A Generalization of Exponential Class and its Applications

TL;DR

This paper introduces a new function space generalizing exponential classes, proves its Banach space properties, and applies it to analyze weak solutions of $a0\mathcal{A}$-harmonic equations, including boundedness of key operators.

Contribution

It defines the space $L^{\theta,\infty)}(\Omega)$, proves its Banach space structure, and applies it to extend results on weak solutions and operator boundedness.

Findings

$L^{\theta,\infty)}(\Omega)$ is a Banach space.

Weak monotonicity property for $a0\mathcal{A}$-harmonic solutions is established.

Boundedness of Hardy-Littlewood maximal and Calderf3n-Zygmund operators on the weighted space.

Abstract

A function space, $L^{\theta,\infty)}(\Omega)$, $0 \leq \theta <\infty$, is defined. It is proved that $L^{\theta,\infty)}(\Omega)$ is a Banach space which is a generalization of exponential class. An alternative definition of $L^{\theta,\infty)}(\Omega)$ space is given. As an application, we obtain weak monotonicity property for very weak solutions of $\mathcal{A}$-harmonic equation with variable coefficients under some suitable conditions related to $L^{\theta,\infty)}(\Omega)$, which provides a generalization of a known result due to Moscariello. A weighted space $L^{\theta,\infty)}_w(\Omega)$) is also defined, and the boundedness for the Hardy-Littlewood maximal operator $M_w$ and a Calder\'{o}n-Zygmund operator $T$ with respect to $L^{\theta,\infty)}_w(\Omega)$ are obtained.

Equations

$$\Vert f{\Vert }_{p),\Omega }=\underset{0<\epsilon \le p-1}{\sup }{\left(\epsilon -{\int }_{\Omega }|f{|}^{p-\epsilon }dx\right)}^{\frac{1}{p-\epsilon }}<\infty ,$$

$$\Vert u{\Vert }_{W_0^{1,p)}}=\underset{0<\epsilon \le p-1}{\sup }{\left(\epsilon -{\int }_{\Omega }|\nabla f{|}^{p-\epsilon }dx\right)}^{\frac{1}{p-\epsilon }}<\infty .$$

$$\Vert f{\Vert }_{L^{(p}}\approx {\int }_0^1(1-\ln t{)}^{-\frac{1}{p}}{\left({\int }_0^t[f^{\ast }(s){]}^{p}ds\right)}^{\frac{1}{p}}\frac{dt}{t},1<p<\infty ,$$

$$\Vert f{\Vert }_{L^{p)}}\approx \underset{0<t<1}{\sup }(1-\ln t{)}^{-\frac{1}{p}}{\left({\int }_t^1[f^{\ast }(s){]}^{p}ds\right)}^{\frac{1}{p}},1<p<\infty .$$

$$\Vert f{\Vert }_{\theta ,p)}=\underset{0<\epsilon \le p-1}{\sup }{\epsilon }^{\frac{\theta }{p}}\Vert f{\Vert }_{p-\epsilon }<\infty ,$$

$$\Vert f{\Vert }_{p-\epsilon }={\left(-{\int }_{\Omega }|f{|}^{p-\epsilon }dx\right)}^{\frac{1}{p-\epsilon }}.$$

$$W^{\theta ,p)}(\Omega )=\left\{f\in \bigcap _{0<\epsilon \le p-1}{W}^{1,p-\epsilon }(\Omega ):\nabla f\in L^{\theta ,p)}(\Omega )\right\}.$$

$${\int }_{\Omega }{e}^{\lambda |f|}dx<\infty$$

$$\Vert f{\Vert }_{EXP}=\inf \left\{\lambda >0:{\int }_{\Omega }{e}^{|f|/\lambda }dx\le 2\right\}.$$

$$L^{\theta ,\infty )}(\Omega )=\left\{f(x)\in \bigcap _{1\le p<\infty }{L}^p(\Omega ):\underset{1\le p<\infty }{\sup }\frac{1}{p^{\theta }}{\left(-{\int }_{\Omega }|f(x){|}^{p}dx\right)}^{\frac{1}{p}}<\infty \right\}.$$

$$L^{\theta ,\infty )}(\Omega )=\left\{g(x)\in \bigcap _{1\le p<\infty }{L}^p(\Omega ):{\mathrm{limsup}}_{p\to \infty }\frac{1}{p^{\theta }}{\left(-{\int }_{\Omega }|g(x){|}^{p}dx\right)}^{\frac{1}{p}}<\infty \right\}.$$

$$L^{0,\infty )}(\Omega )=\left\{f(x)\in \bigcap _{1\le p<\infty }{L}^p(\Omega ):\underset{1\le p<\infty }{\sup }{\left(-{\int }_{\Omega }|f(x){|}^{p}dx\right)}^{\frac{1}{p}}<\infty \right\}.$$

$$L^{\infty }(\Omega )=\left\{f\in \bigcap _{1\le p<\infty }{L}^p(\Omega ):\underset{p\to \infty }{\lim }\Vert f{\Vert }_p<\infty \right\},$$

$$L^{\infty }(\Omega )\subset L^{\theta ,\infty )}(\Omega )\subset L^{{\theta }^{\prime },\infty )}(\Omega )\subset L^q(\Omega ).$$

$$\underset{1\le p<\infty }{\sup }\frac{1}{p^{\theta }}{\left(-{\int }_{\Omega }|f(x){|}^{p}dx\right)}^{\frac{1}{p}}\le \underset{1\le p<\infty }{\sup }\frac{M}{p^{\theta }}=M<\infty ,$$

$$\begin{array}[]{llll}\displaystyle\int_{0}^{1}(-\ln x)^{m}dx&=&\displaystyle\left.x(-\ln x)^{m}\right|_{0}^{1}-\int_{0}^{1}xd(-\ln x)^{m}\\ &=&\displaystyle-\lim_{x\rightarrow 0^{+}}x(-\ln x)^{m}+m\int_{0}^{1}(-\ln x)^{m-1}dx.\end{array}$$

$$\underset{x\to 0^{+}}{\lim }x(-\ln x{)}^m=\underset{x\to 0^{+}}{\lim }\frac{(-\ln x{)}^m}{\frac{1}{x}}=\underset{x\to 0^{+}}{\lim }\frac{m(-\ln x{)}^{m-1}}{\frac{1}{x}}=\cdots =m!\underset{x\to 0^{+}}{\lim }x=0.$$

$${\int }_0^1(-\ln x{)}^{m}dx=m{\int }_0^1(-\ln x{)}^{m-1}dx.$$

$${\int }_0^1{f}^m(x)dx=m{\int }_0^1(-\ln x{)}^{m-1}dx=\cdots =m!{\int }_0^{1}dx=m!.$$

$$p\mapsto {\left(-{\int }_{\Omega }|f(x){|}^{p}dx\right)}^{\frac{1}{p}}$$

$$\begin{array}[]{llll}&\displaystyle\sup_{1\leq p<\infty}\frac{1}{p^{\theta}}\left(-\int_{0}^{1}|f(x)|^{p}dx\right)^{\frac{1}{p}}\\ =&\displaystyle\sup_{1\leq p<\infty}\left[\frac{1}{p}\left(\int_{0}^{1}(-\ln x)^{p\theta}dx\right)^{\frac{1}{p\theta}}\right]^{\theta}\\ \leq&\displaystyle\sup_{1\leq p<\infty}\left[\frac{1}{p}\left(\int_{0}^{1}(-\ln x)^{[p\theta]+1}dx\right)^{\frac{1}{[p\theta]+1}}\right]^{\theta}\\ =&\displaystyle\sup_{1\leq p<\infty}\left[\frac{([p\theta]+1)!^{\frac{1}{[p\theta]+1}}}{p}\right]^{\theta}\leq\sup_{1\leq p<\infty}\left[\frac{[p\theta]+1}{p}\right]^{\theta}\leq 2,\end{array}$$

$$\Vert f{\Vert }_{\theta ,\infty ),\Omega }=\underset{1\le p<\infty }{\sup }\frac{1}{p^{\theta }}{\left(-{\int }_{\Omega }|f(x){|}^{p}dx\right)}^{\frac{1}{p}}.$$

$$\begin{array}[]{llll}\|f_{1}+f_{2}\|_{\theta,\infty)}&=&\displaystyle\sup_{1\leq p<\infty}\frac{1}{p^{\theta}}\left(-\int_{\Omega}|f_{1}+f_{2}|^{p}dx\right)^{\frac{1}{p}}\\ &\leq&\displaystyle\sup_{1\leq p<\infty}\frac{1}{p^{\theta}}\left[\left(-\int_{\Omega}|f_{1}|^{p}dx\right)^{\frac{1}{p}}+\left(-\int_{\Omega}|f_{2}|^{p}dx\right)^{\frac{1}{p}}\right]\\ &\leq&\displaystyle\sup_{1\leq p<\infty}\frac{1}{p^{\theta}}\left(-\int_{\Omega}|f_{1}|^{p}dx\right)^{\frac{1}{p}}+\sup_{1\leq p<\infty}\frac{1}{p^{\theta}}\left(-\int_{\Omega}|f_{2}|^{p}dx\right)^{\frac{1}{p}}\\ &=&\displaystyle\|f_{1}\|_{\theta,\infty)}+\|f_{2}\|_{\theta,\infty)};\end{array}$$

$$\Vert f_{n+p}-f_n{\Vert }_{\theta ,\infty )}\to 0,n\to \infty .$$

$${\int }_{{\Omega }_m}|f_{n+p}(x)-f_n(x)|dx\to 0,n\to \infty .$$

$$f_n(x)\to f^{(m)}(x),n\to \infty ,\text{mbox}in{L}^1({\Omega }_m).$$

$$f_n^{(m)}(x)\to f^{(m)}(x),n\to \infty ,\text{mbox}a.e.x\in {\Omega }_m.$$

$$f(x)=f^{(m)}(x),x\in {\Omega }_m,m=1,2,\cdots ,$$

$$f_n^{(n)}(x)\to f(x),n\to \infty ,\text{mbox}a.e.x\in \Omega .$$

$$f_n(x)\to f(x),n\to \infty ,\text{mbox}a.e.x\in \Omega .$$

Full text

A Generalization of Exponential Class and Its ApplicationsCorresponding author: GAO Hongya, E-mail: ghy@hbu.cn, TEL: 863125079658, FAX: 863125079638.

GAO Hongya LIU Chao  TIAN Hong

College of Mathematics and Computer Science, Hebei University, Baoding, 071002, China

Abstract.  A function space, $L^{\theta,\infty)}(\Omega)$, $0\leq\theta<\infty$, is defined. It is proved that $L^{\theta,\infty)}(\Omega)$ is a Banach space which is a generalization of exponential class. An alternative definition of $L^{\theta,\infty)}(\Omega)$ space is given. As an application, we obtain weak monotonicity property for very weak solutions of $\cal A$-harmonic equation with variable coefficients under some suitable conditions related to $L^{\theta,\infty)}(\Omega)$, which provides a generalization of a known result due to Moscariello. A weighted space $L^{\theta,\infty)}_{w}(\Omega)$ is also defined, and the boundedness for the Hardy-Littlewood maximal operator $M_{w}$ and a Calderón-Zygmund operator $T$ with respect to $L^{\theta,\infty)}_{w}(\Omega)$ are obtained.

**AMS Subject Classification: ** 46E30, 35J70.

Keywords: Exponential class, weak monotonicity, very weak solution, $\cal A$-harmonic equation, Hardy-Littlewood maximal operator, Calderón-Zygmund operator.

§1   Introduction

For $1<p<\infty$ and a bounded open subset $\Omega\subset\mbox{R}^{n}$, the grand Lebesgue space $L^{p)}(\Omega)$ consists of all functions $f(x)\in\bigcap_{0<\varepsilon\leq p-1}L^{p-\varepsilon}(\Omega)$ such that

[TABLE]

where $-\int_{\Omega}=\frac{1}{|\Omega|}\int_{\Omega}$ stands for the integral mean over $\Omega$. The grand Sobolev space $W_{0}^{1,p)}(\Omega)$ consists of all functions $u\in\bigcap_{0<\varepsilon\leq p-1}W_{0}^{1,p-\varepsilon}(\Omega)$ such that

[TABLE]

These two spaces, slightly larger than $L^{p}(\Omega)$ and $W_{0}^{1,p}(\Omega)$, respectively, were introduced in the paper [1] by Iwaniec and Sbordone in 1992 where they studied the integrability of the Jacobian under minimal hypotheses. For $p=n$ in [2] imbedding theorems of Sobolev type were proved for functions $f\in W_{0}^{1,n)}(\Omega)$. The small Lebesgue space $L^{(p}(\Omega)$ was found by Fiorenza [3] in 2000 as the associate space of the grand Lebesgue space $L^{p)}(\Omega)$. Fiorenza and Karadzhov gave in [4] the following equivalent, explicit expressions for the norms of the small and grand Lebesgue spaces, which depend only on the non-decreasing rearrangement (provided that the underlying measure space has measure 1):

[TABLE]

[TABLE]

In [5], Greco, Iwaniec and Sbordone gave two more general definitions than (1.1) and (1.2) in order to derive existence and uniqueness results for $p$-harmonic operators. For $1<p<\infty$ and $0\leq\theta<\infty$, the grand $L^{p}$ space, denoted by $L^{\theta,p)}(\Omega)$, consists of functions $f\in\bigcap_{0<\varepsilon\leq p-1}L^{p-\varepsilon}(\Omega)$ such that

[TABLE]

where

[TABLE]

The grand Sobolev space $W^{\theta,p)}(\Omega)$ consists of all functions $f$ belonging to $\bigcap_{0<\varepsilon\leq p-1}W^{1,p-\varepsilon}(\Omega)$ and such that $\nabla f\in L^{\theta,p)}(\Omega)$. That is,

[TABLE]

Grand and small Lebesgue spaces are important tools in dealing with regularity properties for very weak solutions of $\cal A$-harmonic equation as well as weakly quasiregular mappings, see [6, 7].

The aim of the present paper is to provide a generalization $L^{\theta,\infty)}(\Omega)$, $0\leq\theta<\infty$, of exponential calss $EXP(\Omega)$, and prove that it is a Banach space. An alternative definition of $L^{\theta,\infty)}(\Omega)$ is given in terms of weak Lebesgue spaces. As an application, we obtain weak monotonicity property for very weak solutions of $\cal A$-harmonic equation with variable coefficients under some suitable conditions related to $L^{\theta,\infty)}(\Omega)$. This paper also consider a weighted space $L_{w}^{\theta,\infty)}(\Omega)$, and some boundedness result for classical operators with respect to this space.

In the sequel, the letter $C$ is used for various constants, and may change from one occurrence to another.

§2   A Generalization of Exponential Class

Recall that $EXP(\Omega)$, the exponential class, consists of all measurable functions $f$ such that

[TABLE]

for some $\lambda>0$. It is a Banach space under the norm

[TABLE]

In this section, we define a space $L^{\theta,\infty)}(\Omega)$, $0\leq\theta<\infty$, which is a generalization of $EXP(\Omega)$, and prove that it is a Banach space.

Definition 2.1**.**

For $\theta\geq 0$, the space $L^{\theta,\infty)}(\Omega)$ is defined by

[TABLE]

It is not difficult to see that

[TABLE]

There are two special cases of $L^{\theta,\infty)}(\Omega)$ that are worth mentioning since they coincide with two known spaces.

Case 1: $\theta=0$. In this case,

[TABLE]

From the fact (see [8, P12])

[TABLE]

we get $L^{0,\infty)}(\Omega)=L^{\infty}(\Omega)$.

Case 2: $\theta=1$. The following proposition shows that $L^{\theta,\infty}(\Omega)$ can be regarded as a generalization of $EXP(\Omega)$.

**Proposition 2.1 ** $L^{1,\infty)}(\Omega)=EXP(\Omega)$.

Proof.

In order to realize that a function in the $L^{1,\infty)}(\Omega)$ space is in $EXP(\Omega)$, it is sufficient to read the last lines of [2]. The vice-versa is also true, see e.g. [9, Chap. VI, exercise no. 17]. ∎

It is clear that for any $0\leq\theta<\theta^{\prime}\leq\infty$ and any $q<\infty$, we have the inclusions

[TABLE]

The following theorem shows that, if $\theta>0$, then $L^{\theta,\infty)}(\Omega)$ is slightly larger than $L^{\infty}(\Omega)$.

Theorem 2.1**.**

For $\theta>0$, the space $L^{\infty}(\Omega)$ is a proper subspace of $L^{\theta,\infty)}(\Omega)$.

Proof.

In the proof of Theorem 2.1 we always assume $\theta>0$. Let $f(x)\in L^{\infty}(\Omega)$, then there exists a constant $M<\infty$, such that $|f(x)|\leq M$, a.e. $\Omega$. Thus,

[TABLE]

which implies $f(x)\in L^{\theta,\infty)}(\Omega)$.

The following example shows that $L^{\infty}(\Omega)\subset L^{\theta,\infty)}(\Omega)$ is a proper subset. Since we have the inclusion (2.2), then it is no loss of generality to assume that $\theta\leq 1$. Consider the function $f(x)=(-\ln x)^{\theta}$ defined in the open interval $(0,1)$. It is obvious that $f(x)\notin L^{\infty}(0,1)$. We now show that $f(x)\in L^{\theta,\infty)}(0,1)$. In fact, for $m$ a positive integer, integration by parts yields

[TABLE]

By L’Hospital’s Law, one has

[TABLE]

This equality together with (2.3) yields

[TABLE]

By induction,

[TABLE]

Recall that the function

[TABLE]

is non-decreasing, thus (2.4) yields

[TABLE]

where we have used the assumption $\theta\leq 1$, and $[p\theta]$ is the integer part of $p\theta$. The proof of Theorem 2.1 has been completed. ∎

For functions $f_{1}(x),f_{2}(x)\in L^{\theta,\infty)}(\Omega)$ and $\alpha\in\mbox{R}$, the addition $f_{1}(x)+f_{2}(x)$ and the multiplication $\alpha f_{1}(x)$ are defined as usual.

Theorem 2.2**.**

$L^{\theta,\infty)}(\Omega)$* is a linear space on R.*

Proof.

This theorem is easy to prove, we omit the details. ∎

For $f(x)\in L^{\theta,\infty)}(\Omega)$, we define

[TABLE]

We drop the subscript $\Omega$ from $\|\cdot\|_{\theta,\infty),\Omega}$ when there is no possibility of confusion.

Theorem 2.3**.**

$\|\cdot\|_{\theta,\infty)}$* is a norm.*

Proof.

(1) It is obvious that $\|f\|_{\theta,\infty)}\geq 0$ and $\|f\|_{\theta,\infty)}=0$ if and only if $f=0$ a.e. $\Omega$;

(2) For any $f_{1}(x),f_{2}(x)\in L^{\theta,\infty)}(\Omega)$, Minkowski inequality in $L^{p}(\Omega)$ yields

[TABLE]

(3) For all $\lambda\in\mbox{R}$ and all $f(x)\in L^{\theta,\infty)}(\Omega)$, it is obvious that $\|\lambda f\|_{\theta,\infty)}=|\lambda|\|f\|_{\theta,\infty)}$. ∎

Theorem 2.4**.**

$\left(L^{\theta,\infty)}(\Omega),\|\cdot\|_{\theta,\infty)}\right)$* is a Banach space.*

Proof.

Suppose that $\{f_{n}\}_{n=1}^{\infty}\subset L^{\theta,\infty)}(\Omega)$, and for any positive integer $p$,

[TABLE]

Since $\Omega$ is $\sigma$-finite, then $\Omega=\bigcup_{m=1}^{\infty}\Omega_{m}$ with $|\Omega_{m}|<\infty$. It is no loss of generality to assume that the $\Omega_{m}$s are disjoint. (2.4) implies that for any positive integer $p$,

[TABLE]

Thus, by the completeness of $L^{1}(\Omega_{m})$, there exists $f^{(m)}(x)\in L^{1}(\Omega_{m})$, such that

[TABLE]

Hence for any positive integer $m$, there exists a subsequence $\{f_{n}^{(m)}(x)\}$ of $\{f^{m-1}_{n}(x)\}$, $\{f_{n}^{(0)}(x)\}=\{f_{n}(x)\}$, such that

[TABLE]

If we let

[TABLE]

then

[TABLE]

It is no loss of generality to assume that the subsequence $\{f_{n}^{(n)}(x)\}$ of $\{f_{n}(x)\}$ is itself, thus

[TABLE]

We now prove $f(x)\in L^{\theta,\infty)}(\Omega)$ and $\|f_{n}-f\|_{\theta,\infty)}\rightarrow 0$, $(n\rightarrow\infty)$. In fact, by (2.6), for any $\varepsilon>0$, there exists $N=N(\varepsilon)$, such that if $n>N$, then

[TABLE]

Let $p\rightarrow\infty$, one has

[TABLE]

Hence $f(x)\in L^{\theta,\infty)}(\Omega)$, and $\|f_{n}(x)-f(x)\|_{\theta,\infty)}\rightarrow 0$, $n\rightarrow\infty$. This completes the proof of Theorem 2.4. ∎

Definition 2.2**.**

The grand Sobolev space $W^{\theta,\infty)}(\Omega)$ consists of all functions $f$ belonging to $\bigcap_{1\leq p<\infty}W^{1,\infty)}(\Omega)$ and such that $\nabla f\in L^{\theta,\infty)}(\Omega)$. That is,

[TABLE]

This definition will be used in Section 4.

§3   An Alternative Definition of $L^{\theta,\infty}(\Omega)$

In this section, we give an alternative definition of $L^{\theta,\infty)}(\Omega)$ in terms of weak Lebesgue spaces. Let us first recall the definition of weak $L^{p}(0<p<\infty)$ spaces, or the Marcinkiewicz spaces, $L_{weak}^{p}(\Omega)$, see [10, Chapter 1, Section 2], [11, Chapter 2, Section 5] or [12, Chapter 2, Section 18].

Definition 3.1**.**

Let $0<p<\infty$. We say that $f\in L_{weak}^{p}(\Omega)$ if and only if there exists a positive constant $k=k(f)$ such that

[TABLE]

for every $t>0$, where $|E|$ is the $n$-dimensional Lebesgue measure of $E\subset\mbox{R}^{n}$, and $f_{*}(t)=|\{x\in\Omega:|f(x)|>t\}|$ denotes the distribution function of $f$.

For $p>1$, we recall that if $f\in L_{weak}^{p}(\Omega)$, then $f\in L^{q}(\Omega)$ for every $1\leq q<p$, and $f\in L_{weak}^{p}(\Omega)$ if and only if for every measurable set $E\subset\Omega$, the following inequality holds

[TABLE]

for some constant $c>0$.

(3.1) is equivalent to

[TABLE]

Recall also that

[TABLE]

Definition 3.2**.**

For $\theta\geq 0$, the weak space $L_{weak}^{\theta,\infty}(\Omega)$ is defined by

[TABLE]

The following theorem shows that $L_{weak}^{\theta,\infty}(\Omega)=L^{\theta,\infty)}(\Omega)$, thus $L_{weak}^{\theta,\infty}(\Omega)$ can be regarded as an alternative definition of the space $L^{\theta,\infty)}(\Omega)$.

Theorem 3.1**.**

[TABLE]

Proof.

We divided the proof into two steps.

**Step 1 ** $L_{weak}^{\theta,\infty}(\Omega)\subset L^{\theta,\infty)}(\Omega)$.

If $1\leq s<p$, for each $a>0$, one can split the integral in the right-hand side of (3.3) to obtain

[TABLE]

The second integral has been estimated by the inequality $f_{*}(t)\leq|\Omega|t^{-p}M_{p}^{p}(f)$, which is a direct consequence of the definition of the constant $M_{p}(f)$ (see (3.2)). Setting $a=M_{p}(f)$ we arrive at

[TABLE]

This implies

[TABLE]

Therefore

[TABLE]

here we have used (3.6) and the definition of $L_{weak}^{\infty}(\Omega)$.

**Step 2 ** $L^{\theta,\infty)}(\Omega)\subset L_{weak}^{\infty}(\Omega)$.

Since for any $t>0$,

[TABLE]

then

[TABLE]

This implies

[TABLE]

Thus

[TABLE]

The proof of Theorem 3.1 has been completed. ∎

§4   An Application

In this section, we give an application of the space $L^{\theta,\infty}(\Omega)$ to monotonicity property of very weak solutions of the $\cal A$-harmonic equation

[TABLE]

where ${\cal A}:\Omega\times\mbox{R}^{n}\rightarrow\mbox{R}^{n}$ be a mapping satisfying the following assumptions:

(1) the mapping $x\mapsto{\cal A}(x,\xi)$ is measurable for all $\xi\in\mbox{R}^{n}$,

(2) the mapping $\xi\mapsto{\cal A}(x,\xi)$ is continuous for a.e. $x\in\mbox{R}^{n}$,

for all $\xi\in\mbox{R}^{n}$, and a.e. $x\in\mbox{R}^{n}$,

(3)

[TABLE]

(4)

[TABLE]

where $1<p<\infty$, $0<\gamma(x)\leq\tau(x)<\infty$, a.e. $\Omega$.

Conditions (1) and (2) insure that the composed mapping $x\mapsto{\cal A}(x,g(x))$ is measurable whenever $g$ is measurable. The degenerate ellipticity of the equation is described by condition (3). Finally, condition (4) guarantees that, for any $0\leq\theta<\infty$ and any $\varepsilon>0$, ${\cal A}(x,\nabla u)$ can be integrated for $u\in W^{\theta,p}(\Omega)$ against functions in $W^{1,\frac{p-\varepsilon}{1-p\varepsilon}}(\Omega)$ with compact support.

Definition 4.1**.**

A function $u\in W_{loc}^{1,r}(\Omega)$, $\max\{1,p-1\}<r\leq p$, is called a very weak solution of (4.1), if

[TABLE]

for all $\varphi\in W_{0}^{1,\frac{r}{r-p+1}}(\Omega)$.

A fruitful idea in dealing with the continuity properties of Sobolev functions is the notion of monotonicity. In one dimension a function $u:\Omega\rightarrow\mbox{R}$ is monotone if it satisfies both a maximum and minimum principle on every subinterval. Equivalently, we have the oscillation bounds $\mbox{osc}_{I}u\leq\mbox{osc}_{\partial I}u$ for every interval $I\subset\Omega$. The definition of monotonicity in higher dimensions closely follows this observation.

A continuous function $u:\Omega\rightarrow\mbox{R}^{n}$ defined in a domain $\Omega\subset\mbox{R}^{n}$ is monotone if

[TABLE]

for every ball $B\subset\mbox{R}^{n}$. This definition in fact goes back to Lebesgue [13] in 1907 where he first showed the relevance of the notion of monotonicity in the study of elliptic PDEs in the plane. In order to handle very weak solutions of $\cal A$-harmonic equation, we need to extend this concept, dropping the assumption of continuity. The following definition can be found in [14], see also [6, 7].

Definition 4.2**.**

A real-valued function $u\in W_{loc}^{1,1}(\Omega)$ is said to be weakly monotone if, for every ball $B\subset\Omega$ and all constants $m\leq M$ such that

[TABLE]

we have

[TABLE]

for almost every $x\in B$.

For continuous functions (4.2) holds if and only if $m\leq u(x)\leq M$ on $\partial B$. Then (4.3) says we want the same condition in $B$, that is the maximum and minimum principles.

Manfredi’s paper [14] should be mentioned as the beginning of the systematic study of weakly monotone functions. Koskela, Manfredi and Villamor obtained in [15] that $\cal A$-harmonic functions are weakly monotone. In [16], the first author obtained a result which states that very weak solutions $u\in W_{loc}^{1,p-\varepsilon}(\Omega)$ of the $\cal A$-harmonic equation are weakly monotone provided $\varepsilon$ is small enough. The objective of this section is to extend the operator $\cal A$ to spaces slightly larger than $L^{p}(\Omega)$.

Theorem 4.1**.**

Let $\gamma(x)>0$, a.e. $\Omega$, $\tau(x)\in L^{\theta_{1},\infty)}(\Omega)$. If $u\in W^{\theta_{2},p)}(\Omega)$ is a very weak solution to (4.1), then it is weakly monotone in $\Omega$ provided that $\theta_{1}+\theta_{2}<1$.

Proof.

For any ball $B\subset\Omega$ and $0<\varepsilon<1$, let

[TABLE]

It is obvious that

[TABLE]

Consider the Hodge decomposition (see [6]),

[TABLE]

The following estimate holds

[TABLE]

Definition 4.1 with $\varphi$ acting as a test function yields

[TABLE]

Hölder’s inequality together with the conditions (3), (4), (4.4) and (4.5) yields

[TABLE]

The condition $\tau\in L^{\theta_{1},\infty)}(\Omega)$ implies

[TABLE]

Since $u\in W^{\theta_{2},p)}(\Omega)$, then

[TABLE]

By $\theta_{1}+\theta_{2}<1$, we have

[TABLE]

Combining (4.6)-(4.9), and taking into account the assumption $\gamma(x)>0$, a.e. $\Omega$, we arrive at $\nabla u=0$, a.e. $E$. This implies that $(u-M)^{+}-(m-u)^{+}$ vanishes a.e. in $B$, and thus $(u-M)^{+}-(m-u)^{+}$ must be the zero function in $B$, completing the proof of Theorem 4.1. ∎

Remark 4.1**.**

We remark that the result in Theorem 4.1 is a generalization of a result due to Moscariello, see [17, Corollary 4.1].

§5   A Weighted Version

A weight is a locally integrable function on $\mbox{R}^{n}$ which takes values in $(0,\infty)$ almost everywhere. For a weight $w$ and a measurable set $E$, we define $w(E)=\int_{E}w(x)dx$ and the Lebesgue measure of $E$ by $|E|$. The weighted Lebesgue spaces with respect to the measure $w(x)dx$ are denoted by $L^{p}_{w}$ with $0<p<\infty$. Given a weight $w$, we say that $w$ satisfies the doubling condition if there exists a constant $C>0$ such that for any cube $Q$, we have $w(2Q)\leq Cw(Q)$, where $2Q$ denotes the cube with the same center as $Q$ whose side length is 2 times that of $Q$. When $w$ satisfies this condition, we denote $w\in\Delta_{2}$, for short.

A weight function $w$ is in the Muckenhoupt class $A_{p}$ with $1<p<\infty$ if there exists $C>1$ such that for any cube $Q$

[TABLE]

where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. We define $A_{\infty}=\bigcup_{1<p<\infty}A_{p}$.

Let $w$ be a weight. The Hardy-Littlewood maximal operator with respect to the measure $w(x)dx$ is defined by

[TABLE]

We say that $T$ is a Calderón-Zygmund operator if there exists a function $K$ which satisfies the following conditions:

[TABLE]

[TABLE]

For $w$ a weight and $0\leq\theta<\infty$, we define the space $L_{w}^{\theta,\infty)}(\Omega)$ as follows

[TABLE]

where

[TABLE]

The following lemma comes from [18].

Lemma 5.1**.**

If $1<p<\infty$ and $w\in\Delta_{2}$, then the operator $M_{w}$ is bounded on $L^{p}_{w}(\Omega)$.

Theorem 5.1**.**

The operator $M_{w}$ is bounded on $L^{\theta,\infty)}_{w}(\Omega)$ for $0\leq\theta<\infty$ and $w\in\Delta_{2}$.

Proof.

By Lemma 5.1, since for $1<p<\infty$ and $w\in\Delta_{2}$, the operator $M_{w}$ is bounded on $L^{p}_{w}(\Omega)$, then

[TABLE]

This implies

[TABLE]

completing the proof of Theorem 5.1 ∎

The following lemma can be found in [19].

Lemma 5.2**.**

If $w\in A_{\infty}$, then there exists $q\in(1,\infty)$ such that $w\in A_{q}$.

The following lemma can be found in [20, 21].

Lemma 5.3**.**

If $1<p<\infty$ and $w\in A_{p}$, then a Calderón-Zygmund operator $T$ is bounded on $L^{p}_{w}(\Omega)$.

Theorem 5.2**.**

A Calderón-Zygmund operator $T$ is bounded on $L^{\theta,\infty)}_{w}(\Omega)$ for $0\leq\theta<\infty$ and $w\in A_{\infty}$.

Proof.

By $w\in A_{\infty}$ and Lemma 5.2, one has $w\in A_{q}$ for some $q\in(1,\infty)$. For $1<p<q<\infty$, Hölder’s inequality yields

[TABLE]

Thus

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Lemma 5.3 yields

[TABLE]

As desired. ∎

**Acknowledgement ** This study was funded by NSFC (10971224) and NSF of Hebei Province (A2011201011).