# Singular integrals of the compositions of Laplace-Beltrami and Green's operators

## Abstract

We establish the Poincaré-type inequalities for the composition of the Laplace-Beltrami operator and the Green's operator applied to the solutions of the non-homogeneous A-harmonic equation in the John domain. We also obtain some estimates for the integrals of the composite operator with a singular density.

## 1 Introduction

The purpose of the article is to develop the Poincaré-type inequalities for the composition of the Laplace-Beltrami operator Δ = dd* + d*d and Green's operator G over the δ-John domain. Both operators play an important role in many fields, including partial differential equations, harmonic analysis, quasiconformal mappings and physics . We first give a general estimate of the composite operator Δ G. Then, we consider the composite operator with a singular factor. The consideration was motivated from physics. For instance, when calculating an electric field, we will deal with the integral $E\left(r\right)=\frac{1}{4\pi {\epsilon }_{0}}{\int }_{D}\rho \left(x\right)\frac{r-x}{\parallel r-x{\parallel }^{3}}dx$, where ρ(x) is a charge density and x is the integral variable. It is singular if r D. Obviously, the singular integrals are more interesting to us because of their wide applications in different fields of mathematics and physics.

In this article, we assume that M is a bounded, convex domain and B is a ball in n , n ≥ 2. We use σB to denote the ball with the same center as B and with diam (σB) = σ diam(B), σ > 0. We do not distinguish the balls from cubes in this article. We use |E| to denote the Lebesgue measure of a set E n . We call ω a weight if $\omega \in {L}_{loc}^{1}\left({ℝ}^{n}\right)$ and ω > 0 a.e. Differential forms are extensions of functions in n . For example, the function u(x1, x2,..., x n ) is called a 0-form. Moreover, if u(x1, x2,..., x n ) is differentiable, then it is called a differential 0-form. The 1-form u(x) in n can be written as $u\left(x\right)={\sum }_{i=1}^{n}{u}_{i}\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)d{x}_{i}$. If the coefficient functions u i (x1, x2,..., x n ), i = 1, 2,..., n, are differentiable, then u(x) is called a differential l-form. Similarly, a differential k-form u(x) is generated by $\left\{d{x}_{{i}_{1}}\wedge d{x}_{{i}_{2}}\wedge \cdots \wedge d{x}_{{i}_{k}}\right\}$, k = 1, 2,..., n, that is, $u\left(x\right)={\sum }_{I}{u}_{I}\left(x\right)d{x}_{I}=\sum {u}_{{i}_{1}{i}_{2}\dots {i}_{k}}\left(x\right)d{x}_{{i}_{1}}\wedge d{x}_{{i}_{2}}\wedge \dots \wedge d{x}_{{i}_{k}}$, where I = (i1, i2,..., i k ), 1 ≤ i1 < i2 < < i k n. Let l = l ( n ) be the set of all l-forms in n , D'(M, l ) be the space of all differential l-forms on M and Lp (M, l ) be the l-forms $u\left(x\right)={\sum }_{I}{u}_{I}\left(x\right)d{x}_{I}$ on M satisfying ${\int }_{M}\mid {u}_{I}{\mid }^{p}<\infty$ for all ordered l-tuples I, l = 1, 2,..., n. We denote the exterior derivative by d : D'(M, l ) → D'(M, l+1) for l = 0, 1,..., n - 1, and define the Hodge star operator * : kn-kas follows, If $u={u}_{{i}_{1}{i}_{2}\dots {i}_{k}}\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)d{x}_{{i}_{1}}\wedge d{x}_{{i}_{2}}\wedge \cdots \wedge d{x}_{{i}_{k}}={u}_{I}d{x}_{I}$, i1 < i2 < < i k , is a differential k-form, then $\ast u=\ast \left({u}_{{i}_{1}{i}_{2}\dots {i}_{k}}\text{d}{x}_{{i}_{1}}\wedge \text{d}{x}_{{i}_{2}}\wedge \cdots \wedge \text{d}{x}_{{i}_{k}}\right)=\left(-{1\right)}^{\sum \left(I\right)}{u}_{I}\text{d}{x}_{J}$, where I = (i1, i2, i k ), J = {1, 2,..., n} - I, and $\sum \left(I\right)=\frac{k\left(k+1\right)}{2}+{\sum }_{i=1}^{k}{i}_{j}$. The Hodge codifferential operator d* : D'(M, l+1) → D'(M, l ) is given by d* = (-1)nl+1* d* on D'(M, l+1), l = 0, 1,..., n - 1. and the Laplace-Beltrami operator Δ is defined by Δ = dd* + d* d. We write $\parallel u{\parallel }_{s,M}=\left({\int }_{M}|u{|}^{s}{\right)}^{1/s}$ and $\parallel u{\parallel }_{s,M,\omega }=\left({\int }_{M}|u{|}^{s}\omega \left(x\right)\text{d}x{\right)}^{1/s}$, where ω(x) is a weight. Let lM be the l-th exterior power of the cotangent bundle, C( lM) be the space of smooth l-forms on M and $\mathcal{W}\left({\wedge }^{l}M\right)=\left\{u\in {L}_{loc}^{1}\left({\wedge }^{l}M\right):u\phantom{\rule{0.5em}{0ex}}\text{has}\phantom{\rule{0.5em}{0ex}}\text{generalized}\phantom{\rule{0.5em}{0ex}}\text{gradient}}$. The harmonic l-fields are defined by $\mathcal{H}\left({\wedge }^{l}M\right)=\left\{u\in \mathcal{W}\left({\wedge }^{l}M\right):du={d}^{*}u=0,u\in {L}^{p}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{some}\phantom{\rule{2.77695pt}{0ex}}1. The orthogonal complement of $\mathcal{H}$ in L1 is defined by ${\mathcal{H}}^{\perp }=\left\{u\in {L}^{1}:=0\phantom{\rule{0.3em}{0ex}}\text{for all}\phantom{\rule{2.77695pt}{0ex}}h\in \mathcal{H}\right\}$. Then, the Green's operator G is defined as $G:{C}^{\infty }\left({\wedge }^{l}M\right)\to {\mathcal{H}}^{\perp }\cap {C}^{\infty }\left({\wedge }^{l}M\right)$ by assigning G(u) be the unique element of ${\mathcal{H}}^{\perp }\cap {C}^{\infty }\left({\wedge }^{l}M\right)$ satisfying Poisson's equation ΔG(u) = u - H(u), where H is the harmonic projection operator that maps C( lM) onto $\mathcal{H}$ so that H(u) is the harmonic part of u [[7, 8], for more properties of these operators]. The differential forms can be used to describe various systems of PDEs and to express different geometric structures on manifolds. For instance, some kinds of differential forms are often utilized in studying deformations of elastic bodies, the related extrema for variational integrals, and certain geometric invariance [9, 10].

We are particularly interested in a class of differential forms satisfying the well known non-homogeneous A-harmonic equation

${d}^{*}A\left(x,du\right)=B\left(x,du\right),$
(1.1)

where A : M × l ( n ) → l ( n ) and B : M × l ( n ) → l-1( n ) satisfy the conditions:

$\mid A\left(x,\xi \right)\mid \le a\mid \xi {\mid }^{p-1},\phantom{\rule{1em}{0ex}}A\left(x,\xi \right)\cdot \xi \ge \mid \xi {\mid }^{p},\phantom{\rule{1em}{0ex}}\mid B\left(x,\xi \right)\mid \le b\mid \xi {\mid }^{p-1}$
(1.2)

for almost every x M and all ξ l ( n ). Here a > 0 and b > 0 are constants and 1 < p < ∞ is a fixed exponent associated with the Equation (1.1). If the operator B = 0, Equation (1.1) becomes d* A(x, du) = 0, which is called the homogeneous A-harmonic equation. A solution to (1.1) is an element of the Sobolev space ${W}_{loc}^{1,p}\left(M,{\wedge }^{l-1}\right)$ such that ${\int }_{M}A\left(x,du\right)\cdot d\phi +B\left(x,du\right)\cdot \phi =0$ for all $\phi \in {W}_{loc}^{1,p}\left(M,{\wedge }^{l-1}\right)$ with compact support. Let A : M × l ( n ) → l ( n ) be defined by A(x, ξ) = ξ|ξ|p-2with p > 1. Then, A satisfies the required conditions and d* A(x, du) = 0 becomes the p-harmonic equation

${d}^{*}\left(du\mid du{\mid }^{p-2}\right)=0$
(1.3)

for differential forms. If u is a function (0-form), the equation (1.3) reduces to the usual p-harmonic equation div(u|u|p-2) = 0 for functions. Some results have been obtained in recent years about different versions of the A-harmonic equation [8, 1116].

## 2 Main results and proofs

We first introduce the following definition and lemmas that will be used in this article.

Definition 2.1 A proper subdomain Ω n is called a δ-John domain, δ > 0, if there exists a point x0 Ω which can be joined with any other point x Ω by a continuous curve γ Ω so that

$d\left(\xi ,\partial \Omega \right)\ge \delta \mid x-\xi \mid$

for each ξ γ. Here d(ξ, ∂Ω) is the Euclidean distance between ξ and ∂Ω.

Lemma 2.1Let ϕ be a strictly increasing convex function on [0, ∞) with ϕ(0) = 0, and D be a domain in n . Assume that u is a function in D such that ϕ(|u|) L1(D, μ) and μ({x D : |u - c| > 0}) > 0 for any constant c, where μ is a Radon measure defined by dμ(x) = ω(x)dx for a weight ω(x). Then, we have

${\int }_{D}\varphi \left(\frac{a}{2}\mid u-{u}_{D,\mu }\mid \right)d\mu \le \underset{D}{\int }\varphi \left(a\mid u\mid \right)d\mu$

for any positive constant a, where${u}_{D,\mu }=\frac{1}{\mu \left(D\right)}{\int }_{D}ud\mu$.

Lemma 2.2 [ 3] Let u ClM) and l = 1, 2,..., n, 1 < s < ∞. Then, there exists a positive constant C, independent of u, such that

$\parallel d{d}^{*}G\left(u\right){\parallel }_{s,M}+\parallel {d}^{*}dG\left(u\right){\parallel }_{s,M}+\parallel dG\left(u\right){\parallel }_{s,M}+\parallel {d}^{*}G\left(u\right){\parallel }_{s,M}+\parallel G\left(u\right){\parallel }_{s,M}\le C\parallel u{\parallel }_{s,M}$

Lemma 2.3Each Ω has a modified Whitney cover of cubes$\mathcal{V}=\left\{{Q}_{i}\right\}$such that i Q i = Ω, ${\sum }_{{Q}_{i}\in \mathcal{V}}{\chi }_{\sqrt{\frac{5}{4}}{Q}_{i}}\le N{\chi }_{\Omega }$and some N > 1, and if Q i Q j , then there exists a cube R (this cube need not be a member of$\mathcal{V}$) in Q i Q j such that Q i Q j NR. Moreover, if Ω is δ-John, 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 Q0, Q1,..., Q k = Q from$\mathcal{V}$and such that Q ρQ i , i = 0, 1, 2,..., k, for some ρ = ρ (n, δ).

Lemma 2.4 Let$u\in {L}_{loc}^{s}\left(M,{\Lambda }^{l}\right)$, l = 1, 2,..., n, 1 < s < ∞, G be the Green's operator and Δ be the Laplace-Beltrami operator. Then, there exists a constant C, independent of u, such that

$\parallel \Delta G\left(u\right){\parallel }_{s,B}\le C\parallel u{\parallel }_{s,B}$
(2.1)

for all balls B M.

Proof By using Lemma 2.2, we have

$\parallel \Delta G\left(u\right){\parallel }_{s,B}=\parallel \left(d{d}^{*}+{d}^{*}d\right)G\left(u\right){\parallel }_{s,B}\le \parallel d{d}^{*}G\left(u\right){\parallel }_{s,B}+\parallel {d}^{*}dG\left(u\right){\parallel }_{s,B}\le C\parallel u{\parallel }_{s,B}.$
(2.2)

This ends the proof of Lemma 2.4. □

Lemma 2.5 Let$u\in {L}_{loc}^{s}\left(M,{\Lambda }^{l}\right)$, l = 1, 2,..., n, 1 < s < ∞, be a solution of the non-homogeneous A-harmonic equation in a bound and convex domain M, G be the Green's operator and Δ be the Laplace-Beltrami operator. Then, there exists a constant C independent of u, such that

${\left(\underset{B}{\int }|\Delta G\left(u{\right)|}^{s}\frac{1}{\text{d}{\left(x,\partial M\right)}^{\alpha }}\text{d}x\right)}^{1/s}\le C{\left(\underset{\sigma B}{\int }|u{|}^{s}\frac{1}{|x-{x}_{B}{|}^{\lambda }}\text{d}x\right)}^{1/s}$
(2.3)

for all balls B with σB M and diam(B) ≥ d0> 0, where d0is a constant, σ > 1, and any real number α and λ with α > λ ≥ 0. Here x B is the center of the ball B.

Proof Let ε (0, 1) be small enough such that εn < α - λ and B M be any ball with center x B and radius r B . Also, let δ > 0 be small enough, B δ = {x B : |x - x B | ≤ δ} and D δ = B \B δ . Choose t = s/(1 - ε), then, t > s. Write β = t/(t - s). Using the Hölder inequality and Lemma 2.4, we have

$\begin{array}{l}{\left(\underset{{D}_{\delta }}{\int }|\Delta G\left(u{\right)|}^{s}\frac{1}{\text{d}{\left(x,\partial M\right)}^{\alpha }}\text{d}x\right)}^{1/s}={\left(\underset{{D}_{\delta }}{\int }{\left(|\Delta G\left(u\right)|\frac{1}{\text{d}{\left(x,\partial M\right)}^{\alpha /s}}\right)}^{s}\text{d}x\right)}^{1/s}\\ \le \parallel \Delta G\left(u\right)|{|}_{t,{D}_{\delta }}{\left(\underset{{D}_{\delta }}{\int }{\left(\frac{1}{\text{d}\left(x,\partial M\right)}\right)}^{t\alpha /\left(t-s\right)}\text{d}x\right)}^{\left(t-s\right)/st}\\ =\parallel \Delta G\left(u\right)|{|}_{t,{D}_{\delta }}{\left(\underset{{D}_{\delta }}{\int }{\left(\frac{1}{\text{d}\left(x,\partial M\right)}\right)}^{\alpha \beta }\text{d}x\right)}^{1/\beta s}\\ \le \parallel \Delta G\left(u\right)|{|}_{t,B}{\left(\underset{{D}_{\delta }}{\int }{\left(\frac{1}{\text{d}\left(x,\partial M\right)}\right)}^{\alpha \beta }\text{d}x\right)}^{1/\beta s}\\ \le {C}_{1}\parallel u|{|}_{t,B}{‖\frac{1}{\text{d}{\left(x,\partial M\right)}^{\alpha }}‱}_{\beta ,{D}_{\delta }}^{1/s}.\end{array}$
(2.4)

We may assume that x B = 0. Otherwise, we can move the center to the origin by a simple transformation. Then, $\frac{1}{d\left(x,\partial M\right)}\le \frac{1}{{r}_{B}-\mid x\mid }$ for any x B, we have

${\left(\underset{{D}_{\delta }}{\int }{\left(\frac{1}{d\left(x,\partial M\right)}\right)}^{\alpha \beta }dx\right)}^{1∕\beta s}\le {\left(\underset{{D}_{\delta }}{\int }\frac{1}{\mid x-{x}_{B}{\mid }^{\alpha \beta }}dx\right)}^{1∕\beta s}.$
(2.5)

Therefore, for any x B, |x - x B | ≥ |x|- |x B | = |x|. By using the polar coordinate substitution, we have

$\begin{array}{ll}\hfill {\left(\underset{{D}_{\delta }}{\int }\frac{1}{\mid x-{x}_{B}{\mid }^{\alpha \beta }}\mathsf{\text{d}}x\right)}^{1∕\beta s}& \le {\left({C}_{2}\underset{\delta }{\overset{{r}_{B}}{\int }}{\rho }^{-\alpha \beta }{\rho }^{n-1}\mathsf{\text{d}}\rho \right)}^{1∕\beta s}\phantom{\rule{2em}{0ex}}\\ ={\left|\frac{{C}_{2}}{n-\alpha \beta }\left({{r}_{B}}^{n-\alpha \beta }-{\delta }^{n-\alpha \beta }\right)\right|}^{1∕\beta s}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\le {C}_{3}\mid {{r}_{B}}^{n-\alpha \beta }-{\delta }^{n-\alpha \beta }{\mid }^{1∕\beta s}.\phantom{\rule{2em}{0ex}}\end{array}$
(2.6)

Choose m = nst/(ns + αt - λt), then 0 < m < s. By the reverse Hölder inequality, we find that

$\parallel u{\parallel }_{t,B}\le {C}_{4}\mid B{\mid }^{\frac{m-t}{mt}}\parallel u{\parallel }_{m,\sigma B},$
(2.7)

where σ > 1 is a constant. By the Hölder inequality again, we obtain

$\begin{array}{ll}\hfill \parallel u{\parallel }_{m,\sigma B}& ={\left(\underset{\sigma B}{\int }{\left(\mid u\mid \mid x-{x}_{B}{\mid }^{-\lambda ∕s}\mid x-{x}_{B}{\mid }^{\lambda ∕s}\right)}^{m}dx\right)}^{1∕m}\phantom{\rule{2em}{0ex}}\\ \le {\left(\underset{\sigma B}{\int }{\left(\mid u\mid \mid x-{x}_{B}{\mid }^{-\lambda ∕s}\right)}^{s}dx\right)}^{1∕s}{\left(\underset{\sigma B}{\int }{\left(\mid x-{x}_{B}{\mid }^{\lambda ∕s}\right)}^{\frac{ms}{s-m}}dx\right)}^{\frac{s-m}{ms}}\phantom{\rule{2em}{0ex}}\\ \le {\left(\underset{\sigma B}{\int }\mid u{\mid }^{s}\mid x-{x}_{B}{\mid }^{-\lambda }dx\right)}^{1∕s}{C}_{5}{\left(\sigma {r}_{B}\right)}^{\lambda ∕s+n\left(s-m\right)∕ms}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\le {C}_{6}{\left(\underset{\sigma B}{\int }\mid u{\mid }^{s}\mid x-{x}_{B}{\mid }^{-\lambda }dx\right)}^{1∕s}{\left({r}_{B}\right)}^{\lambda ∕s+n\left(s-m\right)∕ms}.\phantom{\rule{2em}{0ex}}\end{array}$
(2.8)

By a simple calculation, we find that n - αβ + λβ + (s - m)/m = 0. Substituting (2.6)-(2.8) in (2.4), we have

$\begin{array}{c}{\left(\underset{{D}_{\delta }}{\int }\mid \Delta G\left(u\right){\mid }^{s}\frac{1}{d{\left(x,\partial M\right)}^{\alpha }}dx\right)}^{1∕s}\\ \phantom{\rule{1em}{0ex}}\le {C}_{7}\mid B{\mid }^{\frac{m-t}{mt}}{\left(\underset{\sigma B}{\int }\mid u{\mid }^{s}\mid x-{x}_{B}{\mid }^{-\lambda }dx\right)}^{1∕s}{\left({r}_{B}\right)}^{\frac{\lambda }{s}+\frac{n\left(s-m\right)}{ms}}\mid {r}_{B}^{n-\alpha \beta }-{\delta }^{n-\alpha \beta }{\mid }^{1∕\beta s}\\ \phantom{\rule{1em}{0ex}}={C}_{7}\mid B{\mid }^{\frac{m-t}{mt}}{\left(\underset{\sigma B}{\int }\mid u{\mid }^{s}\mid x-{x}_{B}{\mid }^{-\lambda }dx\right)}^{1∕s}{\left[{{r}_{B}}^{\left(\frac{\lambda }{s}+\frac{n\left(s-m\right)}{ms}\right){\beta }_{s}}\left|{r}_{B}^{n-\alpha \beta }-{\delta }^{n-\alpha \beta }\right|\right]}^{1∕\beta s}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}={C}_{7}\mid B{\mid }^{\frac{m-t}{mt}}{\left(\underset{\sigma B}{\int }\mid u{\mid }^{s}\mid x-{x}_{B}{\mid }^{-\lambda }dx\right)}^{1∕s}{\left|{C}_{8}{r}_{B}^{n-\alpha \beta +\lambda \beta +\frac{n\beta \left(s-m\right)}{m}}-{\delta }^{n-\alpha \beta }{r}_{B}^{\lambda \beta +\frac{n\beta \left(s-m\right)}{m}}\right|}^{1∕\beta s}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\le {C}_{7}\mid B{\mid }^{\frac{m-t}{mt}}{\left(\underset{\sigma B}{\int }\mid u{\mid }^{s}\mid x-{x}_{B}{\mid }^{-\lambda }dx\right)}^{1∕s}{\left[{C}_{8}{r}_{B}^{n-\alpha \beta +\lambda \beta +\frac{n\beta \left(s-m\right)}{m}}-{\delta }^{n-\alpha \beta }{\delta }^{\lambda \beta +\frac{n\beta \left(s-m\right)}{m}}\right]}^{1∕\beta s}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\le {C}_{7}\mid B{\mid }^{\frac{m-t}{mt}}{\left(\underset{\sigma B}{\int }\mid u{\mid }^{s}\mid x-{x}_{B}{\mid }^{-\lambda }dx\right)}^{1∕s}{\left[{C}_{8}{r}_{B}^{n-\alpha \beta +\lambda \beta +\frac{n\beta \left(s-m\right)}{m}}+{\delta }^{n-\alpha \beta +\lambda \beta +\frac{n\beta \left(s-m\right)}{m}}\right]}^{1∕\beta s}\\ \phantom{\rule{1em}{0ex}}\le {C}_{9}\mid B{\mid }^{\frac{\lambda -\alpha }{ns}}{\left(\underset{\sigma B}{\int }\mid u{\mid }^{s}\mid x-{x}_{B}{\mid }^{-\lambda }dx\right)}^{1∕s}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\le {C}_{10}{\left(\underset{\sigma B}{\int }\mid u{\mid }^{s}\mid x-{x}_{B}{\mid }^{-\lambda }dx\right)}^{1∕s},\end{array}$
(2.9)

thus is,

${\left(\underset{{D}_{\delta }}{\int }\mid \Delta G\left(u\right){\mid }^{s}\frac{1}{d{\left(x,\partial M\right)}^{\alpha }}dx\right)}^{1∕s}\le {C}_{10}{\left(\underset{\sigma B}{\int }\mid u{\mid }^{s}\frac{1}{{|x-{x}_{B}|}^{\lambda }}dx\right)}^{1∕s}.$
(2.10)

Notice that ${\mathrm{lim}}_{\delta \to 0}{\left(\underset{{D}_{\delta }}{\int }|\Delta G\left(u{\right)|}^{s}\frac{1}{\text{d}{\left(x,\partial M\right)}^{\alpha }}\text{d}x\right)}^{1/s}={\left(\underset{B}{\int }|\Delta G\left(u{\right)|}^{s}\frac{1}{\text{d}{\left(x,\partial M\right)}^{\alpha }}\text{d}x\right)}^{1/s}.$ letting δ → 0 in (2.10), we obtain (2.3). we have completed the proof of Lemma 2.5. □

Theorem 2.6 Let u D'(Ω, Λ l ) be a solution of the A-harmonic equation (1.1), G be the Green's operator and Δ be the Laplace-Beltrami operator. Assume that s is a fixed exponent associated with the non-homogeneous A-harmonic equation. Then, there exists a constant C, independent of u, such that

${\left(\underset{\Omega }{\int }\mid \Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{\mid }^{s}\frac{1}{d{\left(x,\partial \Omega \right)}^{\alpha }}dx\right)}^{1∕s}\le C{\left(\underset{\Omega }{\int }\mid u{\mid }^{s}g\left(x\right)dx\right)}^{1∕s}$
(2.11)

for any bounded and convex δ-John domain Ω n, where$g\left(x\right)={\sum }_{i}{\chi }_{{Q}_{i}}\frac{1}{\mid x-{x}_{{Q}_{i}}{\mid }^{\lambda }}$, ${x}_{{Q}_{i}}$is the center of Q i with Ω = i Q i . Here α and λ are constants with 0 ≤ λ < α < n, and the fixed cube Q0 Ω, the constant N > 1 and the cubes Q i Ω appeared in Lemma 2.3, ${x}_{{Q}_{i}}$is the center of Q i .

Proof We use the notation appearing in Lemma 2.3. There is a modified Whitney cover of cubes $\mathcal{V}=\left\{{Q}_{i}\right\}$ for Ω such that Ω = Q i , and ${\sum }_{{Q}_{i}\in \mathcal{V}}{\chi }_{\sqrt{\frac{5}{4}}{Q}_{i}}\le N{\chi }_{\Omega }$ for some N > 1. For each ${Q}_{i}\in \mathcal{V}$, if diam(Q i ) ≥ d0 (where d0 is the constant appearing in Lemma 2.5), it is fine and we keep Q i in the collection $\mathcal{V}$. Otherwise, if diam(Q i ) < d0, we replace Q i by a new cube ${Q}_{i}^{*}$ with the same center as Q i and $diam\left({Q}_{i}^{*}\right)={d}_{0}$. Thus, we obtain a modified collection ${\mathcal{V}}^{*}$ consisting of all cubes ${Q}_{i}^{*}$, and ${\mathcal{V}}^{*}$ has the same properties as $\mathcal{V}$. Moreover, diam $\left({Q}_{i}^{*}\right)\ge {d}_{0}$ for any ${Q}_{i}^{*}\in {\mathcal{V}}^{*}$. Let ${\Omega }^{*}=\cup {Q}_{i}^{*}$. Also, we may extend the definition of u to Ω* such that u(x) = 0 if x Ω* - Ω. Hence, without loss of generality, we assume that diam(Q i ) ≥ d0 for any ${Q}_{i}\in \mathcal{V}$. Thus, $\mid {Q}_{i}\mid \ge K{d}_{0}^{n}$ for any ${Q}_{i}\in \mathcal{V}$ and some constant K > 0. Since Ω = Q i , for any x Ω, it follows that x Q i for some i. Applying Lemma 2.5 to Q i , we have

${\left(\underset{{Q}_{i}}{\int }\mid \Delta G\left(u\right){\mid }^{s}\frac{1}{d{\left(x,\partial \Omega \right)}^{\alpha }}dx\right)}^{1∕s}\le {C}_{1}{\left(\underset{\sigma {Q}_{i}}{\int }\mid u{\mid }^{s}\frac{1}{d{\left(x,{x}_{{Q}_{i}}\right)}^{\lambda }}dx\right)}^{1∕s},$
(2.12)

where σ > 1 is a constant. Let μ(x) and μ1(x) be the Radon measure defined by $d\mu =\frac{1}{d{\left(x,\partial \Omega \right)}^{\alpha }}dx$ and dμ1(x) = g(x)dx, respectively. Then,

$\mu \left(Q\right)=\underset{Q}{\int }\frac{1}{d{\left(x,\partial \Omega \right)}^{\alpha }}dx\ge \underset{Q}{\int }\frac{1}{{\left(diam\left(\Omega \right)\right)}^{\alpha }}dx=P\mid Q\mid ,$
(2.13)

where P is a positive constant. Then, by the elementary in equality (a + b) s ≤ 2 s (|a| s + |b| s ), s ≥ 0, we have

$\begin{array}{l}{\left(\underset{\Omega }{\int }|\Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{|}^{s}\frac{1}{\text{d}{\left(x,\partial \Omega \right)}^{\alpha }}\text{d}x\right)}^{1/s}\\ ={\left(\underset{\cup {Q}_{i}}{\int }|\Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{|}^{s}\text{d}\mu \right)}^{1/s}\\ \phantom{\rule{0.5em}{0ex}}\le {\left(\sum _{{Q}_{i}\in \mathcal{V}}\left({2}^{s}\underset{{Q}_{i}}{\int }|\Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{i}}{|}^{s}\text{d}\mu +{2}^{s}\underset{{Q}_{i}}{\int }|\left(\Delta G\left(u{\right)\right)}_{{Q}_{i}}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{|}^{s}\text{d}\mu \right)\right)}^{1/s}\\ \le {C}_{2}{\left(\left(\sum _{{Q}_{i}\in \mathcal{V}}\underset{{Q}_{i}}{\int }|\Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{i}}{|}^{s}\text{d}\mu \right)\right)}^{1/s}\\ \phantom{\rule{0.5em}{0ex}}+{\left(\sum _{{Q}_{i}\in \mathcal{V}}\underset{{Q}_{i}}{\int }|\left(\Delta G\left(u{\right)\right)}_{{Q}_{i}}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{|}^{s}\text{d}\mu \right)}^{1/s}\right)\end{array}$
(2.14)

for a fixed Q0 Ω. The first sum in (2.14) can be estimated by using Lemma 2.1 with φ = ts , a = 2, and Lemma 2.5

$\begin{array}{c}\sum _{{Q}_{i}\in \mathcal{V}}\underset{{Q}_{i}}{\int }|\Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{i}}{|}^{s}\text{d}\mu \le \sum _{{Q}_{i}\in \mathcal{V}}\underset{{Q}_{i}}{\int }{2}^{s}|\Delta G\left(u{\right)|}^{s}\text{d}\mu \\ \le {C}_{3}\sum _{{Q}_{i}\in \mathcal{V}}\underset{\sigma {Q}_{i}}{\int }|u{|}^{s}\text{d}{\mu }_{1}\\ \le {C}_{4}\sum _{{Q}_{i}\in \mathcal{V}}\underset{\Omega }{\int }\left(|u{|}^{s}\text{d}{\mu }_{1}\right){\chi }_{\sigma {Q}_{i}}\\ \le {C}_{5}\underset{\Omega }{\int }|u{|}^{s}\text{d}{\mu }_{1}\\ ={C}_{5}\underset{\Omega }{\int }|u{|}^{s}g\left(x\right)dx.\end{array}$
(2.15)

To estimate the second sum in (2.14), we need to use the property of δ-John domain. Fix a cube $Q\in \mathcal{V}$ and let Q0, Q1,..., Q k = Q be the chain in Lemma 2.3.

$|\left(\Delta G\left(u{\right)\right)}_{Q}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}|\le \sum _{i=0}^{k-1}|\left(\Delta G{\left(u\right)}_{{Q}_{i}}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{i+1}}|.$
(2.16)

The chain {Q i } also has property that, for each i, i = 0, 1,..., k - 1, with Q i Qi+1, there exists a cube D i such that D i Q i Qi+1and Q i Qi+1 ND i , N > 1.

$\frac{max\left\{\mid {Q}_{i}\mid ,\mid {Q}_{i+1}\mid \right\}}{\mid {Q}_{i}\cap {Q}_{i+1}\mid }\le \frac{max\left\{\mid {Q}_{i}\mid ,\mid {Q}_{i+1}\mid \right\}}{\mid {D}_{i}\mid }\le {C}_{6}.$

For such D j , j = 0, 1,..., k - 1, Let |D*| = min{|D0|, |D1|,..., |D k - 1|} then

$\frac{max\left\{\mid {Q}_{i}\mid ,\mid {Q}_{i+1}\mid \right\}}{\mid {Q}_{i}\cap {Q}_{i+1}\mid }\le \frac{max\left\{\mid {Q}_{i}\mid ,\mid {Q}_{i+1}\mid \right\}}{\mid {D}^{*}\mid }\le {C}_{7}.$
(2.17)

By (2.13), (2.17) and Lemma 2.5, we have

$\begin{array}{c}|\left(\Delta G\left(u\right){\right)}_{{Q}_{i}}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{i+1}}{|}^{s}=\frac{1}{\mu \left({Q}_{i}\cap {Q}_{i+1}\right)}\underset{{Q}_{i}\cap {Q}_{i+1}}{\int }|\left(\Delta G\left(u{\right)\right)}_{{Q}_{i}}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{i+1}}{|}^{s}\frac{\text{d}x}{\text{d}{\left(x,\partial \Omega \right)}^{\alpha }}\\ \le \frac{{C}_{8}}{|{Q}_{i}\cap {Q}_{i+1}|}\underset{{Q}_{i}\cap {Q}_{i+1}}{\int }|\left(\Delta G\left(u{\right)\right)}_{{Q}_{i}}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{i+1}}{|}^{s}\frac{\text{d}x}{\text{d}{\left(x,\partial \Omega \right)}^{\alpha }}\\ \le \frac{{C}_{8}{C}_{7}}{\mathrm{max}\left\{|{Q}_{i}|,|{Q}_{i+1}|\right\}}\underset{{Q}_{i}\cap {Q}_{i+1}}{\int }|\left(\Delta G\left(u{\right)\right)}_{{Q}_{i}}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{i+1}}{|}^{s}\text{d}\mu \\ \le {C}_{9}\sum _{j=i}^{i+1}\frac{1}{|{Q}_{j}|}\underset{{Q}_{j}}{\int }|\Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{j}}{|}^{s}\text{d}\mu \\ \le {C}_{10}\sum _{j=i}^{i+1}\frac{1}{|{Q}_{j}|}\underset{\sigma {Q}_{j}}{\int }|u{|}^{s}\text{d}{\mu }_{1}\\ ={C}_{10}\sum _{j=i}^{i+1}|{Q}_{j}{|}^{-1}\underset{\sigma {Q}_{j}}{\int }|u{|}^{s}\text{d}{\mu }_{1}.\end{array}$
(2.18)

Since Q NQ j for j = i, i + 1, 0 ≤ ik - 1, from (2.18)

$\begin{array}{c}|\left(\Delta G\left(u\right){\right)}_{{Q}_{i}}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{i+1}}{|}^{s}{\chi }_{Q}\left(x\right)\le {C}_{11}\sum _{j=i}^{i+1}{\chi }_{N{Q}_{j}}\left(x\right)|{Q}_{j}{|}^{-1}\underset{\sigma {Q}_{j}}{\int }|u{|}^{s}\text{d}{\mu }_{1}\\ \le {C}_{12}\sum _{j=i}^{i+1}{\chi }_{N{Q}_{j}}\left(x\right)\frac{1}{{d}_{0}^{n}}\underset{\sigma {Q}_{j}}{\int }|u{|}^{s}\text{d}{\mu }_{1}\\ \le {C}_{13}\sum _{j=i}^{i+1}{\chi }_{N{Q}_{j}}\left(x\right)\underset{\sigma {Q}_{j}}{\int }|u{|}^{s}\text{d}{\mu }_{1}.\end{array}$
(2.19)

Using (a + b)1/s≤ 21/s(|a|1/s+ |b|1/s), (2.16) and (2.19), we obtain

$|\left(\Delta G\left(u{\right)\right)}_{Q}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}|{\chi }_{Q}\left(x\right)\le {C}_{14}\sum _{{D}_{i}\in \mathcal{V}}{\left(\underset{\sigma {D}_{i}}{\int }|u{|}^{s}\text{d}{\mu }_{1}\right)}^{1/s}\cdot {\chi }_{N{D}_{i}}\left(x\right)$

for every x n . Then

$\sum _{Q\in \mathcal{V}}\underset{Q}{\int }|\left(\Delta G\left(u{\right)\right)}_{Q}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{|}^{s}\text{d}\mu \le {C}_{14}\underset{{}^{{ℝ}^{n}}}{\int }|\sum _{{D}_{i}\in \mathcal{V}}{\left(\underset{\sigma {D}_{i}}{\int }|u{|}^{s}\text{d}{\mu }_{1}\right)}^{1/s}{\chi }_{N{D}_{i}}\left(x{\right)|}^{s}\text{d}\mu .$

Notice that

$\sum _{{D}_{i}\in \mathcal{V}}{\chi }_{N{D}_{i}}\left(x\right)\le \sum _{{D}_{i}\in \mathcal{V}}{\chi }_{\sigma N{D}_{i}}\left(x\right)\le N{\chi }_{\Omega }\left(x\right).$

Using elementary inequality $\mid {\sum }_{i=1}^{M}{t}_{i}{\mid }^{s}\le {M}^{s-1}{\sum }_{i=1}^{M}\mid {t}_{i}{\mid }^{s}$ for s > 1, we finally have

$\begin{array}{c}\sum _{Q\in \mathcal{V}}\underset{Q}{\int }|\left(\Delta G\left(u{\right)\right)}_{Q}-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{|}^{s}\text{d}\mu \le {C}_{15}\underset{{ℝ}^{n}}{\int }\left(\sum _{{D}_{i}\in \mathcal{V}}\left(\underset{\sigma {D}_{i}}{\int }|u{|}^{s}\text{d}{\mu }_{1}\right){\chi }_{N{D}_{i}}\left(x\right)\right)\text{d}\mu \\ ={C}_{15}\sum _{{D}_{i}\in \mathcal{V}}\left(\underset{\sigma {D}_{i}}{\int }|u{|}^{s}\text{d}{\mu }_{1}\right)\\ \le {C}_{16}\underset{\Omega }{\int }|u{|}^{s}g\left(x\right)\text{d}x.\end{array}$
(2.20)

Substituting (2.15) and (2.20) in (2.14), we have completed the proof of Theorem 2.6.

Using Lemma 2.2, we obtain

$\begin{array}{c}\parallel \nabla \left(\Delta G\left(u\right){\parallel }_{s,B}=\parallel \text{d}\left(\Delta G\left(u\right)\right){\parallel }_{s,B}\\ =\parallel \Delta G\left(du\right){\parallel }_{s,B}\\ =\parallel \left(d{d}^{*}+{d}^{*}d\right)\left(G\left(du\right)\right){\parallel }_{s,B}\\ \le \parallel d{d}^{*}\left(G\left(du\right)\right){\parallel }_{s,B}+\parallel {d}^{*}\text{d}\left(G\left(du\right)\right){\parallel }_{s,B}\\ \le {C}_{1}\parallel du{\parallel }_{s,B}+{C}_{2}\parallel du{\parallel }_{s,B}\\ \le {C}_{3}\parallel du{\parallel }_{s,B}\\ \le {C}_{4}{\left(diam\left(B\right)\right)}^{-1}||u|{|}_{s,\sigma B}\\ \le {C}_{5}\parallel u{\parallel }_{s,\sigma B},\end{array}$
(2.21)

where σ > 1 is a constant. Using (2.21), we have the following Lemma 2.7 whose proof is similar to the proof of Lemma 2.5. □

Lemma 2.7 Let$u\in {L}_{loc}^{s}\left(M,{\Lambda }^{l}\right)$, l = 1, 2,..., n, 1 < s < ∞, be a solution of the non-homogeneous A-harmonic equation in a bounded and convex domain M, G be the Green's operator and Δ be the Laplace-Beltrami operator. Then, there exists a constant C independent of u, such that

${\left(\underset{B}{\int }|\nabla \left(\Delta G\left(u{\right)\right)|}^{s}\frac{1}{\text{d}{\left(x,\partial M\right)}^{\alpha }}\text{d}x\right)}^{1/s}\le C{\left(\underset{\rho B}{\int }|u{|}^{s}\frac{1}{|x-{x}_{B}{|}^{\lambda }}\text{d}x\right)}^{1/s}$
(2.22)

for all balls B with ρB M and diam(B) ≥ d0> 0, where d0is a constant, ρ > 1, any real number α and λ with α > λ ≥ 0. Here, x B is the center of the ball.

Notice that (2.22) can also be written as

$\parallel \nabla \left(\Delta G\left(u\right)\right){\parallel }_{s,B,{\omega }_{1}}\le C\parallel u{\parallel }_{s,\rho B,{\omega }_{2}.}$
(2.22a)

Next, we prove the imbedding inequality with a singular factor in the John domain.

Theorem 2.8 Let u D'(Ω, Λ l ) be a solution of the A-harmonic equation (1.1), G be the Green's operator and Δ be the Laplace-Beltrami operator. Assume that s is a fixed exponent associated with the non-homogeneous A-harmonic equation. Then, there exists a constant C, independent of u, such that

$\parallel \nabla \left(\Delta G\left(u\right)\right){\parallel }_{s,\Omega ,{\omega }_{1}}\le C\parallel u{\parallel }_{s,\Omega ,{\omega }_{2}},$
(2.23)
$\parallel \Delta G\left(u\right){\parallel }_{{W}^{1,s}\left(\Omega \right),{\omega }_{1}}\le C\parallel u{\parallel }_{s,\Omega ,{\omega }_{2}}$
(2.24)

for any bounded and convex δ-John domain Ω n. Here, the weights are defined by${\omega }_{1}\left(x\right)=\frac{1}{d{\left(x,\partial \Omega \right)}^{\alpha }}$and${\omega }_{2}\left(x\right)={\sum }_{i}{\chi }_{{Q}_{i}}\frac{1}{\mid x-{x}_{{Q}_{i}}{\mid }^{\lambda }}$, respectively, α and λ are constants with 0 ≤ λ < α.

Proof Applying the Covering Lemma 2.3 and Lemma 2.7, we have (2.23) immediately. For inequality (2.24), using Lemma 2.5 and the Covering Lemma 2.3, we have

$\parallel \Delta G\left(u\right){\parallel }_{s,\Omega ,{\omega }_{1}}\le {C}_{1}\parallel u{\parallel }_{s,\Omega ,{\omega }_{2}}.$
(2.25)

By the definition of the $\parallel \cdot {\parallel }_{{W}^{1,s}\left(\Omega \right),{\omega }_{1}}$ norm, we know that

(2.26)

Substituting (2.23) and (2.25) into (2.26) yields

$\parallel \Delta G\left(u\right){\parallel }_{{W}^{1,s}\left(\Omega \right),{\omega }_{1}}\le {C}_{2}\parallel u{\parallel }_{s,\Omega ,{\omega }_{2}}.$

We have completed the proof of the Theorem 2.8. □

Theorem 2.9 Let u D'(Ω, Λ l ) be a solution of the A-harmonic equation (1.1), G be the Green's operator and Δ be the Laplace-Beltrami operator. Assume that s is a fixed exponent associated with the non-homogeneous A-harmonic equation. Then, there exists a constant C, independent of u, such that

$\parallel \Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{\parallel }_{{W}^{1,s}\left(\Omega \right),{\omega }_{1}}\le C\parallel u{\parallel }_{s,\Omega ,{\omega }_{2}}$
(2.27)

for any bounded and convex δ-John domain Ω n. Here the weights are defined by${\omega }_{1}\left(x\right)=\frac{1}{d{\left(x,\partial \Omega \right)}^{\alpha }}$and${\omega }_{2}\left(x\right)={\sum }_{i}{\chi }_{{Q}_{i}}\frac{1}{\mid x-{x}_{{Q}_{i}}{\mid }^{\lambda }}$, α and λ are constants with 0 ≤ λ < α, and the fixed cube Q0 Ω and the constant N > 1 appeared in Lemma 2.3.

Proof Since ${\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}$ is a closed form, $\nabla \left(\left(\Delta G\left(u{\right)\right)}_{{Q}_{0}}\right)=\text{d}\left(\left(\Delta G\left(u{\right)\right)}_{{Q}_{0}}\right)=0$. Thus, by using Theorem 2.6 and (2.23), we have

$\begin{array}{c}\parallel \Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{\parallel }_{{W}^{1,s}\left(\Omega \right),{\omega }_{1}}\\ =diam{\left(\Omega \right)}^{-1}\parallel \Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{\parallel }_{s,\Omega ,{\omega }_{1}}+\parallel \nabla \left(\Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}\right){\parallel }_{s,\Omega ,{\omega }_{1}}\\ =diam{\left(\Omega \right)}^{-1}\parallel \Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{\parallel }_{s,\Omega ,{\omega }_{1}}+\parallel \nabla \left(\Delta G\left(u\right)\right){\parallel }_{s,\Omega ,{\omega }_{1}}\\ \le {C}_{1}\parallel u{\parallel }_{s,\Omega ,{\omega }_{2}}+{C}_{2}\parallel u{\parallel }_{s,\Omega ,{\omega }_{2}}\\ \le {C}_{3}\parallel u{\parallel }_{s,\Omega ,{\omega }_{2}}.\end{array}$

Thus, (2.27) holds. The proof of Theorem 2.9 has been completed. □

As applications of our main results, we consider the following example.

Example 1 Let B = 0, A(x, ξ) = ξ|ξ|p-2, p > 1, and u be a function(0-form) in (1.1). Then, the operator A satisfies the required conditions and the non-homogeneous A-harmonic equation(1.1) reduces to the usual p-harmonic equation

$div\left(\nabla u\mid \nabla u{\mid }^{p-2}\right)=0$
(2.28)

which is equivalent to

$\left(p-2\right)\sum _{k=1}^{n}\sum _{i=1}^{n}{u}_{{x}_{k}}{u}_{{x}_{i}}{u}_{{x}_{k}{x}_{i}}+\mid \nabla u{\mid }^{2}\Delta u=0.$
(2.29)

If we choose p = 2 in (2.28), we have Laplace equation Δu = 0 for functions. Hence, the Equations (2.28), (2.29) and the Δu = 0 are the special cases of the non-homogeneous A-harmonic equation (1.1). Therefore, all results proved in Theorem 2.6, 2.8, and 2.9 are still true for u that satisfies one of the above three equations.

Example 2 Let f : Ω → n , f = (f1,..., fn ), be a mapping of the Sobolev class ${W}_{loc}^{1,p}\left(\Omega ,{ℝ}^{n}\right)$, 1 < p < ∞, whose distributional differential Df = [∂fi /∂x j ] : Ω → GL(n) is a locally integrable function in Ω with values in the space GL(n) of all n × n-matrices, i, j = 1, 2,..., n. we use

$J\left(x,f\right)=\mathrm{det}Df\left(x\right)=|\begin{array}{ccccc}{f}_{{x}_{1}}^{1}& {f}_{{x}_{2}}^{1}& {f}_{{x}_{3}}^{1}& \cdots & {f}_{{x}_{n}}^{1}\\ {f}_{{x}_{1}}^{2}& {f}_{{x}_{2}}^{2}& {f}_{{x}_{3}}^{2}& \cdots & {f}_{{x}_{n}}^{2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {f}_{{x}_{1}}^{n}& {f}_{{x}_{2}}^{n}& {f}_{{x}_{3}}^{n}& \cdots & {f}_{{x}_{n}}^{n}\end{array}|$

to denote the Jacobian determinant of f. A homeomorphism f : Ω → n of the Sobolev class ${W}_{loc}^{1,n}\left(\Omega ,{ℝ}^{n}\right)$ is said to be K-quasiconformal, 1 ≤ K < ∞, if its differential matrix Df(x) and the Jacobian determinant J(x, f) satisfy

$\mid Df\left(x\right){\mid }^{n}\le KJ\left(x,f\right),$
(2.30)

where |Df(x)| = max |Df(x)h| : |h| = 1 denotes the norm of the Jacobi matrix Df(x). It is well known that if the differential matrix Df(x) = [∂fi / ∂x j ], i, j = 1, 2,..., n, of a homeomorphism f(x) = (f1, f2,..., fn ) : Ω → n satisfies (2.30), then, each of the functions

$u={f}^{i}\left(x\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}i=1,2,...,n,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}or\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}u=log\mid f\left(x\right)\mid ,$
(2.31)

is a generalized solution of the quasilinear elliptic equation

$div\phantom{\rule{0.3em}{0ex}}A\left(x,\nabla u\right)=0,$
(2.32)

in Ω - f-1(0), where

$A=\left({A}_{1},{A}_{2},\dots ,{A}_{n}\right),A\left(x,\xi \right)=\frac{\partial }{\partial {\xi }_{i}}{\left(\sum _{i,j=1}^{n}{\theta }_{i,j}\left(x\right){\xi }_{i}{\xi }_{j}\right)}^{n∕2}$

and θi,jare some functions, which can be expressed in terms of the differential matrix Df(x) and satisfy

${C}_{1}\left(K\right)\mid \xi {\mid }^{2}\le \sum _{i,j=1}^{n}{\theta }_{i,j}\left(x\right){\xi }_{i}{\xi }_{j}\le {C}_{2}\left(K\right)\mid \xi {\mid }^{2}$
(2.33)

for some constants C1(K), C2(K) > 0. Choosing u is defined in (2.31) and applying Theorems (2.6), (2.8) and (2.9) to u, respectively, we have the following theorems.

Theorem 3.0 Let u = fi (x) or u = log |f(x)| D'(Ω, Λ l ), i = 1, 2,..., n, be a solution of the quasilinear elliptic equation (2.32), where f : Ω → n , f = (f1,..., fn ) be a K-quasiconformal mapping of the Sobolev class${W}_{loc}^{1,p}\left(\Omega ,{ℝ}^{n}\right)$, 1 < p < ∞, G be the Green's operator and Δ be the Laplace-Beltrami operator. Assume that s is a fixed exponent associated with the non-homogeneous A-harmonic equation. Then, there exists a constant C, independent of u, such that

${\left(\underset{\Omega }{\int }\mid \Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{\mid }^{s}\frac{1}{d{\left(x,\partial \Omega \right)}^{\alpha }}dx\right)}^{1∕s}\le C{\left(\underset{\Omega }{\int }\mid u{\mid }^{s}g\left(x\right)dx\right)}^{1∕s}$
(2.34)

for any bounded and convex δ-John domain Ω n, where$g\left(x\right)={\sum }_{i}{\chi }_{{Q}_{i}}\frac{1}{\mid x-{x}_{{Q}_{i}}{\mid }^{\lambda }}$, ${x}_{{Q}_{i}}$is the center of Q i with Ω = i Q i . Here α and λ are constants with 0 ≤ λ < α < n, and the fixed cube Q0 Ω, the constant N > 1 and the cubes Q i Ω appeared in Lemma 2.3, ${x}_{{Q}_{i}}$is the center of Q i .

Theorem 3.1 Let u = fi(x) or u = log |f(x)| D'(Ω, Λ l ), i = 1, 2,..., n, be a solution of the quasilinear elliptic equation (2.32), where f : Ω → n, f = (f1,..., fn) be a K-quasiconformal mapping of the Sobolev class${W}_{loc}^{1,p}\left(\Omega ,{ℝ}^{n}\right)$, 1 < p < ∞, G be the Green's operator and Δ be the Laplace-Beltrami operator. Assume that s is a fixed exponent associated with the non-homogeneous A-harmonic equation. Then, there exists a constant C, independent of u, such that

$\parallel \nabla \left(\Delta G\left(u\right)\right){\parallel }_{s,\Omega ,{\omega }_{1}}\le C\parallel u{\parallel }_{s,\Omega ,{\omega }_{2}},$
(2.35)
$\parallel \Delta G\left(u\right){\parallel }_{{W}^{1,s}\left(\Omega \right),{\omega }_{1}}\le C\parallel u{\parallel }_{s,\Omega ,{\omega }_{2}}$
(2.36)

for any bounded and convex δ-John domain Ω n. Here, the weights are defined by${\omega }_{1}\left(x\right)=\frac{1}{d{\left(x,\partial \Omega \right)}^{\alpha }}$and${\omega }_{2}\left(x\right)={\sum }_{i}{\chi }_{{Q}_{i}}\frac{1}{\mid x-{x}_{{Q}_{i}}{\mid }^{\lambda }}$, respectively, α and λ are constants with 0 ≤ λ < α < n.

Theorem 3.2 Let u = f i (x) or u = log |f(x)| D'(Ω, Λ l ), i = 1, 2,..., n, be a solution of the quasilinear elliptic equation (2.32), where f : Ω → n , f = (f1,..., fn ) be a K-quasiconformal mapping of the Sobolev class${W}_{loc}^{1,p}\left(\Omega ,{ℝ}^{n}\right)$, 1 < p < ∞, G be the Green's operator and Δ be the Laplace-Beltrami operator. Assume that s is a fixed exponent associated with the non-homogeneous A-harmonic equation. Then, there exists a constant C, independent of u, such that

$\parallel \Delta G\left(u\right)-{\left(\Delta G\left(u\right)\right)}_{{Q}_{0}}{\parallel }_{{W}^{1,s}\left(\Omega \right),{\omega }_{1}}\le C\parallel u{\parallel }_{s,\Omega ,{\omega }_{2}}$
(2.37)

for any bounded and convex δ-John domain Ω n. Here, the weights are defined by${\omega }_{1}\left(x\right)=\frac{1}{d{\left(x,\partial \Omega \right)}^{\alpha }}$and${\omega }_{2}\left(x\right)={\sum }_{i}{\chi }_{{Q}_{i}}\frac{1}{\mid x-{x}_{{Q}_{i}}{\mid }^{\lambda }}$, α and λ are constants with 0 ≤ λ < α < n, and the fixed cube Q0 Ω and the constant N > 1 appeared in Lemma 2.3.

Our results can be applied to all differential forms or functions satisfying some version of the A-harmonic equation, the usual p-harmonic equation or the Laplace equation [1, 9, 10, for more applications].

## References

1. Agarwal RP, Ding S, Nolder CA: Inequalities for differential forms. Springer, New York; 2009.

2. Ding S, Liu B: Singular integral of the composite operators. Appl Math Lett 2009, 22: 1271–1275. 10.1016/j.aml.2009.01.041

3. Scott C: Lp - theory of differential forms on manifolds. Trans Am Soc 1995, 347: 2075–2096. 10.2307/2154923

4. Cartan H: Differential forms. Houghton Mifflin Co, Boston; 1970.

5. Warner FW: Foundations of differentiable manifolds and Lie groups. Springer, New York; 1983.

6. Xing Y: Weighted Poincaré-type estimates for conjugate A-harmonic tensors. J Inequal Appl 2005, 1: 1–6.

7. Ding S: Integral estimates for the Laplace-Beltrami and Green's operators applied to differential forms on manifolds. J Inequal Appl 2003,22(4):939–957.

8. Ding S: Two-weight caccioppoli inequalities for solutions of nonhomogeneous A-harmonic equations on Riemannian manifolds. Proc Am Math Soc 2004, 132: 2367–2375. 10.1090/S0002-9939-04-07347-2

9. Westenholz C: Differential forms in mathematical physics. North-Holland Publishing, Amsterdam; 1978.

10. Sachs SK, Wu H: General relativity for mathematicians. Springer, New York; 1977.

11. Xing Y: Two-weight imbedding inequalities for solutions to the A-harmonic equation. J Math Anal Appl 2005, 307: 555–564. 10.1016/j.jmaa.2005.03.019

12. Ding S, Nolder CA: Weighted Poincaré-type inequalities for solutions to the A-harmonic equation. Ill. J Math 2002, 2: 199–205.

13. Liu B: ${A}_{r}^{\lambda }\left(\Omega \right)$- weighted imbedding inequalities for A-harmonic tensions. J Math Anal Appl 2002, 273: 667–676. 10.1016/S0022-247X(02)00331-1

14. Wang Y, Wu C: Sobolev imbedding theorems and Poincaré inequalities for Green's operator on solutions of the nonhomogeneous A-harmonic equation. Comput Math Appl 2004, 47: 1545–1554. 10.1016/j.camwa.2004.06.006

15. Xing Y, Wu C: Global weighted inequalities for operators and harmonic forms on manifolds. J Math Anal Appl 2004, 294: 294–309. 10.1016/j.jmaa.2004.02.018

16. Xing Y: Weighted integral inequalities for solutions of the A-harmonic equation. J Math Anal Appl 2003, 279: 350–363. 10.1016/S0022-247X(03)00036-2

17. Ding S: Lφ ( μ ) averaging domains and the quasihyperbolic metric. Comput Math Appl 2004, 47: 1611–1618. 10.1016/j.camwa.2004.06.016

18. Nolder CA: Hardy-Littlewood theorems for A-harmonic tensors. Ill. J Math 1999, 43: 613–631.

## Author information

Authors

### Corresponding author

Correspondence to Ru Fang.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

RF and SD jointly contributed to the main results and RF drafted the manuscript. All authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

Fang, R., Ding, S. Singular integrals of the compositions of Laplace-Beltrami and Green's operators. J Inequal Appl 2011, 74 (2011). https://doi.org/10.1186/1029-242X-2011-74

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1029-242X-2011-74

### Keywords

• Poincaré-type inequalities
• differential forms
• A-harmonic equations
• the Laplace-Beltrami operator
• Green's operator 