数理解析研究所講究録 1293 巻 2002 年 18-26

18Lindel\"of type theorems for monotone Sobolev

functions on half spaces
広島大学理学研究科
広島大学総合科学部
広島大学教育学研究科(Toshihide Futamura)(Yoshihiro Mizuta)(Tetsu Shimomura)

二村俊英
水田義弘
下村哲

Abstract
This paper deals with Lindel\"of type theorems for monotone functions inweighted Sobolev spaces.

1

Introduction

Let
denote the -dimensional Euclidean space. We use the notation
denote the upper half space of, that is,$\mathrm{R}^{n}(n\geq 2)$$n$$\mathrm{D}$to$\mathrm{R}^{n}$$\mathrm{D}=\{x=(x’, x_{n})\in \mathrm{R}^{n-1}\cross \mathrm{R}:x_{n}>0\}$.

We denote by
the distance of from the boundary, that is,for $x=(x’, x_{n})$ . Denote by $B(x, r)$ the open ball centered at with radius , and set$\sigma B(x, r)=B(x, \sigma r)$ for $\sigma>0$ and $S(x, r)=\partial B(x, r)$ .
Acontinuous function on is called monotone in the sense of Lebesgue (see [6])if the equalities
and$x$$\rho_{\mathrm{D}}(x)$$\partial \mathrm{D}$

$x$

$u$

$\rho_{\mathrm{D}}(x)=|x_{n}|$$r$$\mathrm{D}$$\mathrm{m}_{\frac{\mathrm{a}}{G}}\mathrm{x}u=\max u\partial G$$\mathrm{m}_{\frac{\mathrm{i}}{G}}\mathrm{n}u=\min_{\partial G}u$hold whenever $G$ is adomain with compact closure
function in and $p>n-1$ , then$\overline{G}\subset \mathrm{D}$. If is amonotone Sobolev$u$$\mathrm{D}$$|u(x)-u(y)| \leq Mr(\frac{1}{r^{n}}\int_{2B}|\nabla u(z)|^{p}dz)^{1/p}$(1)

for all , $y\in B$ , where $B$ is an arbitrary ball of radius with(see [7, Theorem
1] and [5, Theorem 2.8] . For further results of monotone functions, we refer to [3],[14] and [16].
$x$

$r$

$2B\subset \mathrm{D}$$)$2000 Mathematics Subject Classification :Primary
Key words and phrases :monotone functions, Sobolev functions, Lindelof theorem, Hausdorff
measures, weighted pcapacity$31\mathrm{B}25(46\mathrm{E}35)$

19 19
Our aim in the present note is to extend the second author’s result [13, Theorem
2] to weighted case.
Let

$\mu$

be aBorel measure on$\mathrm{R}^{n}$satisfying the doubling condition :$\mu(2B)\leq c_{1}\mu(B)$for every ball

$B\subset \mathrm{R}^{n}$. We further assume that(2)$\frac{\mu(B’)}{\mu(B)}\geq c_{2}(\frac{r’}{r})^{s}$for all $B’=B(\xi’, r’)$ and $B=B(\xi, r)$ with

THEOREM 1. Let$u$$\xi’$,$\xi\in\partial \mathrm{D}$be aSobolev function on$\mathrm{D}$and $B’\subset B$ , where $s>1$ .

satisfying(3)$|u(x)-u(y)|\leq M\rho_{\mathrm{D}}(z)(f_{\sigma B}|\nabla u(z)|^{p}d\mu)^{1/p}$

for every , $y\in B=B(z, \rho_{\mathrm{D}}(z)/(2\sigma))$ with$x$$z\in \mathrm{D}$and$\int_{\mathrm{D}}|\nabla u(z)|^{p}d\mu(z)<\infty$.

Define$E_{1}=\{\xi\in\partial \mathrm{D}$: $\int_{B(\xi,1)\cap \mathrm{D}}|\xi-y|^{1-n}|\nabla u(y)|dy=\infty\}$and$E_{2}=\{\xi\in\partial \mathrm{D}$

Then

$u$

:

$\lim_{rarrow}\sup_{0}(r^{-p}\mu(B(\xi, r)))^{-1}\int_{B(\xi,r)\cap \mathrm{D}}|\nabla u(y)|^{p}d\mu(y)>0\}$

has anontangential limit at every$\xi\in\partial \mathrm{D}\backslash (E_{1}\cup E_{2})$.

.

-capacity zero. In Manfredi-Villamor
is ofRemark 1. Note here that[9], the exceptional sets are characterized by Hausdorff dimension, so that their result
follows from this nontangential limit result.
$E_{1}\cup E_{2}$THEOREM 2. Let be afunction on for which there exist anonnegative function
such that, $M>0$ and$\mathrm{D}$

$u$

$g$

$C_{1,p,\mu}$$\in L_{loc}^{p}(\mathrm{D};\mu)$$\sigma\geq 1$$|u(x)-u(y)|\leq M\rho_{\mathrm{D}}(z)(\mathrm{f}_{\sigma B}^{g^{p}d\mu})^{1/p}$

for every , $y\in B=B(z, \rho_{\mathrm{D}}(z)/(2\sigma))$ with$x$$z\in \mathrm{D}$

(4)

and

$\int_{\mathrm{D}}g(z)^{p}d\mu(z)<\infty$

.

(5)

20
Suppose $p>s-1$ and set$E=\{\xi\in\partial \mathrm{D}$: $\lim_{rarrow}\sup_{0}(r^{-p}\mu(B(\xi, r)))^{-1}\int_{B(\xi,r)\cap \mathrm{D}}g(z)^{p}d\mu(z)>0\}$ .

If
and there exists a
limit , then has anontangential limit$\xi\in\partial \mathrm{D}\backslash E$$c\mathrm{u}r\iota^{r}e\gamma \mathrm{j}n\mathrm{D}$$u$$\beta$$\beta$tending to

$\xi$

at 4.

along which

has afinite$u$

For $\alpha>-1$ , we consider$d\nu(x)=|x_{n}|^{\alpha}dx$

as ameasure, which satisfies
for all$\nu(B(\xi, r))=\nu(B(0, 1))r^{n+\alpha}$$\xi\in\partial \mathrm{D}$and $r>0$ .

Then we obtain the following result.

COROLLARY 1. Let

be amonotone Sobolev function on$u$$\mathrm{D}$satisfying$\int_{\mathrm{D}}|\nabla u(z)|^{p}z_{n}^{\alpha}dz<\infty$

for $p>n-1and-1<\alpha<p-n+1$ . Consider the set$E_{n+\alpha-p}=\{\xi\in\partial \mathrm{D}$If:$\lim_{rarrow}\sup_{0}r^{p-\alpha-n}\int_{B(\xi,r)\cap \mathrm{D}}|\nabla u(z)|^{p}z_{n}^{\alpha}dz>0\}$

and there exists a curve intending to
,
limit
then
has
anontangential
limit
at
finite4.
$\xi\in\partial \mathrm{D}\backslash E_{n+\alpha-p}$$\beta$$\mathrm{D}$$\gamma$

$u$

$\xi$

.

along which

$u$

has a

$\beta$REMARK 2. We know that
, wheredenotes the ddimensional Hausdorff measure, and hence it is of-capacity zero; for theseresults, see Meyers $[10, 11]$ and the second author’s book [14].
$\mathcal{H}^{n+\alpha-p}(E_{n+\alpha-p})=0$$\mathcal{H}^{d}$$C_{1-\alpha/p,p}$2

Proof of Theorem 2

Asequence$\{x_{j}\}$is called regular at$\xi$$\in\partial \mathrm{D}$if$x_{j}arrow\xi$and$|x_{j+1}-\xi|<|x_{j}-\xi|<c|x_{j+1}-\xi|$for some constant $c>1$ .
First we give the following result, which can be proved by (4).

LEMMA 1. Let and be as in Theorem 2. If
regular sequencewith $x_{j}=\xi+$ 0, , 0,such that!, then has anontangential limit 4at 4.
$u$$\{x_{j}\}\subset \mathrm{D}$

$g$

$\xi$

$($

$\ldots$$u$$r_{j})$$\in\partial \mathrm{D}\backslash E$

and there exists a
has afinite limit$u(x_{j})$21
PROOF OF THEOREM 2:For $r>0$ sufficiently small, take $C(r)\in\gamma\cap S(\xi, r)$ .
Letting $C_{1}(r)=\xi+(0, \ldots, 0, r)$ , take an end pointof aquarter of circle
containing
and $C(r)$ .
( $N$ may depend on ) with the,,
We take afinite chain of balls
following properties:$C_{2}(r)\in\partial \mathrm{D}$$C_{1}(r)$$B_{2},\ldots$$B_{1}$

(i)
(ii)

(iii)

$B_{j}=B(z_{j}, \rho_{\mathrm{D}}(z_{j})/(2\sigma))$$\rho_{\mathrm{D}}(z_{j})\leq\rho_{\mathrm{D}}(z_{j+1})$$B_{j}\cap B_{j+1}\neq\emptyset$

and

$r$

with $z_{j}\in C(\hat{r)C_{1}}(r), z_{1}=C(r)$ and $z_{N}=C_{1}(r)$ ;$z_{j+1}\not\in B_{j}$;for each ;$j$(iv) $|C_{2}(r)-z|\leq 3\rho_{\mathrm{D}}(z)$ for(v)$B_{N}$$z \in A(\xi, r)=\bigcup_{j=1}^{N}\sigma B_{j}\subset B(\xi, 2r)\cap \mathrm{D}$;denotes the characteristic function of
constant depending only on and ;$\sum_{j}\chi_{\sigma B_{j}}\leq c_{3}$, where$\chi_{A}$

$A$

and

$c_{3}$

is a$\sigma$$c_{1}$

see Heinonen [2] and Hajlasz-Koskela [1].

Pick $x_{j}\in B_{j+1}\cap B_{j}$ for $1\leq j\leq N-1$ . By (4), we see that$|u(x_{j})-u(x_{j-1})|\leq M\rho_{\mathrm{D}}(z_{j})(\mathrm{f}\sigma B_{\mathrm{j}}g(z)^{p}d\mu(z))1/p$

for $1\leq j\leq N$ , where $x_{0}=C(r)$ and $x_{N}=C_{1}(r)$ .
Since $p>s-1$ by our assumption, there is $\delta>0$ such that $s-p<\delta<1$ . We
have by H\"older’s inequality$|u(C_{1}(r))-u(C(r))|$$\leq$$|u(x_{1})-u(x_{0})|+|u(x_{2}\}-u(x_{1})|+\cdots+|u(x_{N})-u(x_{N-1})|$$\leq$$M \sum_{j=1}^{N}\rho_{\mathrm{D}}(z_{j})^{1+\delta/p}\mu(\sigma B_{j})^{-1/p}(\int_{\sigma B_{\mathrm{j}}}g(z)^{p}\rho_{\mathrm{D}}(z)^{-\delta}d\mu(z))1/p$$\leq$$M( \sum_{j=1}^{N}\rho_{\mathrm{D}}(z_{j})^{p’(1+\delta/p)}\mu(\sigma B_{j})^{-p’/p)^{1/p’}}(\int_{A(\xi,r)}g(z)^{p}\rho_{\mathrm{D}}(z)^{-\delta}d\mu(z))^{1/p}$$\leq$$M( \sum_{j=1}^{N}\rho_{\mathrm{D}}(z_{j})^{p’(1+\delta/p)}\mu(\sigma B_{j})^{-p’/p)^{1/p’}}(\int_{B(\xi,2\tau)\cap \mathrm{D}}g(z)^{p}|C_{2}(r)-z|^{-\delta}d\mu(z))^{1/p}$

where $1/p+1/p’=1$ . Here note that
positive constant depending only on the doubling constantfrom (2) that$\mu(B(C_{2}(r), \rho_{\mathrm{D}}(z_{j})))\leq c_{4}\mu(\sigma B_{j})$$\sum_{j=1}^{N}\rho_{\mathrm{D}}(z_{j})^{p’(p+\delta)/p}\mu(\sigma B_{j})^{-p’/p}$$\leq$$c_{1}$. Since, where$\delta>s-p$ ,$c_{4}$is a

we see$M \sum_{j=1}^{N}\rho_{\mathrm{D}}(z_{j})^{p’(p+\delta)/p}\mu(B(C_{2}(r), \rho_{\mathrm{D}}(z_{j})))^{-p’/p}$22$\leq$$M \sum_{j=1}^{N}\rho_{\mathrm{D}}(z_{j})^{p’(p+\delta)/p}(\frac{\rho_{\mathrm{D}}(z_{j})}{2r})^{-sp’/p}\mu(B(\xi, 2r))^{-p’/}$$\leq$$Mr^{sp/p’} \mu(B(\xi, r))^{-p’/p}\sum_{j=1}^{N}\rho_{\mathrm{D}}(z_{j})^{p’(p+\delta-s)/p}$$\leq$$Mr^{sp’/p} \mu(B(\xi, r))^{-p’/p}\int_{0}^{f}t^{p’(p+\delta-s)/p}dt/t$$\leq$$Mr^{\delta p’/p}(r^{-p}\mu(B(\xi, r)))^{-p’/p}$Moreover, since $0<\delta<1$ , we note that$\int_{2^{-j}}^{2^{-j+1}}|C_{2}(r)-z|^{-\delta}dr\leq\int_{2^{-j}}^{2^{-j+1}}|r-|z||^{-\delta}dr\leq M2^{-j(1-\delta)}$.

(6)

Hence it follows from (6) that$\int_{2^{-j}}^{2^{-\mathrm{j}+1}}|u(C_{1}(r))-u(C(r))|^{p}\frac{dr}{r}$$M \int_{2^{-j}}^{2^{-j+1}}r^{\delta}(r^{-p}\mu(B(\xi, r)))^{-1}(\int_{B(\xi,2r)\cap \mathrm{D}}g(z)^{p}|C_{2}(r)-z|^{-\delta}d\mu(z))\frac{dr}{r}$$\leq$$\leq$$M2^{-j(p+\delta-1)} \mu(B(\xi, 2^{-j}))^{-1}\int_{B(\xi,)\cap \mathrm{D}}2^{-\mathrm{j}+2}g(z)^{p}(\int_{2^{-j}}^{2^{-j+1}}|C_{2}(r)-z|^{-\delta}dr)d\mu(z)$$\leq$$M(2^{jp} \mu(B(\xi, 2^{-j})))^{-1}\int_{B(\xi,)\cap \mathrm{D}}2^{-j+2}g(z)^{p}d\mu(z)$

Since

$\xi$

$\in\partial \mathrm{D}\backslash E$, we can find asequence$\{r_{j}\}$.

such that $2^{-j}<r_{j}<2^{-j+1}$ and$\lim_{jarrow\infty}|u(C_{1}(r_{j}))-u(C(r_{j}))|=0$.

By our assumption we see that $u(C_{1}(rj))$ has afinite limit as $jarrow\infty$ . If we note that$\{C_{1}(rj)\}$ is regular at, then Lemma 1 proves the required conclusion of the theorem.
$\beta$

$\xi$

3
Let

$A_{q}$$w$

weights

be aMuckenhoupt$A_{q}$

weight, and define$d\nu(y)=w(y)dy$ .

Let

$u$

be amonotone Sobolev function on$\mathrm{D}$such that.

$\int_{\mathrm{D}}|\nabla u(x)|^{p}d\nu(x)<\infty$

23
Suppose that $1<q<p/(n-1)$ . Since $p_{1}=p/q>n-1$ , applying inequality (1) we
obtain$|u(x)-u(y)| \leq Mr(\frac{1}{r^{n}}\int_{2B}|\nabla u(z)|^{p1}dz)^{1/p_{1}}$for all , $y\in B$ , where $B$ is an arbitrary ball of radiusof Theorem 2we insist that
$x$

$r$

with

$2B\subset \mathrm{D}$. As in the proof$\int_{2^{-j}}^{2^{-\mathrm{j}+1}}|u(C_{1}(r))-u(C(r))|^{p_{1}}\frac{dr}{r}\leq M2^{-jp1}|B(\xi, 2^{-j})|^{-1}\int_{B(\xi,)\cap \mathrm{D}}2^{-j+2}|\nabla u(z)|^{p_{1}}dz$

Using Holder inequality and$A_{q}$-condition of$w$.

, we have$\int_{2^{-j}}^{2^{-j+1}}|u(C_{1}(r))-u(C(r))|^{p1}\frac{dr}{r}$$\leq$$\leq$$M2^{-jp1}|B( \xi, 2^{-j})|^{-1}(\int_{B(\xi,2^{-\mathrm{j}+2})\cap \mathrm{D}}|\nabla u(z)|^{p_{1}q}w(z)dz)^{1/q}(\int_{B(\xi,)}2^{-j+2}w(z)^{-q’/q}dz)^{1/\emptyset}$$M((2^{jp} \nu(B(\xi, 2^{-j})))^{-1}\int_{B(\xi,)\cap \mathrm{D}}2^{-j+2}|\nabla u(z)|^{p}d\nu(z))^{1/q}$,

where $1/q+1/q’=1$ . Thus we obtain the following result (cf. Manfredi-Villamor [9]),as in the proof of Theorem 2.

COROLLARY 2. Let $1\leq q<p/(n-1)$ . Let $w\in A_{q}$ and set $d\nu(y)=w(y)dy$ .
satisfying
Suppose that is amonotone Sobolev function on$\mathrm{D}$$u$$\int_{\mathrm{D}}|\nabla u(z)|^{p}d\nu(z)<\infty$

(7)

.

Set
$E=\{\xi\in\partial \mathrm{D}$:$\lim_{rarrow}\sup_{0}(r^{-p}\nu(B(\xi, r)))^{-1}\int_{B(\xi,r)\cap \mathrm{D}}|\nabla u(z)|^{p}d\nu(z)>0\}$tending to 4along which
and there exists acurve inIflimit , then has anontangential limit at .
$\mathrm{D}$$\xi\in\partial \mathrm{D}\backslash E$$\beta$$u$$\beta$$u$$\gamma$.

has afinite$\xi$

REMARK 3. Let $1\leq q<p/(n-1)$ . Let
define$w$

be aMuckenhoupt$A_{q}$

weight, and$d\nu(y)=w(y)dy$ .

Suppose that is amonotone Sobolev function on satisfying (7). Applying H\"older’sinequality to (1) with replaced by $p/q$ , we see that$\mathrm{D}$

$u$

$p$

$|u(x)-u(y)|\leq Mr(f_{2B}|\nabla u(z)|^{p}d\nu(z))^{1/p}$

for all$x$, $y\in B$ , where $B$ is an arbitrary ball of radius

$r$

with

$2B\subset \mathrm{D}$(see also

Manfredi-Villamor [9] .
$)$$\mathrm{i}\mathrm{f}-1<\alpha<q-1$ .
REMARK 4. Consider $w(y)=|y_{n}|^{\alpha}$ . Then $w\in A_{q}$ if and only
In this case, Corollary 2does not imply Corollary 1 when $n\geq 3$ .

24
4Generalizations of Lindel\"of theorems

For an integer , $1\leq d<n$ , let$d$$P_{d}$:$\mathrm{R}^{n}arrow \mathrm{R}^{d}$be the projection, that is,for $x=(x_{1}, x_{2}, \ldots, x_{n})$ .

$P_{d}(x)=(x_{1}, \ldots, x_{d}, 0, \ldots, 0)$We say thatis a( , A2, )-approach set at 4, whereand, ifthere exists asequence of positive numbers
tending to zero such that $r_{j+1}<r_{j}<$and$\Gamma\subset \mathrm{D}$$d$$\lambda_{1}$$\lambda_{1}\geq 1$$\lambda_{2}>0$$\{r_{j}\}$$\lambda_{1}r_{j+1}$$\mathcal{H}^{d}(P_{d}(\Gamma\cap(B(\xi, r_{j})\backslash B(\xi, r_{j+1}))))\geq\lambda_{2}r_{j}^{d}$

Theorem 3. Let$u$be afunction on$\mathrm{D}$

with

$g$

.

satisfying (4) and$\int_{\mathrm{D}}g(z)^{p}d\mu(z)<\infty$.

Suppose $p>s-d$ , and de $ine$$E=\{\xi\in\partial \mathrm{D}$If: $\lim_{rarrow}\sup_{0}(r^{-p}\mu(B(\xi, r)))^{-1}\int_{B(\xi,r)\cap \mathrm{D}}g(z)^{p}d\mu(z)>0\}$ .

and there exists-approach set
,
limit
at
then has anontangential limit at .
afinite$\xi\in\partial \mathrm{D}\backslash E$$\beta$$\mathrm{p}_{\mathrm{R}\mathrm{O}\mathrm{O}\mathrm{F}}$$a(\lambda_{1}, \lambda_{2}, d)$$\Gamma\subset \mathrm{D}$

$u$

$\xi$

$\beta$. By our assumption, we can take$\delta>0$

at

$\xi$

along wiich

$u$

has

$\xi$

such that $s-p<\delta<d$. Set$G_{j}=P_{d}(\Gamma\cap(B(\xi, r_{j})\backslash B(\xi, r_{j+1})))$.

For $X\in Gj$ , take $C(X)\in\Gamma\cap(B(\xi, rj)\backslash B(\xi, rj+1))$ , and set $r(X)=r=|\xi-\mathrm{C}(\mathrm{X})$ .
Let $C_{1}(X)=\xi+(0, \ldots, 0, r)$ and $D(X)=P_{n-1}(C(X))$ .
We take afinite chain of balls ,,with the following properties$B_{1}$

(i)
(ii)
(iii)

$B_{j}=B(z_{j}, \rho_{\mathrm{D}}(z_{j})/(2\sigma))$$\rho_{\mathrm{D}}(z_{j})\leq\rho_{\mathrm{D}}(z_{j+1})$$B_{\grave{J}}\cap B_{j+1}\neq\emptyset$

and$\sum_{j}\chi_{\sigma B_{\mathrm{j}}}\leq c_{3}$$B_{N}$

with $z_{j}\in C(X\hat{)C_{1}}(X), z_{1}=C(X)$ and $z_{N}=C_{1}(X)$ ;$z_{j+1}\not\in B_{j}$;for each ;$j$(iv) $|D(X)-z|\leq 3\rho_{\mathrm{D}}(z)$ for(v)$B_{2},\ldots$$z$:$A( \xi, r)=\bigcup_{j=1}^{N}\sigma B_{j}\subset \mathrm{B}(\mathrm{f}, 2r)\cap \mathrm{D}$;.

Since $\delta>s-p$ , we have as in the proof of Theorem 2$|u(C_{1}(X))-u(C(X))|^{p} \leq Mr^{\delta}(r^{-p}\mu(B(\xi, r)))^{-1}\int_{B(\xi,2\mathrm{r})\cap \mathrm{D}}g(z)^{p}|D(X)-z|^{-\delta}d\mu(z)$.

25 25Further, since$P_{d}$

is 1-Lipschitz and $0<\delta<d$ , we see that$\int_{G_{j}}|D(X)-z|^{-\delta}d\mathcal{H}^{d}(X)$$\leq$$\int_{G_{j}}|X-P_{d}(z)|^{-\delta}d\mathcal{H}^{d}(X)$$\leq$$\int_{P_{d}(B(\xi,r_{j}))}|X-P_{d}(z)|^{-\delta}d\mathcal{H}^{d}(X)$$\leq$$Mr_{j}^{d-\delta}$.

Hence we have$f_{G_{j}}|u(C_{1}(X))-u(C(X))|^{p}d \mathcal{H}^{d}(X)\leq M(r_{j}^{-p}\mu(B(\xi, rj))^{-1}\int_{B(\xi,2r_{j})\cap \mathrm{D}}g(z)^{p}d\mu(z)$Thus we can find asequence$\{X_{j}\}$such that$X_{j}\in G_{j}$.

and$\lim_{jarrow\infty}|u(C_{1}(X_{j}))-u(C(Xj))|=0$.

Thus we see that $u(C_{1}(X_{j}))$ has afinite limit as $jarrow\infty$ . Since $\{C_{1}(X_{j})\}$ is regularat 4, we can show that has anontangential limit at 4by Lemma 1.
$\beta$$\beta$$u$Corollary 3. Let u be aharmonic function on D satisfying$\int_{\mathrm{D}\cap B(0,N)}|\nabla u(z)|^{p}z_{n}^{\alpha}dz<\infty$

and there exists a
for every $N>0$ , $and-1<\alpha<p-n+d$ . If (at 4along which has afinite limit at 4, then has( , A2, )-approach set
anontangential limit at .
$\in\partial \mathrm{D}\backslash E_{n+\alpha-p}$$\lambda_{1}$$d$$\Gamma\subset \mathrm{D}$$\beta$$u$$\beta$

$u$

$\xi$

REMARK 5. The conclusion of Corollary 3is still valid for $A$-harmonic functions
and polyharmonic functions.

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[14] Y. Mizuta, Potential theory in Euclidean spaces, Gakk\={o}tosyo, Tokyo, 1996.
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Department of Mathematics
Faculty of Science
Hiroshima University
Higashi-Hiroshima 739-8526, Japan-mail address: toshi@mis. hiroshima-u. ac.jp

$E$

and
The Division of Mathematical and Information Sciences
Faculty of Integrated Arts and Sciences
Hiroshima University
Higashi-Hiroshima 739-8521, Japan-mail address: mizuta@mis. hiroshima-u. ac.jp
andDepartment of Mathematics
Faculty of Education
Hiroshima University
Higashi-Hiroshima 739-8524, Japan-mail address: tshimo(Uhiroshima-u. ac.jp$E$$E$