# Some Weighted Norm Estimates for the Composition of the Homotopy and Green’s Operator

Authors: Huacan Li; Qunfang Li
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We establish theAr(D)-weighted integral inequality for the composition of the Homotopy T and Green’s operator G on a bounded convex domain and also motivated it to the global domain by the Whitney cover. At the same time, we also obtain some (p,q)-type norm inequalities. Finally, as applications of above results, we obtain the upper bound for the Lp norms of T(G(u)) or (T(G(u)))B in terms of Lq norms of u or du.


---

## Body

## 1. Introduction

Our purpose is to study theLp theory of the composition of the Homotopy T and Green’s operator G acting on differential forms on a bounded convex domain. Both operators play an important role in many fields, including harmonic analysis, potential theory, and partial equations (see [1–6]). In the present paper, we will obtain some (p,q)-type norm inequalities for the composition of the Homotopy T and Green’s operator G and also prove the Ar(D)-weighted integral inequality on a bounded convex domain. These results will provide effective tools for studying behavior of solutions of A-harmonic equations and related differential systems on manifolds.We start this paper by introducing some notations and definitions. LetM be a Riemannian, compact, oriented, and C∞-smooth manifold without boundary on Rn and let Ω be an open subset of Rn. Also, we use G to denote Green’s operator throughout this paper. Furthermore, we use B to denote a ball and ρB to denote the ball with the same center as B and with diameter(ρB)=ρdiameter(B). We do not distinguish balls from cubs in this paper.We assume that∧k=∧k(Rn)(k=0,1,2,…,n) is the linear space of all k-forms ω(x)=∑I‍(x)dxI=∑‍ωi1,i2,…,ikdxi1∧dxi2∧⋯∧dxik with summation over all ordered k-tuples I=(i1,i2,…,ik),  1≤i1≤i2≤⋯≤ik≤n. If the coefficient ωI(x) of k-form ω(x) is differential on M, then we call ω(x) a differential k-form on M. A differential k-form ω(x) on M is a de Rham current (see [7]) on M with values in ∧k(Rn). Let ∧kM be the kth exterior power of the cotangent bundle and C∞(∧kM) be the space of smooth k-forms on M. As usual, we use D′(M,∧k) to denote the space of all differential k-forms and LP(∧kM) to denote the k-form ω(x) with the norm
(1)∥ω(x)∥p,M=(∫M‍|ω(x)|pdx)1/p=(∫M‍(∑I‍|ωI(x)|2)p/2dx)1/p
on M. Thus Lp(∧kM) is a Banach space. As usual, we still use ⋆ to denote the Hodge star operator. Also, we use d:D′(M,∧k)→D′(M,∧k+1) to denote the differential operator and use d⋆:D′(M,∧k+1)→D′(M,∧k) to denote the Hodge codifferential operator which is defined by d⋆=(-1)nk+1⋆d⋆ on D′(M,∧k+1). The n-dimensional Lebesgue measure of a set E⊆Rn is denoted by |E|. We call w a weight if w∈Lloc⁡1(Rn) and w>0,a.e. For 0<p<1, we denote the weighted Lp-norm of a measurable function f over M by
(2)∥f∥p,M,wα=(∫M‍|f|pwαdx)1/p,
where α is a real number.LetD⊂Rn be a bounded, convex domain. Iwaniec and Lutoborski in [1] first introduced a linear operator Ky:C∞(D,∧k)→C∞(D,∧k-1) satisfying that
(3)(Kyω)(x;ξ1,ξ2,…,ξk-1)=∫01‍tk-1ω(tx+y-ty;x-y,ξ1,ξ2,…,ξk-1)dt
and the decomposition ω=d(Kyω)+Ky(dω). Then by averaging Ky over all points y in D, they constructed a Homotopy operator T:C∞(D,∧k)→C∞(D,∧k-1) satisfying that Tω=∫D‍φ(y)Ky(ω)dy, where φ∈C0∞(D) is normalized by ∫D‍φ(y)dy=1. The k-form ωD∈D′(D,∧k) is defined by ωD=(1/|D|)∫D‍ω(y)dy, if k=0, and if k=1,2,…,n, then
(4)ωD=d(Tω)=ω-T(dω),(5)|Tω(x)|≤C∫D‍|ω(y)||y-x|n-1dy.


## 2. Boundedness of the Composition of the Homotopy and Green’s Operator inLp Space

In this section, we will prove theAr(D)-weighted norm inequality for the composition of the Homotopy T and Green’s operator G on a bounded convex domain. Then using the Whitney cover, we develop the local result to the global domain. In [8], Gol’dshtein and Troyanov proved the following lemma.Lemma 1.
LetD⊂Rn be a bounded convex domain. The operator T maps Lp(D,∧k) continuously to Lq(D,∧k-1) in the following cases:
(6)Either1≤p,q≤∞,1p-1q<1n,Or1<p,q≤∞,1p-1q≤1n.From [3], we have the following lemma about Ls-estimates for Green’s operator.Lemma 2.
Letu∈C∞(∧kM)(k=0,1,2,…,n) and 1<s<∞. Then there exists a constant C, independent of u, such that
(7)∥dd⋆G(u)∥s,M+∥d⋆dG(u)∥s,M+∥dG(u)∥s,M+∥d⋆G(u)∥s,M+∥G(u)∥s,M≤C∥u∥s,M.Definition 3.
We say that a weightw(x) satisfies the Ar(D) condition for r>1 and write w(x)∈Ar(D), if w>0 a.e. and
(8)sup⁡B⊂D(1|B|∫B‍wdx)(1|B|∫B‍(1w)1/(r-1)dx)r-1<∞.ForAr(D) weight, we also need the following result which appears in [9].Lemma 4.
Ifw(x)∈Ar(D), then there exist constants β>1 and C, independent of w, such that
(9)∥w∥β,B≤C|B|(1-β)/β∥w∥1,B
for all balls B⊂D.Theorem 5.
LetD⊂Rn be a bounded convex domain, n<p<∞, and let T:Lp(D,∧k)→Lp(D,∧k-1) be the Homotopy operator, k=1,2,…,n. Then there exists a constant C, independent of u, such that
(10)∥T(G(u))∥p,B,w≤C∥u∥p,B,w
for any ball B⊂D, w(x)∈Ar(D), and 1<r<p/n.Proof.
Sincew(x)∈Ar(D), by Lemma 4, there exist constants β>1 and C1, independent of w, such that
(11)∥w∥β,B≤C1|B|(1-β)/β∥w∥1,B
for any ball B⊂D.
Choosingk=βp/(β-1), then by Hölder inequality with 1/k+1/βp=1/p, we have
(12)∥T(G(u))∥p,B,w=(∫B‍|T(G(u))|pw(x)dx)1/p≤(∫B‍|T(G(u))|kdx)1/k(∫B‍wβdx)1/βp=∥T(G(u))∥k,B∥w(x)∥β,B1/p.
Thus, substituting (11) into (12), we obtain
(13)∥T(G(u))∥p,B,w≤C1|B|(1-β)/βp∥T(G(u))∥k,B∥w(x)∥1,B1/p.
Taking m=p/r, it is easy to see that m>1 and (1/m)-(1/k)<(1/m)<(1/n). Hence communicating Lemmas 1 and 2, we have
(14)∥T(G(u))∥k,B≤C2∥G(u)∥m,B≤C3∥u∥m,B.
Combining (13) and (14), we have
(15)∥T(G(u))∥p,B,w≤C4|B|(1-β)/βp∥u∥m,B∥w(x)∥1,B1/p.
Using Hölder inequality with 1/p+(r-1)/p=r/p, we have
(16)∥u∥m,B≤(∫B‍(|u|w1/p)pdx)1/p(∫B‍(1w)1/(r-1)dx)(r-1)/p=∥u∥p,B,w(∫B‍(1w)1/(r-1)dx)(r-1)/p.
Note w(x)∈Ar(D); then,
(17)sup⁡B⊂D(1|B|∫B‍wdx)(1|B|∫B‍(1w)1/(r-1)dx)r-1<C5<∞.
Thus, observing (15) and (16), we immediately obtain that
(18)∥T(G(u))∥p,B,w≤C6|B|(1-β)/βp+(r/p)∥u∥p,B,w≤C6|D|(1-β)/βp+(r/p)∥u∥p,B,w≤C7∥u∥p,B,w.
Here C7 is a constant independent of u. Thus we complete the proof of Theorem 5.Furthermore, ifu is an A-harmonic tensor on D, ρ>1 and 0<s, t<∞, then there exists a constant C, independent of u, such that
(19)∥u∥s,B≤C|B|(t-s)/ts∥u∥t,ρB
for all balls or cubs B with ρB⊂D (for more details about A-harmonic tensors, see [10]). By the property of A-harmonic tensor, using the same method developed in the proof of Theorem 5, we can easily extend into the following Ar(D)-weighted version.Corollary 6.
LetD⊂Rn be a bounded convex domain, n<p<∞, u be an A-harmonic tensor, and T:Lp(D,∧k)→Lp(D,∧k-1) be the Homotopy operator, k=1,2,…,n. Then there exists a constant C, independent of u, such that
(20)∥T(G(u))∥p,B,wα≤C∥u∥p,ρB,wα
for any ball B⊂D, w(x)∈Ar(D), and 1<r<p/n, 0<α≤1, ρ>1.In order to obtain the boundedness of the compositionT∘G, we need the following modified Whitney cover in [10] and see [11] for more details about Whitney cover.Lemma 7.
Each open subsetE⊂Rn has a modified Whitney cover of cubs W={Qi} satisfying ⋃‍iQi=E and ∑‍Qi∈Wχ5/4Qi≤N·χE(x), for all x∈Rn and some N>1, where χE(x) is the characteristic function for the set E.Theorem 8.
LetD⊂Rn be a bounded convex domain, n<p<∞. Then the composite operator T∘G:Lp(D,∧k,w)→Lp(D,∧k-1,w) is bounded, k=1,2,…,n. Here w(x)∈Ar(D) and 1<r<p/n.Proof.
From Lemma7, we know that there exists a sequence of cubs W={Qi} such that ⋃‍iQi=D and ∑‍Qi∈Wχ5/4Qi≤N·χE(x) for all x∈D, where N>1 is some constant. Hence, for u∈Lp(D,∧k,w), we have
(21)∥T(G(u))∥p,D,wp=∫D‍|T(G(u))|pdμ≤∑‍Qi∈W∫Qi‍|T(G(u))|pdμ≤∑‍Qi∈WC1∫Qi‍|u|pdμ≤∑‍Qi∈WC1∫D‍|u|pχQi(x)dμ≤C1∫D‍∑‍Qi∈W|u|pχQi(x)dμ≤C1∫D‍N·|u|pχD(x)dμ≤C1N∫D‍|u|pdμ=C2∫D‍|u|pdμ=C2∥u∥p,D,wp,
where dμ=w(x)dx and C2=C1N is independent of u and each Qi. Thus, we complete the proof of Theorem 8.


## 3. Norm Estimates with Power-Type Weights

LetS⊂Rn be a bounded domain and D be a nonempty of S¯=S⋃‍∂S. If we use dist⁡(x,D) to denote the distance of the point x from the set D, then ω(x)=(dist⁡(x,D))ɛ for ɛ∈R is called power-type weight. In this section, we will establish some strong (p,q)-type norm inequalities with power-type weights for the composition of the Homotopy T and Green’s operator G acting on differential form. In the following proof, we will use the following Lemma which appears in [8].Lemma 9.
The operatorT:Ωp,r(D,∧k)→Ωq,p(D,∧k-1) is bounded provided that
(22)Either1≤p,q,r≤∞,1p-1q<1n,1r-1p<1n,Or1<p,q,r≤∞,1p-1q≤1n,1r-1p≤1n.Theorem 10.
LetD⊂Rn be a bounded convex domain, 1<p, q<∞, 0≤1/p-1/q≤1/n, and let T:Lp(D,∧k)→Lq(D,∧k-1) be the Homotopy operator, k=1,2,…,n. Then there exists a constant C, independent of u, such that
(23)∥T(G(u))-(T(G(u)))D∥q,D≤C(1+diam⁡(D))∥u∥p,D
for any u∈Ωp,p(D,∧k).Proof.
From (4), we have the following decomposition:
(24)G(u)=T(d(G(u)))+d(T(G(u)))
for any differential form u∈Ωp,p(D,∧k), k=1,2,…,n.
Note thatu is an element of Ωp,p(D,∧k), k=1,2,…,n. From (4) and Lemmas 1 and 9, we have
(25)∥T(G(u))-(T(G(u)))D∥q,D=∥T(d(T(G(u))))∥q,D≤C1∥d(T(G(u)))∥p,D.
Here C1 is a constant independent of u. Applying (24) and (5), we have
(26)∥d(T(G(u)))∥p,D=∥G(u)-T(d(G(u)))∥p,D≤∥G(u)∥p,D+∥T(d(G(u)))∥p,D≤∥G(u)∥p,D+C2diam⁡(D)∥d(G(u))∥p,D.
Applying Lemma 2 into (26), we obtain
(27)∥d(T(G(u)))∥p,D≤(C3+C4diam⁡(D))∥u∥p,D.
Thus
(28)∥T(G(u))-(T(G(u)))D∥q,D≤(C5+C6diam⁡(D))∥u∥p,D≤C7(1+diam⁡(D))∥u∥p,D.
Here C7=max⁡{C5,C6} is independent of u. Thus, we complete the proof of Theorem 10.Next, we consider the following norm comparison equipped with power-type weights.Theorem 11.
LetD⊂Rn be a bounded convex domain, 1<p, q<∞, 0≤1/p-1/q≤1/n, let T:Lp(D,∧k)→Lq(D,∧k-1) be the Homotopy operator, k=1,2,…,n, and that continuous functions h and g defined in (0,+∞) satisfy (1)lim⁡t→0h(t)=0; (2)lim⁡t→0g(t)=∞. Then there exists a constant C, independent of u, such that
(29)∥T(G(u))-(T(G(u)))D∥q,D,μ1≤C(1+diam⁡(D))∥u∥p,D,μ2
for any u∈Ωp,p(D,∧k), dμ1=h(dist⁡(x,∂D))dx, dμ2=g(dist⁡(x,∂D))dx.Proof.
From Theorem10, we know that there exists a constant C1, independent of u, such that
(30)∥T(G(u))-(T(G(u)))D∥q,D≤C1(1+diam⁡(D))∥u∥p,D.
Fixing ɛ>0, then there exists δ1(ɛ)>0 such that h(dist⁡(x,∂D))<ɛ for all x∈D with dist⁡(x,∂D)<δ1. Let D1={x∈D,dist⁡(x,∂D)<δ1} and D2=D-D1. Then for all x∈D2, we have
(31)δ1≤dist⁡(x,∂D)<diam⁡(D).
Therefore, by the continuity of h, we know that there exists M1>0, such that
(32)h(dist⁡(x,∂D))<M1
for all x∈D2. Thus we have
(33)∥T(G(u))-(T(G(u)))D∥q,D,μ1=(∫D‍|T(G(u))-(T(G(u)))D|q·h(dist⁡(x,∂D))dx)1/q≤(ɛ∫D1‍|T(G(u))-(T(G(u)))D|qdx+M1∫D2‍|T(G(u))-(T(G(u)))D|qdx)1/q≤C2(∫D‍|T(G(u))-(T(G(u)))D|qdx)1/q.
Here C2=max⁡{ɛ1/q,M11/q}. Communicating (30) and (33), we have
(34)∥T(G(u))-(T(G(u)))D∥q,D,μ1≤C2∥T(G(u))-(T(G(u)))D∥q,D≤C3(1+diam⁡(D))∥u∥p,D.
Note that lim⁡t→0(1/g(t))=0. Then there exists δ2(ɛ)>0 such that 1/g(dist⁡(x,∂D))<ɛ for all x∈D with dist⁡(x,∂D)<δ2. Let D1′={x∈D,dist⁡(x,∂D)<δ2} and D2′=D-D1′. Then for all x∈D2′, we have
(35)δ2≤dist⁡(x,∂D)<diam⁡(D).
Therefore, by the continuity of g, we know that there exists M2>0, such that
(36)1g(dist⁡(x,∂D))<M2
for all x∈D2′. Therefore, we obtain
(37)∥u∥p,D=(∫D‍|u|p1g(dist⁡(x,∂D))dμ2)1/p≤(ɛ∫D1′‍|u|pdμ2+M2∫D2′‍|u|pdμ2)1/p≤C4(∫D‍|u|pdμ2)1/p=C4∥u∥p,D,μ2.
Here C4=max⁡{ɛ1/p,M21/p}. By (34) and (37), we have
(38)∥T(G(u))-(T(G(u)))D∥q,D,μ1≤C5(1+diam⁡(D))∥u∥p,D,μ2.
Here C5 is independent of u. Thus, we complete the proof of Theorem 11.In Theorem11, if we choose h(t)=tr and g(t)=t-s, 0<r, s<∞, we can easily obtain the following corollary.Corollary 12.
LetD⊂Rn be a bounded convex domain, 1<p, q<∞, 0≤1/p-1/q≤1/n, and let T:Lp(D,∧k)→Lq(D,∧k-1) be the Homotopy operator, k=1,2,…,n. Then there exists a constant C, independent of u, such that
(39)∫D‍|T(G(u))-(T(G(u)))D|q·(dist⁡(x,∂D))rdx≤C(1+diam⁡(D))(∫D‍|u|p1(dist⁡(x,∂D))sdx)1/p.
Here 0<r, s<∞.Note that, in the proof of Theorem11, if we let the composite operator T∘G act on the solution of nonhomogeneous A-harmonic equation, then we can drop lim⁡t→0h(t)=0. Next, we state the result as follows.Corollary 13.
LetD⊂Rn be a bounded convex domain, 1<p, q<∞, 0≤1/p-1/q≤1/n, let T:Lp(D,∧k)→Lq(D,∧k-1) be the Homotopy operator, and u∈Ωp,p(D,∧k) is a solution of nonhomogeneous A-harmonic equation, k=1,2,…,n. If continuous functions h and g defined in (0,+∞) satisfy that lim⁡t→0g(t)=∞,dμ1=h(dist⁡(x,∂D))dx and dμ2=g(dist⁡(x,∂D))dx. Then there exists a constant C, independent of u, such that
(40)∥T(G(u))-(T(G(u)))D∥q,B,μ1≤C(1+diam⁡(D))∥u∥p,ρB,μ2
for all balls B with ρB⊂D. Here ρ>1 is some constant.It is easy to find that the above corollary does not hold for ballsB⊂D with ∂B⋂‍∂D≠Φ but holds for those balls with ρB⊂D. Next, we introduce the following singular integral inequality.Theorem 14.
LetD⊂Rn be a bounded convex domain, 1<p, q<∞, 0≤1/p-1/q≤1/n, let T:Lp(D,∧k)→Lq(D,∧k-1) be the Homotopy operator, and u∈Ωp,p(D,∧k) is a solution of nonhomogeneous A-harmonic equation, k=1,2,…,n. If continuous functions h and g defined in (0,+∞) and h(t) is an increasing function, then there exists a constant C, independent of u, such that
(41)(∫B‍|T(G(u))-(T(G(u)))B|q1g(dist⁡(x,∂D))dx)1/q≤C(1+diam⁡(B))|ρB|(p-q)/pq×(∫ρB‍|u|p(h(dist⁡(x,∂D)))λdx)1/p
for all balls B with ρB⊂D and 0<λ<1. Here ρ>1 is some constant.Proof.
Letk=q/(1-λ). From 0<λ<1, it is easy to see that k>q. Using the Hölder inequality, we have
(42)(∫B‍|T(G(u))-(T(G(u)))B|q1g(dist⁡(x,∂D))dx)1/q≤(∫B‍|T(G(u))-(T(G(u)))B|kdx)1/k×(∫B‍1(g(dist⁡(x,∂D)))k/(k-q)dx)(k-q)/kq=∥T(G(u))-(T(G(u)))B∥k,B×(∫B‍1(g(dist⁡(x,∂D)))k/(k-q)dx)(k-q)/kq.
Note that ρB⊂D. Therefore, there exists a positive number c such that
(43)c<dist⁡(x,∂D)≤diam⁡(D)
for all x∈B. Furthermore, by the continuity of function g in (0,+∞), g(dist⁡(x,∂D)) has a positive lower bound M in B. Thus, from Theorem 10 and (42), we have
(44)(∫B‍|T(G(u))-(T(G(u)))B|q1g(dist⁡(x,∂D))dx)1/q≤(1M)1/q|B|(k-q)/kq∥T(G(u))-(T(G(u)))B∥k,B≤C1|B|(k-q)/kq(1+diam⁡(B))∥u∥k,B≤C1|B|(k-q)/kq(1+diam⁡(B))∥u∥k,ρ1B,
where ρ1>1 is a constant. Let ɛ∈(1/p,1) and m=ɛp. Since u is the solution of nonhomogenous A-harmonic equation. By (19), we know
(45)∥u∥k,ρ1B≤C2|ρ1B|(m-k)/mk∥u∥m,ρB,
where ρ>ρ1>1 is a constant. It is easy to find that 1<m<p. Using the Hölder inequality, we have
(46)∥u∥m,ρB=(∫ρB‍|u|m1(h(dist⁡(x,∂D)))mλ/p·(h(dist⁡(x,∂D)))mλ/pdx∫ρB‍|u|m1(h(dist⁡(x,∂D)))mλ/p)1/m≤(∫ρB‍|u|p(h(dist⁡(x,∂D)))λdx)1/p×(∫ρB‍((h(dist⁡(x,∂D)))λ/p)mp/(p-m)dx)(p-m)/mp.
The continuity and monotonicity of function h imply that
(47)(∫ρB‍((h(dist⁡(x,∂D)))λ/p)mp/(p-m)dx)(p-m)/mp=(∫ρB‍(h(dist⁡(x,∂D)))ɛλ/(1-ɛ)dx)(1-ɛ)/ɛp≤|ρB|(1-ɛ)/ɛp(h(diam⁡(D)))λ/p.
Hence, combining (41)–(47), we have
(48)(∫B‍|T(G(u))-(T(G(u)))B|q1g(dist⁡(x,∂D))dx)1/q≤C3|B|(k-q)/kq(1+diam⁡(B))|ρ1B|(m-k)/mk|ρB|(1-ɛ)/ɛp×(h(diam⁡(D)))λ/p(∫ρB‍|u|p(h(dist⁡(x,∂D)))λdx)1/p≤C4(1+diam⁡(B))|ρB|(p-q)/pq×(∫ρB‍|u|p(h(dist⁡(x,∂D)))λdx)1/p.
Here C4 is dependent of B and h but independent of u. Thus, we complete the proof of Theorem 11.


## 4. Application

In this section, we will use the estimates in Section3 to obtain the upper bound for the Lp norms of T(G(u)) or (T(G(u)))B in terms of Lq norms of u or du.Example 15.
Forn≥2, let u be a (n-1)-form defined in Rn by
(49)u=x1x12+x22+⋯+xn2dx2∧dx3∧⋯∧dxn-x2x12+x22+⋯+xn2dx1∧dx3∧⋯∧dxn+⋯+(-1)n-1×xnx12+x22+⋯+xn2dx1∧dx2∧⋯∧dxn-1.
It is easy to find that
(50)|u|=1,du=n-1x12+x22+⋯+xn2dx1∧dx2∧⋯∧dxn.
If we choose the usual (p,p)-type norm inequality to estimate T(G(u))-(T(G(u)))B and take p=n, where B=B(O,r)⊂Rn is a ball, then by Theorem 10, we have
(51)(∫B‍|T(G(u))-(T(G(u)))B|ndx)1/n≤C1(1+diam⁡(B))(∫B‍|u|ndx)1/n=C1(1+diam⁡(B))|B|1/n.
However, if we choose the (p,q)-type norm inequality to estimate T(G(u))-(T(G(u)))B and take p=n-1, q=n, then p, q satisfy the condition 0≤1/p-1/q≤1/n. Hence by using Theorem 10, we obtain
(52)(∫B‍|T(G(u))-(T(G(u)))B|ndx)1/n≤C2(1+diam⁡(B))(∫B‍|u|n-1dx)1/(n-1)=C2(1+diam⁡(B))|B|1/(n-1).
Compare (51) and (52), we can easily find that if we choose different (p,q)-type norm inequality to estimate the oscillation T(G(u))-(T(G(u)))B, we also obtain the different upper bound.Example 16.
InR2, consider that
(53)u(x,y)=arctanyx-1-arctanyx+1.
It is easy to check that u(x,y) is harmonic in the upper half plane. Note that
(54)du=∂u∂xdx+∂u∂ydy,*du=∂u∂xdy-∂u∂ydx.
Therefore, we have
(55)d*du=(∂2u∂x2+∂2u∂y2)dx∧dy=0,
which implies that *du is a closed form and hence is a solution of nonhomogenous A-harmonic equation. It is easy to see that
(56)|*du|=1((x-1)2+y2)((x+1)2+y2).
Let D denote a bound convex domain in the upper half plane and let σB¯⊂D be a closed ball without the points (-1,0) and (1,0). If σB¯ and D satisfy that dist⁡(σB,∂D)=M>0, then both |*du| and (dist⁡(x,∂D))-1 have the upper bounds in σB¯. Thus, for the term
(57)∫B‍|T(G(u))-(T(G(u)))B|p1g(dist⁡(x,∂D))dx,
it is usually not easy to be estimated due to the complexity of the compositions T(G(u)) and the function g. However, by Theorem 14, (57) can be controlled by the term
(58)∫ρB‍|u|p(h(dist⁡(x,∂D)))λdx.
Thus, we obtain an upper bound of (57).



---

*Source: 941658-2014-03-02.xml*