We first establish the local Poincaré inequality with L𝒜-averaging domains for the composition of the sharp maximal operator and potential operator, applied to the nonhomogenous A-harmonic equation. Then, according to the definition of L𝒜-averaging domains and relative properties, we demonstrate the global Poincaré inequality with L𝒜-averaging domains. Finally, we give some illustrations for these theorems.
1. Introduction
Poincaré inequality applied to differential forms has a vital role in PDEs, nonlinear analysis, and other related fields. With the further research conducted, we have established various versions of Poincaré inequality under different conditions. From [1–8], we have obtained the Poincaré inequality for the solution to the A-harmonic equation in uniformly bounded domain, John domains, and Ls-averaging domains. Nevertheless, most of these Poincaré inequalities are developed in Ls-averaging domains. In this paper, we will establish the Poincaré inequality for the composition of the sharp maximal operator and potential operator in L𝒜-averaging domains. As we all know, both the uniformly bounded domain and John domains are special Ls-averaging domains, and the Ls-averaging domains are also particular L𝒜-averaging domains, so the following results are the generalizations of the Poincaré inequality in Ls-averaging domains.
For convenience, we firstly introduce some notations and terminologies. Except for special instructions, E⊆ℝm is a bounded domain, |E| denotes the Lebesgue measure of E, and m≥2. The constant K and C can be varied at each step of the proof. Suppose that Bxr is a ball, with a radius r, centered at x. For any ρ>0, B⊆E and ρB⊆E have the same center and satisfy diam(ρB)=ρdiam(B). Let Λl(ℝm) be the space of all l-forms in ℝm, which is expanded by the exterior product of eℬ=ei1∧ei2∧⋯∧eil, where ℬ=(i1,…,il),1≤i1<⋯<il≤m, l=1,2,…,m. C∞(ΛlE) is the space of a smooth l-form on E. We use D′(E,Λl) to denote the space of all differential l-forms on E; that is, w(x) belongs to D′(E,Λl) if and only if there exist some lth-differential functions wℬ in E such that w(x)=∑ℬwℬ(x)dxℬ=∑wi1i2⋯il(x)dxi1∧dxi2∧⋯∧dxil. Lp(E,Λl) is a Banach space with the norm equipped by ∥w(x)∥p,E=(∫E|w(x)|pdx)1/p, where w(x)∈D′(E,Λl) and every coefficient function wℬ∈Lp(E),0<p<∞. In fact, w(x) on E is the Schwartz distribution. If ω(x)>0 a.e. and ω(x)∈Lloc1(ℝm), ω(x) is called a weight. Let dμ=ω(x)dx; then Lp(E,Λl,ω) is a weighted Banach space with the norm expressed by ∥w(x)∥p,E,ω=(∫E|w(x)|pω(x)dx)1/p. In this notation, the exterior derivative is denoted by d and Hodge codifferential operator is expressed by d⋆. Search [9] for more details.
Considering our purpose, we intend to give a brief discussion about the A-harmonic equation for the differential form. The following equation is called a nonhomogeneous A-harmonic equation:
(1)d⋆A(x,dw)=B(x,dw),
where A:E×∧l(ℝm)→∧l(ℝm) and B:E×∧l(ℝm)→∧l-1(ℝm) satisfy the conditions:
(2)|A(x,ξ)|≤a|ξ|p-1,A(x,ξ)·ξ≥|ξ|p,|B(x,ξ)|≤b|ξ|p-1,
for almost every x∈E and all ξ∈∧l(ℝm). Here, a,b>0 are constants and 1<p<∞ is a fixed exponent associated with (1). If B=0, the equation d⋆A(x,dw)=0 is called a homogenous A-harmonic equation. See [9] for more information.
In order to describe it easily, we first give some definitions in this part.
Definition 1.
Let E⊆ℝm be a bounded domain and w(x)∈Lp(E,Λl); the sharp maximal operator 𝕄s♯ is equipped with
(3)𝕄s♯(w)=𝕄s♯w=𝕄s♯w(x)=supr>0(1|Bxr|∫Bxr|w(t)-wBxr|sdt)1/s,
where Bxr is the ball of radius r, centered at x, 1≤s≤p, p≥1.
Especially, if we take s=1, denote 𝕄s♯≜𝕄♯.
Definition 2 (see [10]).
Suppose that w(x) is a differential l-form; the potential operator P is expressed by
(4)Pw(x)=∑ℬ∫EK(x,y)wℬ(y)dydxβ,
where the nonnegative and measurable function K(x,y), defined on the set {(x,y)∣x≠y,x,y∈ℝm}, is a kernel function, and the summation is over all ordered l-tuple ℬ.
Definition 3.
Take an increasingly continuous function 𝒜:[0,+∞)→[0,+∞) as a convex function with 𝒜(0)=0, and E⊆ℝm is a bounded domain, for any w(x)∈Lp(E); the Orlicz norm for differential form is denoted by
(5)∥w∥LE,μ𝒜=inf{λ>0∣1μ(E)∫E𝒜(λ-1|u|dμ)<1},
where measure μ satisfies dμ=ω(x)dx,ω(x) is a weight.
We call 𝒜 an Orlicz function if 𝒜:[0,+∞)→[0,+∞) is an increasingly continuous function and satisfies 𝒜(0)=0 and 𝒜(∞)=∞. Meanwhile, if the Orlicz function 𝒜(t) is a convex function, it is called a Young function.
Based on the above definition, we get the notation of L𝒜-averaging domains.
Definition 4 (see [3]).
Let 𝒜 be a Young function; the proper domain E⊆ℝm is called the L𝒜-averaging domains if μ(E)<∞ and there exists a constant C>0 such that for any B0⊆E and 𝒜(|w|)∈Lloc1(E,μ), w satisfies
(6)1μ(E)∫E𝒜(τ|w-wB0|)dμ≤Csup4B⊂E1μ(B)∫B𝒜(σ|w-wB|)dμ,
where the measure μ is denoted by dμ=ω(x)dx, ω(x) is a weight, σ and τ are constants with 0<τ, σ<1, and the supremum is over all balls B⊂E with 4B⊂E.
Notice that if we let 𝒜(t)=ts, L𝒜-averaging domains become the Ls-averaging domains, so L𝒜-averaging domains are the generalization of Ls-averaging domains.
Definition 5 (see [11]).
We call w(x)∈D′(E,Λl) belongs to the WRH(Λl,E)-class, l=0,1,…,m, if for any constants 0<s, t<∞ and any ball B⊂E with ρB⊂E, there exists a constant C>0 such that w(x) satisfies
(7)∥w∥s,B≤C|B|(t-s)/st∥w∥t,ρB,
where ρ>1 is a constant.
Remark 6.
If w(x) is a solution to the A-harmonic equation, we can prove that w(x) belongs to the WRH(Λl,E)-class.
2. Main Results
Before the main results are given, we need to impose some restrictions on the kernel function K(x,y) and Young function 𝒜. Firstly, let the kernel function satisfy the standard estimates; it is equal to say that if there exist 0<δ<1 and a constant c>0 such that for any point z∈{z:|x-z|<(1/2)|x-y|,x,y∈ℝm}, the kernel function K(x,y) satisfies that
K(x,y)≤c|x-y|-m, x≠y;
|K(x,y)-K(z,y)|≤c|x-z|δ|x-y|-m-δ, x≠y;
|K(y,x)-K(y,z)|≤c|x-z|δ|x-y|-m-δ, x≠y,
where function K(x,y):ℝm×ℝm→ℝ, m≥1.
With regard to the Young function 𝒜, we let the Young function 𝒜 belong to the G(p,q,C)-class (1≤p<q<∞,C≥1); that is, for any t>0, the Young function 𝒜 satisfies that
1/C≤𝒜(t1/p)/f(t)≤C;
1/C≤𝒜(t1/q)/g(t)≤C,
where f and g are the increasingly convex and concave functions defined on [0,∞], respectively.
Now, we establish these two important theorems based on the above conditions.
Theorem 7.
Suppose that the Young function 𝒜 belongs to the G(p,q,C)-class, w∈C∞(ΛlE) is a solution to the nonhomogenous A-harmonic equation, the sharp maximal operator is noted by 𝕄s♯, P is the potential operator with its kernel function K(x,y) satisfying the standard estimates, 1≤s<p,q<∞, and the bounded subset E⊆ℝm is the L𝒜-averaging domains. Then, for any ball B⊆E, one gets
(8)∥𝕄s♯(P(w))-(𝕄s♯(P(w)))B∥LB𝒜≤K∥w∥LρB𝒜,
where B and ρB⊆E and the constant ρ>1.
Based on the above theorem, we can establish the following theorem for the global Poincaré inequality in L𝒜-averaging domains.
Theorem 8.
Suppose that the Young function 𝒜 belongs to the G(p,q,C)-class, w∈C∞(ΛlE) is a solution to the nonhomogenous A-harmonic equation, the sharp maximal operator is denoted by 𝕄s♯, P is the potential operator with its kernel function K(x,y) satisfying the standard estimates, 1≤s<p, q<∞, and the bounded subset E⊆ℝm is the L𝒜-averaging domains. Then, one has
(9)∥𝕄s♯(P(w))-(𝕄s♯(P(w)))B0∥LE𝒜≤K∥w∥LE𝒜,
where B0⊆B is a fixed ball, which appears in Definition 4.
3. Preliminary Results
For proving the theorems in Section 2, we will show and demonstrate some lemmas in this part.
Lemma 9 (see [9]).
Let 0<p, q<∞, and 1/t=(1/p)+(1/q), if f and g are the measurable functions defined on ℝm, then
(10)∥fg∥t,I≤∥f∥p,I·∥g∥q,I,
for any I⊆ℝm.
Lemma 10 (see [5]).
Let P be the potential operator applied on a differential form with E⊆ℝm, w(x)∈WRH(Λl,E), and assume that the weight ω(x) belongs to A(α,β,E) with α,β>0. Then, there exists a constant C, independent of w(x) such that
(11)∥P(w)-(P(w))B∥s,B,ω≤C|B|diam(B)∥w∥s,B,ω,
for any B⊂E, where s>1 is a constant.
Remark 11.
If we take ω(x)≡1, we get
(12)∥P(w)-(P(w))B∥s,B≤C|B|diam(B)∥w∥s,B.
Lemma 12 (see [3]).
Take Ψ defined on [0,+∞) to be a strictly increasing convex function, Ψ(0)=0, and E⊂ℝm is a domain. Assume that w(x)∈D′(E,Λl) satisfies Ψ(|w|)∈L1(E,μ) and, for any constant c,
(13)μ{x∈E:|w-c|>0}>0,
where μ is a Radon measure defined by dμ(x)=ω(x)dx with a weight ω(x); then for any a>0, one obtains
(14)∫EΨ(a2|w-wE|)dμ≤∫EΨ(a|w|)dμ.
Lemma 13.
If ω(x)∈Ar(E), then there exist constants α>1 and K, not dependent on ω, such that
(15)∥ω∥α,B≤K|B|(1/α)-1∥ω∥1,B,
for all balls B contained in E.
Lemma 14.
The sharp maximal operator 𝕄s♯ is denoted by Definition 1, and the potential operator P is defined by Definition 2 with the kernel function K(x,y) satisfying the standard estimates, w(x)∈Lt(E,Λl)∩C∞(ΛlE)(l=1,2,…,m), t≥1. Then, there exists a constant K>0, independent of w, such that
(16)∫B|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|tdx≤K|B|1+(1/m)∥w∥t,B,
for all balls B⊂E.
Proof .
Let B be a ball in E, using Lemma 10 on any Bxr⊂B, we have
(17)(1|Bxr|∫Bxr|P(w)-(P(w))Bxr|sdx)1/s≤K|Bxr|diam(Bxr)|Bxr|-1/s∥w∥s,Bxr≤K|Bxr|1-(1/s)diam(Bxr)∥w∥s,B≤K|B|1-(1/s)+(1/m)∥w∥s,B.
From Lemma 3.5 in [7], it follows that
(18)∥w∥s,B≤|B|(1/s)-(1/t)∥w∥t,B,
where 0<s≤t<∞. Substituting (18) into (17) yields
(19)(1|Bxr|∫Bxr|P(w)-(P(w))Bxr|sdx)1/s≤K|B|1-(1/t)+(1/m)∥w∥t,B.
Taking the supremum for r, we get that
(20)supr>0(1|Bxr|∫Bxr|P(w)-(P(w))Bxr|sdx)1/s≤supr>0K|B|1-(1/t)+(1/m)∥w∥t,B=K|B|1-(1/t)+(1/m)∥w∥t,B.
That is,
(21)𝕄s♯(P(w))≤K|B|1-(1/t)+(1/m)∥w∥t,B.
According to the definition of Lt(E) norm and formula (21), it yields
(22)∥𝕄s♯(P(w))∥t,B=(∫B|𝕄s♯(P(w))|tdx)1/t≤(∫B|K|B|1-(1/t)+(1/m)∥w∥t,B|tdx)1/t=K|B|1+(1/m)∥w∥t,B.
Choosing Ψ(t)=2t, a=2, and ω(x)≡1 in Lemma 12, we have
(23)∥𝕄s♯(P(w))-(𝕄s♯(P(w)))B∥t,B=(∫B|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|tdx)1/t≤(∫B2t|𝕄s♯(P(w))|tdx)1/t≤K|B|1+(1/m)∥w∥t,B.
The proof of Lemma 14 has been completed.
Lemma 15.
Suppose that w(x)∈C∞(ΛlE) is a solution to the A-harmonic equation, E⊂ℝm is a bounded domain, P is a potential operator with the kernel function K(x,y) satisfying the standard estimates, and the sharp maximal operator 𝕄s♯ is expressed by Definition 1, 1≤s<p, q<∞. Then, there exists a constant K>0, such that
(24)∥𝕄s♯(P(w))-(𝕄s♯(P(w)))B∥q,B≤K∥w∥p,ρB,
where the ball B⊂E with ρB⊂E, constant ρ>1, the measure μ is defined by dμ=ω(x)dx, weight ω(x)∈Ar(E), ω(x)≥δ>0, for some r>1 and a constant δ.
Proof .
Because 1/q=((α-1)/αq)+(1/αq), for any B with ρB contained in E, using Lemmas 9 and 14, we have
(25)(∫B|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|qdμ)1/q=(∫B(|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|ω1/q)qdx)1/q≤(∫B(|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|αq/(α-1))dx)(α-1)/αq×(∫Bωαdx)1/qα≤K|B|(1-α)/αq∥ω∥1,B1/q∥𝕄s♯(P(w))-(𝕄s♯(P(w)))B∥qα/(α-1),B.
According to Lemma 14 and Definition 5, letting p=r×z, we get
(26)∥𝕄s♯(P(w))-(𝕄s♯(P(w)))B∥αq/(α-1),B≤K|B|1+(1/m)∥w∥αq/(α-1),B=K|B|1+(1/m)+((z(α-1)-αq)/αqz)∥w∥z,ρB.
Therefore, we know that
(27)(∫B|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|qdμ)1/q≤K|B|(1-α)/qα∥ω∥1,B1/q|B|1+(1/m)+((z(α-1)-αq)/αqz)∥w∥z,ρB=K|B|1+(1/m)-(1/z)(μ(B))1/q∥w∥z,ρB.
Because of 1/z=(1/p)+((p-z)/zp), and using generalized Hölder’s inequality, we get
(28)∥w∥z,ρB=(∫ρB(|w|ω1/pω-1/p)zdx)1/z≤(∫ρB|w|pωdx)1/p·(∫ρBω1/(r-1)dx)(r-1)/p=∥w∥p,ρB,ω∥ω-1∥1/(r-1),ρB1/p.
In the light of ω∈Ar(E), finding details in [9], we know
(29)supB⊂E(1|B|∫Bωdx)(1|B|∫B(1ω)1/(r-1)dx)1/(r-1)<∞.
Therefore, we can see that
(30)∥1ω∥1/(r-1),ρB1/p=∥ω∥1,ρB-1/p·∥ω∥1,ρB1/p·∥1ω∥1/(r-1),ρB1/p≤K(μ(ρB))-1/p|ρB|1/z≤K(μ(ρB))-1/p|B|1/z.
In addition, considering ω≥δ>0, so we have that
(31)μ(ρB)=∫ρBdμ≥∫ρBδdx=δ|ρB|.
Combining (27), (28), and (31), we obtain
(32)(∫B|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|qdμ)1/q≤K|ρB|1+(1/m)-(1/p)(μ(B))1/q(∫ρB|w|1/pdμ)1/q≤K|E|1+(1/m)-(1/p)(μ(E))1/q(∫ρB|w|1/pdμ)1/q≤K∥w∥p,ρB.
Therefore, we finish the proof of this lemma.
4. Demonstration of Main Results
According to the above definitions and lemmas, we will prove these two theorems in detail. Firstly, let us prove Theorem 7.
Proof of Theorem 7.
Let B and ρB⊆E, f and g are, respectively, convex and concave increasing function, use Lemma 15, and take ω(x)≡1; then
(33)𝒜(λ-1(∫B|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|qdx))1/q≤𝒜(λ-1K(∫ρB|w|pdx)1/p)=𝒜((λ-pKp∫ρB|w|pdx)1/p)≤Cf(λ-pKp∫ρB|w|pdx)≤C∫ρBf(λ-pKp|w|p)dx.
Because f(t)≤C𝒜(t1/p), we know that
(34)∫ρBf(λ-pKp|w|p)dx≤C∫ρB𝒜(Kλ-1|w|)dx.
Furthermore, we obtain
(35)𝒜(λ-1(∫B|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|qdx))1/q≤C∫ρB𝒜(Kλ-1|w|)dx.
For function g, using Jensen's inequality, we get
(36)∫B𝒜(λ-1|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|)dx≤g(∫Bg-1(𝒜(λ-1|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|))dx)≤g(C∫B(λ-1|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|)qdx)≤C𝒜((C∫B(λ-1|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|)qdx)1/q)=C𝒜(λ-1(C∫B(|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|)qdx)1/q).
Using the doubling property of 𝒜 for the above the formula, we have
(37)∫B𝒜(λ-1|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|)dx≤K𝒜(λ-1(∫B|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|qdx)1/q)≤K∫ρB𝒜(λ-1|w|)dx.
The proof of Theorem 7 has been finished.
Now, we will use Definition 4 and Theorem 7 to prove Theorem 8.
Proof of Theorem 8.
According to Definition 4, we can know
(38)∫E𝒜(λ-1|𝕄s♯(P(w))-(𝕄s♯(P(w)))B0|)dx≤KsupB⊆E∫B𝒜(λ-1|𝕄s♯(P(w))-(𝕄s♯(P(w)))B|)dx≤KsupB⊆E∫ρB𝒜(λ-1|w|)dx≤KsupB⊆E∫E𝒜(λ-1|w|)dx.
Because supB⊆E∫E𝒜(λ-1|w|)dx is independent on the ball B, we obtain that
(39)∥𝕄s♯(P(w))-(𝕄s♯(P(w)))B0∥LE𝒜≤K∥w∥LE𝒜.
We finish the proof of Theorem 8.
5. Applications
In this part, we firstly use Theorem 8 to do an estimate for a solution to the Laplace equation Δu=0.
Example 16.
Let u be a differential 2-form in ℝm, and
(40)u=1ϑ(x1dx2∧dx3+x2dx1∧dx3+x3dx1∧dx2),
where ϑ=x12+x22+x32. It is very easy to obtain that |u|=1 and du=0, so u is a solution for the Laplace equation Δu=0. If we take
(41)𝒜(t)=tlog+t={t,t≤etlogt,t>e,
then 𝒜(t) is a Young function and belongs to the G(p,q,C)-class, with 𝒜(|u|)∈L1(E). According to Theorem 8, we get that, for any fixed B0⊂E, there exists a constant K>0 such that
(42)∥𝕄s♯(P(u))-(𝕄s♯(P(u)))B0∥LE𝒜≤K∥1∥LE𝒜,
where ∥1∥LE𝒜=inf{λ>0∣(1/|E|)∫E𝒜(λ-1log+λ)dt<1}.
Now, our aim is to prove the following corollary by using Theorem 7.
Corollary 17.
Suppose that the Young function 𝒜 belongs to the G(p,q,C)-class, and w∈C∞(ΛlE) is a solution to the nonhomogenous A-harmonic equation. The sharp maximal operator is noted by Ms♯, P is the potential operator with its kernel function K(x,y) satisfying the standard estimates (1≤s<p, q<∞), and the bounded E⊆ℜm is the L𝒜-averaging domains. Then, there exists a constant K>0, such that
(43)∥𝕄s♯(P(w))∥LB𝒜≤K∥w∥LρB𝒜,
where B⊂E with ρB⊆E, and the constant ρ>1.
Proof .
By using Minkowski inequality, we know that
(44)∥𝕄s♯(P(w))∥LB𝒜≤∥𝕄s♯(P(w))-(𝕄s♯(P(w)))B∥LB𝒜+∥(𝕄s♯(P(w)))B∥LB𝒜.
From [12] and formula (22), we have
(45)∫B𝒜(λ-1|(𝕄s♯(P(w)))B|)dt=f·f-1(∫B𝒜(λ-1|(𝕄s♯(P(w)))B|)dt)≤f(K∫Bf-1(𝒜(λ-1|(𝕄s♯(P(w)))B|))dt)≤f(K∫Bλ-q|(𝕄s♯(P(w)))B|qdt)≤f(K∫Bλ-q|(𝕄s♯(P(w)))|qdt)≤f(Kλ-q∫B|w|qdt)≤K∫Bf(λ-q|w|q)dt≤K∫B𝒜(λ-1|w|)dt≤K∥w∥LρB𝒜,
where ρ>1. In addition, according to Theorem 7, there exists a constant K such that
(46)∥𝕄s♯(P(w))-(𝕄s♯(P(w)))B∥LB𝒜≤K∥w∥LρB𝒜.
Substituting (45) and (46) into (44), we conclude that there exists a constant K>0, independent of w, such that
(47)∥𝕄s♯(P(w))∥LB𝒜≤K∥w∥LρB𝒜.
The proof of Corollary 17 has been completed.
Virtually, we can obtain a global estimate about the composition operator by using Definition 4.
Corollary 18.
Suppose that the Young function 𝒜 belongs to the G(p,q,C)-class, and w∈C∞(ΛlE) is a solution to the nonhomogenous A-harmonic equation. Let the sharp maximal operator be noted by Ms♯, P is the potential operator with its kernel function K(x,y) satisfying standard estimates (1≤s<p, q<∞), and the bounded E⊆ℜm is the L𝒜-averaging domains. Then, there exists a constant K>0 such that
(48)∥𝕄s♯(P(w))∥LE𝒜≤K∥w∥LE𝒜,
where B0 is a fixed ball.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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