of the standard estimates. Assume that \phi (u)\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega}), and u is a solution of the nonhomogeneous Aharmonic equation (1.1) in Ω. Then there exists a constant C, independent of u, such that
for all balls B with \sigma B\subset \mathrm{\Omega}.
Proof Using (1.10), we have
for any differential form u and q>1. Using (1.15) and the fact that d(Tu)={u}_{B}, and noticing that
holds for any differential form u, we obtain
for all balls B with B\subset \mathrm{\Omega}. From (2.13) and (2.14), it follows that
By Lemma 2.2, for any positive numbers p and q, it follows that
where σ is a constant \sigma >1. Using Jensen’s inequality for {h}^{1}, (2.2), (2.15), (2.16), (i) in Definition 2.1, and noticing the fact that φ and h are doubling, and φ is an increasing function, we obtain
Since p\ge 1, then 1+\frac{pq}{pq}>0. Hence, we have
Note that φ is doubling, we obtain
Combining (2.17) and (2.18) yields
Since each of φ, g and h in Definition 2.1 is doubling, from (2.19), we have
for all balls B with \sigma B\subset \mathrm{\Omega} and any constant \lambda >0. From (2.1) and (2.20), we have the following inequality with the Luxemburg norm
The proof of Theorem 2.4 has been completed. □
Theorem 2.5 Let φ be a Young function in the class G(p,q,C), 1\le p<q<\mathrm{\infty}, C\ge 1, let Ω be a bounded and convex domain, let T be the homotopy operator defined in (1.6), and let P be the potential operator defined in (1.13) with the kernel K(x,y) satisfying condition (i) of the standard estimates. Assume that \phi (u)\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega}), and u is a solution of the nonhomogeneous Aharmonic equation (1.1) in Ω. Then there exists a constant C, independent of u, such that
for all balls B with \sigma B\subset \mathrm{\Omega}.
Proof Replacing u by d(T(P(u))) in the first inequality in (1.10), we find that
holds for any differential form u and q>1. From (2.14), we have
Combining (2.23) and (2.24) yields
for all balls B with B\subset \mathrm{\Omega}. Starting with (2.25) and using the similar method as we did in the proof of Theorem 2.4, we can obtain
The proof of Theorem 2.5 has been completed. □
Now, we are ready to present and prove the main local theorem, the {L}^{\phi}imbedding theorem, as follows.
Theorem 2.6 Let φ be a Young function in the class G(p,q,C), 1\le p<q<\mathrm{\infty}, C\ge 1, let Ω be a bounded and convex domain, let T be the homotopy operator defined in (1.6), and let P be the potential operator defined in (1.13) with the kernel K(x,y) satisfying condition (i) of the standard estimates. Assume that \phi (u)\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega}), and u is a solution of the nonhomogeneous Aharmonic equation (1.1) in Ω. Then there exists a constant C, independent of u, such that
for all balls B with \sigma B\subset \mathrm{\Omega}.
Proof From (2.1), (2.12) and (2.22), we have
for all balls B with \sigma B\subset \mathrm{\Omega}, where \sigma =max\{{\sigma}_{1},{\sigma}_{2}\}. The proof of Theorem 2.6 has been completed. □
The following version of local imbedding will be used in Section 3 to establish a global imbedding theorem which indicates that the operator T\circ P is bounded.
Theorem 2.7 Let φ be a Young function in the class G(p,q,C), 1\le p<q<\mathrm{\infty}, C\ge 1, let Ω be a bounded and convex domain, let T be the homotopy operator defined in (1.6), and let P be the potential operator defined in (1.13) with the kernel K(x,y) satisfying condition (i) of the standard estimates. Assume that \phi (u)\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega}), and u is a solution of the nonhomogeneous Aharmonic equation (1.1) in Ω. Then there exists a constant C, independent of u, such that
for all balls B with \sigma B\subset \mathrm{\Omega}.
Proof Applying (1.10) to P(u), then using (1.15), we find that
and
for any differential form u and all balls B with B\subset \mathrm{\Omega}, where q>1 is a constant. Starting with (2.30) and (2.31) and using the similar method developed in the proof of Theorem 2.5, we obtain
and
respectively, where {\sigma}_{1} and {\sigma}_{2} are constants. From (2.1), (2.32) and (2.33), we have
where \sigma =max\{{\sigma}_{1},{\sigma}_{2}\}. The proof of Theorem 2.7 has been completed. □
Note that if we choose \phi (t)={t}^{p}{log}_{+}^{\alpha}t or \phi (t)={t}^{p} in Theorems 2.3, 2.4, 2.5, 2.6 and 2.7, we will obtain some {L}^{p}({log}_{+}^{\alpha}L)norm or {L}^{p}norm inequalities, respectively. For example, let \phi (t)={t}^{p}{log}_{+}^{\alpha}t in Theorem 2.6, we have the following imbedding inequalities for T\circ P with the {L}^{p}({log}_{+}^{\alpha}L)norms.
Corollary 2.8 Let \phi (t)={t}^{p}{log}_{+}^{\alpha}t, p\ge 1 and \alpha \in \mathbb{R}. Assume that \phi (u)\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega}), and u is a solution of the nonhomogeneous Aharmonic equation (1.1). Then there exists a constant C, independent of u such that
for all balls B with \sigma B\subset \mathrm{\Omega}, where \sigma >1 is a constant.
Selecting \phi (t)={t}^{p} in Theorem 2.6, we obtain the usual imbedding inequalities T\circ P with the {L}^{p}norms.
for all balls B with \sigma B\subset \mathrm{\Omega}, where \sigma >1 is a constant. Similarly, if we choose \phi (t)={t}^{p}{log}_{+}^{\alpha}t or \phi (t)={t}^{p} in Theorems 2.3, 2.4, 2.5 and 2.7, respectively, we will obtain the corresponding special results.
3 Global imbedding theorem
We have established the local {L}^{\phi}norm and {L}^{\phi}imbedding inequalities for T\circ P and some composite operators related to the imbedding theorem for T\circ P. In this section, we prove the global {L}^{\phi}imbedding theorem in the following {L}^{\phi}averaging domains.
Definition 3.1 [20]
Let φ be an increasing convex function on [0,\mathrm{\infty}) with \phi (0)=0. We call a proper subdomain \mathrm{\Omega}\subset {\mathbb{R}}^{n} an {L}^{\phi}averaging domain if \mathrm{\Omega}<\mathrm{\infty}, and there exists a constant C such that
for some ball {B}_{0}\subset \mathrm{\Omega} and all u such that \phi (u)\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega}), where τ, σ are constants with 0<\tau <\mathrm{\infty}, 0<\sigma <\mathrm{\infty} and the supremum is over all balls B\subset \mathrm{\Omega}.
From the definition above, we see that {L}^{s}averaging domains are special {L}^{\phi}averaging domains when \phi (t)={t}^{s} in Definition 3.1. Also, uniform domains and the John domains are very special {L}^{\phi}averaging domains, see [1] and [20] for more results about the averaging domains.
Lemma 3.2 [19] (Covering lemma)
Each Ω has a modified Whitney cover of cubes \mathcal{V}=\{{Q}_{i}\} such that {\bigcup}_{i}{Q}_{i}=\mathrm{\Omega}, {\sum}_{{Q}_{i}\in \mathcal{V}}{\chi}_{\sqrt{\frac{5}{4}}Q}\le N{\chi}_{\mathrm{\Omega}} and some N>1, and if {Q}_{i}\cap {Q}_{j}\ne \mathrm{\varnothing}, then there exists a cube R (this cube need not be a member of \mathcal{V}) in {Q}_{i}\cap {Q}_{j} such that {Q}_{i}\cup {Q}_{j}\subset NR. Moreover, if Ω is δJohn, then there is a distinguished cube {Q}_{0}\in \mathcal{V}, which can be connected with every cube Q\in \mathcal{V} by a chain of cubes {Q}_{0},{Q}_{1},\dots ,{Q}_{k}=Q from \mathcal{V} and such that Q\subset \rho {Q}_{i}, i=0,1,2,\dots ,k, for some \rho =\rho (n,\delta ).
Now, we are ready to prove another main theorem, the global imbedding theorem with the {L}^{\phi}norm, as follows.
Theorem 3.3 Let φ be a Young function in the class G(p,q,C), 1\le p<q<\mathrm{\infty}, let C\ge 1, Ω be any convex bounded {L}^{\phi}averaging domain, let T be the homotopy operator defined in (1.6), and let P be the potential operator defined in (1.13) with the kernel K(x,y) satisfying condition (i) of the standard estimates. Assume that \phi (v)\in {L}^{1}(\mathrm{\Omega}), and v\in {D}^{\prime}(\mathrm{\Omega},{\wedge}^{1}) is a solution of the nonhomogeneous Aharmonic equation (1.1) in Ω. Then there exists a constant C, independent of v, such that
where {B}_{0}\subset \mathrm{\Omega} is some fixed ball.
Proof Since v\in {\wedge}^{1}, it follows that T(P(v))\in {\wedge}^{0}, and hence \mathrm{\nabla}({(TP(v))}_{{B}_{0}})=d({(TP(v))}_{{B}_{0}}). Note that {(TP(v))}_{{B}_{0}} is a closed form, then d({(TP(v))}_{{B}_{0}})=0. Thus,
Applying the first inequality in (1.10) to P(v), we have
for any ball B and q>1. Starting from (3.4), and using the similar method to the proof of Theorem 2.4, we obtain
where \sigma >1 is a constant. From the covering lemma and (3.5), it follows that
where N is a positive integer appearing in the covering lemma. Letting u=T(P(v)) and using (2.11), we find that
From (2.1), (3.6) and (3.7), we have
We have completed the proof of Theorem 3.3. □
It is well known that any John domain is a special {L}^{\phi}averaging domain [1]. Hence, we have the following global {L}^{\phi}imbedding theorem for the John domains.
Theorem 3.4 Let φ be a Young function in the class G(p,q,C), 1\le p<q<\mathrm{\infty}, let C\ge 1, Ω be any convex bounded John domain, let T be the homotopy operator defined in (1.6), and let P be the potential operator defined in (1.13) with the kernel K(x,y) satisfying condition (i) of the standard estimates. Assume that \phi (v)\in {L}^{1}(\mathrm{\Omega}), and v\in {D}^{\prime}(\mathrm{\Omega},{\wedge}^{1}) is a solution of the nonhomogeneous Aharmonic equation (1.1) in Ω. Then there exists a constant C, independent of v, such that
where {B}_{0}\subset \mathrm{\Omega} is some fixed ball.
Choosing \phi (t)={t}^{p}{log}_{+}^{\alpha}t in Theorems 3.3, we obtain the following imbedding inequality with the {L}^{p}({log}_{+}^{\alpha}L)norms.
Corollary 3.5 Let \phi (t)={t}^{p}{log}_{+}^{\alpha}t, p\ge 1, \alpha \in \mathbb{R}, Ω be any convex bounded {L}^{\phi}averaging domain, let T be the homotopy operator defined in (1.6), and let P be the potential operator defined in (1.13) with the kernel K(x,y) satisfying condition (i) of the standard estimates. Assume that \phi (v)\in {L}^{1}(\mathrm{\Omega}), and v\in {D}^{\prime}(\mathrm{\Omega},{\wedge}^{1}) is a solution of the nonhomogeneous Aharmonic equation (1.1) in Ω. Then there exists a constant C, independent of v, such that
where {B}_{0}\subset \mathrm{\Omega} is some fixed ball.
Next, let S be the set of all solutions of the nonhomogeneous Aharmonic equation in Ω. We have the following version of imbedding theorem with {L}^{\phi} norm for any bounded domain, which says that the composite operator T\circ P maps {W}^{1,\phi}(\mathrm{\Omega},{\wedge}^{1})\cap S continuously into {L}^{\phi}(\mathrm{\Omega}).
Theorem 3.6 Let φ be a Young function in the class G(p,q,C), 1\le p<q<\mathrm{\infty}, C\ge 1, let T be the homotopy operator defined in (1.6), and let P be the potential operator defined in (1.13) with the kernel K(x,y) satisfying condition (i) of the standard estimates. Assume that \phi (v)\in {L}^{1}(\mathrm{\Omega}), and v\in {D}^{\prime}(\mathrm{\Omega},{\wedge}^{1})\cap S in Ω. Then the composite operator T\circ P maps {W}^{1,\phi}(\mathrm{\Omega},{\wedge}^{1})\cap S continuously into {L}^{\phi}(\mathrm{\Omega}). Furthermore, there exists a constant C, independent of v, such that
holds for any bounded domain Ω.
Proof Let v\in {D}^{\prime}(\mathrm{\Omega},{\wedge}^{1}) be a solution of equation (1.1). Since the composite operator T\circ P is continuous if and only if it is bounded, we only need to prove that (3.11) holds. Using (2.29) and the Lemma 3.2, we obtain
Hence, inequality (3.11) holds. We have completed the proof of Theorem 3.6. □
Selecting \phi (t)={t}^{p} in Theorems 3.3, we have the following version of the imbedding inequality with {L}^{p}norms.
Corollary 3.7 Let \phi (t)={t}^{p}, p\ge 1, let T be the homotopy operator defined in (1.6) and P be the potential operator defined in (1.13). Assume that \phi (v)\in {L}^{1}(\mathrm{\Omega}), and v\in {D}^{\prime}(\mathrm{\Omega},{\wedge}^{1}) is a solution of the nonhomogeneous Aharmonic equation (1.1) in Ω. Then there exists a constant C, independent of v, such that
holds for any bounded domain Ω.
4 Examples
In this last section, we will present two examples to show applications of our imbedding theorems. All of our local and global inequalities work for these two examples. We should note that functions are 0forms. Thus, all of our theorems proved in this paper will work for harmonic functions. For example, choose u to be a function (0form) in the homogeneous Aharmonic equation (1.5), then (1.5) reduces to the following Aharmonic equation
for functions. Assume that A(x,\xi )=\xi {\xi }^{p2} with p>1. Then, the operator A:\mathrm{\Omega}\times {\wedge}^{l}({\mathbb{R}}^{n})\to {\wedge}^{l}({\mathbb{R}}^{n}) satisfies the required conditions (1.2) and the equation (4.1) becomes the usual pharmonic equation for functions
which is equivalent to
If we choose p=2 in (4.2), we have the Laplace equation \mathrm{\Delta}u=0 for functions. Thus, from Theorem 3.3, we have the following inequality for harmonic functions.
Example 4.1 Let u be a solution of the usual Aharmonic equation (4.1) or the pharmonic equation (4.2), let φ be a Young function in the class G(p,q,C), 1\le p<q<\mathrm{\infty}, C\ge 1, and let Ω be any bounded {L}^{\phi}averaging domain. If \phi (u)\in {L}^{1}(\mathrm{\Omega}), then there exists a constant C, independent of u, such that
where {B}_{0}\subset \mathrm{\Omega} is some fixed ball.
Example 4.2 Let u(x,y) be a function (0form) defined in {\mathbb{R}}^{2} by
We can check that u(x,y) satisfies the Laplace equation {u}_{xx}(x,y)+{u}_{yy}(x,y)=0 in the upper half plane, that is, u(x,y) is a harmonic function in the upper half plane. Let r>0 be a constant, let ({x}_{0},{y}_{0}) be a fixed point with {y}_{0}>r and B=\{(x,y):{(x{x}_{0})}^{2}+{(y{y}_{0})}^{2}\le {r}^{2}\}. To obtain the upper bound for the OrliczSobolevnorm {\parallel T(P(u)){(T(P(u)))}_{B}\parallel}_{{W}^{1,\phi}(B,{\wedge}^{l})} directly, it would be very complicated. However, using Theorem 2.6 with n=2, we can easily obtain the upper bound of the OrliczSobolevnorm {\parallel T(P(u)){(T(P(u)))}_{B}\parallel}_{{W}^{1,\phi}(B,{\wedge}^{l})} as follows. First, we know that B=\pi {r}^{2} and
Applying (2.27) and (4.4), we have
Remark

(i)
We know that the {L}^{s}averaging domains are the special {L}^{\phi}averaging domains. Thus, Theorem 3.3 also holds for the {L}^{s}averaging domain;

(ii)
Theorem 3.6 holds for any bounded domain in {\mathbb{R}}^{n}.
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Xing, Y., Ding, S. Imbedding inequalities with {L}^{\phi}norms for composite operators. J Inequal Appl 2013, 544 (2013). https://doi.org/10.1186/1029242X2013544
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DOI: https://doi.org/10.1186/1029242X2013544