数理解析研究所講究録 1293 巻 2002 年 98-109

98
Riesz decomposition and limits at infinity for -precise functions
on ahalf space$p$

広島大総合科学部

水田義弘 (Yoshihiro Mizuta)

広島大教育学部

下村

哲 (Tetsu Shimomura)

1IntroductionLet

be anonnegative superharmonic function on $D=\{x=(x_{1}, \ldots, x_{n-1}, x_{n})\in$, where $n\geqq 2$ . Then it is known (cf. Lelong-Ferrand [6]) that is uniquely
decomposed as$u$$\mathrm{R}^{n};x_{n}>0\}$$u$$u(x)=ax_{n}+ \int_{D}G(x, y)d\mu(y)+f_{\partial D}P(x, y)d\nu(y)$ ,where$\partial D),$

$a$

$G$

is anonnegative number,(resp. ) is anonnegative measure on $D$ (resp.
is the Green function for $D$ and is the Poisson kernel for $D$ . The first author
$\nu$

$\mu$

$P$

showed in [9] that if $0\leqq\beta\leqq 1,1-n\leqq\gamma<1$ and$\int_{D}y_{n}^{\gamma}d\mu(y)+\int_{\partial D}|y|^{\gamma-1}d\nu(y)<\infty$,

then$\lim_{|x|arrow\infty,x\in D-E},$$x_{n}^{-\beta}|x|^{n+\gamma-2+\beta}[u(x)-ax_{n}]=0$with asuitable exceptional set $E’\subset D.$ For related results, we also refer the reader toEss\’en-Jackson[3, Theorem 4.6], Aikawa [1] and MiyamotO-Yoshida [8].
Our main aim in this paper is to establish the analogue of these results for locally-precise functions in $D$ satisfying
$u$

$p$

$\int_{D}|\nabla u(x)|^{p}x_{n}^{\gamma}dx<\infty$

,

(1)

where
denotes the gradient, $1<p<\infty$ and $-1<\gamma<p-1$ (see Ohtsuka [15] andZiemer [17] for locally pprecise functions).
$\nabla$2Fine limits at infinity
Denote by
the space of all locally pprecise functions on $D$ satisfying (1). Consider
the kernel function$\mathrm{D}^{p,\gamma}$$K_{\gamma}(x, y)=|x-y|^{1-n}y_{n}^{-\gamma/p}$.

To evaluate the size of exceptional sets, we use the capacity$C_{K_{\gamma},p}(E;G)= \inf\int_{D}g(y)^{p}dy$,

99
where $E$ is asubset of an open set
measurable functions such that
$g$

$G$

in $D$ and the infimum is taken over all nonnegative
outside and$G$$g=\mathrm{O}$for all $x\in E$ .

$\int_{D}K_{\gamma}(x, y)g(y)dy\geqq 1$

We say that $E\subset D$ is$(K_{\gamma},p)$-thin at infinity if$\sum_{i=1}^{\infty}2^{-i(n+\gamma-p)}C_{K_{\gamma},p}(E_{i;}D_{i})<\infty$(2),

where $E_{i}=\{x\in E : 2^{i}\leqq|x|<2^{i+1}\}$ and $D_{i}=\{x\in D : 2^{i-1}<|x|<2^{i+2}\}$ .
Our first aim in this paper is to establish the following theorem.
,
THEOREM 1(cf. [4]). Let $p>1,$ $-1<\gamma<p-1$ and $n+\gamma-p\geqq 0.$ If-thin at inhnity;
then there exist aset $E\subset D$ and anumber such that is$u\in \mathrm{D}^{p,\gamma}$

$E$

$A$

$(K_{\gamma},p)$$\lim_{|x|arrow\infty,x\in D-E}|x|^{(n+\gamma-p)/p}[u(x)-A]=0$

in case $n+\gamma-p>0$ and$\lim$$(\log|x|)^{-1/p’}[u(x)-A]=0$$|x|arrow\infty,x\in D-E$

in case $n+\gamma-p=0$ , where $p’=p/(p-1)$ .
In fact, if $1\leqq q<p$ and $q<p/(1+\gamma)$ , then H\"older’s inequality gives$\int_{G}|\nabla u(x)|^{q}dx$$\leqq$$( \int_{G}x_{n}^{-\gamma q/(p-q)}dx)^{1-q/p}(\int_{G}|\nabla u(x)|^{p}x_{n}^{\gamma}dx)^{q/p}<\infty$

for every bounded open set $G\subset D$ . Hence we can find alocally -precise extension
for $x_{n}>0$ and $\overline{u}(x’, x_{n})=u(x’, -x_{n})$ for $x_{n}<0$ .
such thatto
We denote by $B(x, r)$ the open ball centered at with radius $r>0$ . In view of [13], we
can find anumber such that$\overline{u}$$q$$\overline{u}(x’, x_{n})=u(x’, x_{n})$$\mathrm{R}^{n}$

$x$

$a$

$\overline{u}(x)=c_{n}\sum_{i=1}^{n}\int_{B(0,1)}\frac{x_{i}-y_{i}}{|x-y|^{n}}\frac{\partial\overline{u}}{\partial y_{i}}(y)dy+c_{n}\sum_{i=1}^{n}\int_{\mathrm{R}^{n}-B(0,1)}(\frac{x_{i}-y_{i}}{|x-y|^{n}}-\frac{-y_{i}}{|y|^{n}})\frac{\partial\overline{u}}{\partial y_{i}}(y)dy+a$

for almost every
that in aset of$x\in \mathrm{R}^{n}$$C_{K_{\gamma},p}$. Here we see that the equality holds for every $x\in D$ except-capacity zero. Now Theorem 1is aconsequence of [4].

3Riesz decomposition
having vertical limit zero at almost
the space of all functions
We denote by$D$ , and by
the space of all harmonic functions on $D$ inevery boundary point of.
. As in Deny-Lions [2], we have the following Riesz decomposition of$u\in \mathrm{D}^{p,\gamma}$$\mathrm{D}_{0}^{p,\gamma}$$\mathrm{H}\mathrm{D}^{p,\gamma}$$u\in \mathrm{D}^{p,\gamma}$$\mathrm{D}^{p,\gamma}$

100
THEOREM 2. A function$u\in \mathrm{D}^{p,\gamma}$

is uniquely represented as(3)$u=u_{0}+h$ ,where$u_{0}\in \mathrm{D}_{0}^{p,\gamma}$

and$h\in \mathrm{H}\mathrm{D}^{p,\gamma}.$More precisely, for fixed $\xi\in D$ ,,$u_{0}(x)$$=$$c_{n} \sum_{i=1}^{n}\int_{D}(\frac{x_{i}-y_{i}}{|x-y|^{n}}-\frac{\overline{x}_{i}-y_{i}}{|\overline{x}-y|^{n}})\frac{\partial u}{\partial y_{i}}(y)dy$$h(x)$$=$$2c_{n} \sum_{i=1}^{n}\int_{D}(\frac{\overline{x}_{i}-y_{i}}{|\overline{x}-y|^{n}}-\frac{\overline{\xi}_{i}-y_{i}}{|\overline{\xi}-y|^{n}})\frac{\partial u}{\partial y_{i}}(y)dy+A$

,

where
constant depending on

for $x=(x_{1}, \ldots, x_{n-1}, x_{n}),$ $c_{n}=\Gamma(n/2)/(2\pi^{n/2})$ andand .

$\overline{x}=(x_{1}, \ldots, x_{n-1}, -x_{n})$
$u$

is a

$\xi$

As applications we are concerned with the limits at infinity of functions in$\mathrm{H}\mathrm{D}^{p,\gamma}$$A$$\mathrm{D}_{0}^{p,\gamma}$

and.

Consider the kernel function$k_{\beta,\gamma}(x, y)=x_{n}^{1-\beta}y_{n}^{-\gamma/p}|x-y|^{1-n}|\overline{x}-y|^{-1}$

for

$x$

and

$y$

in $D$ . To evaluate the size of exceptional sets, we use the capacity$C_{k_{\beta,\gamma},p}(E;G)= \inf\int_{D}g(y)^{p}dy$where $E$ is asubset of an open set
measurable functions such that
$g$

$G$

in $D$ and the infimum is taken over all nonnegative
outside and$g=\mathrm{O}$$G$$\int_{D}k_{\beta,\gamma}(x, y)g(y)dy\geqq 1$

We say that $E\subset D$ is$(k\beta,\gamma’ p)$,for all $x\in E$ .

-thin at infinity if$\sum_{i=1}^{\infty}2^{-i(n+\gamma-(1-\beta)p)}C_{k_{\beta,\gamma},p}(E_{i;}D_{i})<\infty$.

(4)

THEOREM 3. Let $p>1,$ $-1<\gamma<p-1$ and $0\leqq\beta\leqq 1$ . If
-thin at infinity andexists aset $E\subset D$ such that $E$ is$u\in \mathrm{D}_{0}^{p,\gamma}$, then there$(k\beta,\gamma’ p)$$\lim_{|x|arrow\infty,x\in D-E}x_{n}^{-\beta}|x|^{(n+\gamma-(1-\beta)p)/p}u(x)=0$.

THEOREM 4. Let $p>1,$ $-1<\gamma<p-1$ and $n+\gamma-p\geqq 0.$ If
there exist anumber such that$A$$\lim_{|x|arrow\infty,x\in D}x_{n}^{(n+\gamma-p)/p}[h(x)-A]=0$$h\in \mathrm{H}\mathrm{D}^{p,\gamma},$

then

101
in case $n+\gamma-p>\mathrm{O}$ and$\lim_{|x|arrow\infty,x\in D}(\max\{\log(1/x_{n}), \log|x|\})^{-1/p’}[h(x)-A]=0$

in case $n+\gamma-p=0$ .

REMARK 1. Let $p>1,$ $-1<\gamma<p-1$ and $n+\gamma-p>0$ . Then we can find asuch thatfunction$h\in \mathrm{H}\mathrm{D}^{p,\gamma}$$\lim\sup$$|x|^{(n+\gamma-p)/p}h(x)=\infty$$|x|arrow\infty,x\in D$

and$\lim$$x_{n}^{(n+\gamma-p)/p}h(x)=0$.

$|x|arrow\infty,x\in D$

For proofs of these theorems, we refer to [14].

4Examples of thin sets at infinity
We are concerned with the measure condition on sets which are thin at infinity.
, denote by $|E|$ the Lebesgue measure of $E$ . Then we
For ameasurable set
can prove that$E\subset \mathrm{R}^{n}$$|E|^{(1-(1-\beta)/n)p}\leqq MC_{k_{\beta,\gamma\prime}p}(E;D_{0})$(5)$C_{k_{\beta,\gamma},p}(rE;rD_{0})=r^{n+\gamma-(1-\beta)p}C_{k_{\beta,\gamma},p}(E;D_{0})$

(6)

and
whenever $E\subset D\cap B(0,2)-B(0,1)$ and $r>0$ . Hence we have the following result.
$\mathrm{p}_{\mathrm{R}\mathrm{O}\mathrm{P}\mathrm{O}\mathrm{S}\mathrm{I}\mathrm{T}\mathrm{I}\mathrm{O}\mathrm{N}}1$. Let $0\leqq\beta\leqq 1and-1<\gamma<p-1$ . If (4) holds, then$\sum_{i=1}^{\infty}(\frac{|E_{i}|}{|B_{i}|})^{(1-(1-\beta)/n)p}<\infty$,

where $E_{i}=E\cap B_{i+1}-B_{i}$ with $B_{i}=B(0,2^{i})\cap D$ .

If

$E$

is well situated, then we have stronger results as in the following.

, wherePROPOSITION 2. Let $0\leqq\beta\leqq 1$ and $-1<\gamma<p-1$ . Set
If $p<n$ and $F$ is$B_{j}=B(x_{j}, s_{j})$ with $2^{j}\leqq|x_{j}|<2^{j+1}$ and $r_{j}=(x_{j})_{n}>2s_{j}$ .
-thin at inBnity, then
$F= \bigcup_{j=1}^{\infty}B_{j}$$(k_{\beta,\gamma},p)$$\sum_{j=1}^{\infty}(\frac{s_{j}}{2^{j}})^{n-p}(\frac{r_{\mathrm{j}}}{2^{j}})^{\beta p+\gamma}<\infty$

;

(7)

102
conversely, if (7) holds, then $F$ is$(k\beta,\gamma’ p)$-thin at inhnity.

PROOF. First we show that if $p<n$ , then(8)$s^{n-p}r^{\beta p+\gamma}\leqq MC_{k_{\beta,\gamma},p}(B;D_{0})$for $B=B(x_{0}, s)$ with $1\leqq|x_{0}|<2$ and $r=(x_{0})_{n}>2s$ . Let
andoutsidemeasurable function such that$g=\mathrm{O}$$g$

be anonnegative$D_{0}$$\int_{D}k_{\beta,\gamma}(x, y)g(y)dy\geqq 1$

for every $x\in B$ . Then we have by Fubini’s theorem$|B|$$\leqq$$\int_{B}(\int_{D_{0}}k_{\beta,\gamma}(x, y)g(y)dy)dx$$=$$\int_{D_{0}}g(y)y_{n}^{-\gamma/p}(\int_{B}x_{n}^{1-\beta}|x-y|^{1-n}|\overline{x}-y|^{-1}dx)dy$$\leqq$$Mr^{1-\beta} \int_{D_{0}}g(y)y_{n}^{-\gamma/p}(\int_{B}|x-y|^{1-n}|\overline{x}-y|^{-1}dx)dy$.

We set$I(y)= \int_{B}|x-y|^{1-n}|\overline{x}-y|^{-1}dx$

and$J= \int_{D_{0}}g(y)y_{n}^{-\gamma/p}(\int_{B}|x-y|^{1-n}|\overline{x}-y|^{-1}dx)dy$.

If $|y-x_{0}|<3s/2$ , then$I(y) \leqq r^{-1}\int_{B}|x-y|^{1-n}dx\leqq Mr^{-1}s$,so that we have by H\"older’s inequality$J_{1}$$=$$\int_{\{y\in D_{0}:|y-x\mathrm{o}|<3s/2\}}\prime g(y)y_{n}^{-\gamma/p}I(y)dy$$\leqq$$Mr^{-1}s \int_{\{y\in D_{0}:|y-x\mathrm{o}|<3s/2\}}g(y)y_{n}^{-\gamma/p}dy$$\leqq$$Mr^{-1}s( \int_{\{y\in D_{0}:|y-x\mathrm{o}|<3s/2\}}y_{n}^{-\gamma p’/p}dy)^{1/p’}(\int_{D_{0}}g(y)^{p}dy)^{1/p}$$\leqq$$Ms^{n}r^{-1-\gamma/p_{S}1-n/p}( \int_{D_{0}}g(y)^{p}dy)^{1/p}$

103
If $|y-x_{0}|$that:$3s/2$

and$y_{n}\leqq x_{n}/2$$I_{2}(y)$, then $|x-y|\geqq M(|x’-y|+x_{n})\geqq M(|x_{0}’-y|+r),$ so$=$$\int_{\{x\in B:y_{n}\leqq x_{n}/2\}}|x-y|^{1-n}|\overline{x}-y|^{-1}dx$$\leqq$$M(|x_{0}’-y|+r)^{-n_{\mathrm{S}}n}$.

Hence we have by H\"older’s inequality$J_{2}$$=$$\int_{\{y\in D_{0}:|x0-y|\geqq 3s/2\}}g(y)y_{n}^{-\gamma/p}I_{2}(y)dy$$\leqq$$Ms^{n} \int_{D_{0}}g(y)y_{n}^{-\gamma/p}(|x_{0}’-y|+r)^{-n}dy$$\leqq$$g(y)^{p}dy)^{1/p}$$Ms^{n}( \int_{D_{0}}y_{n}^{-\gamma p’/p}(|x_{0}’-y|+r)^{-p’n}dy)^{1/p’}($Do$\leqq$$Ms^{n}r^{-\gamma/p-n/p}(D_{0}g(y)^{p}dy)^{1/p}$$\leqq$$Ms^{n}s^{1-n/p}r^{-1-\gamma/p}( \int_{D_{0}}g(y)^{p}dy)^{1/p}$,

since $p<n$ . If $|y-x_{0}|\geqq 3s/2$ and $y_{n}>x_{n}/2,$ then $|x-y|\geqq M(|x_{0}-y|+s)$ and$|\overline{x}-y|\geqq M(|x_{0}-y|+r)$ , so that
$I_{3}(y)$$=$$\int_{\{x\in B:y_{n}>x_{n}/2\}}|x-y|^{1-n}|\overline{x}-y|^{-1}dx$$\leqq$$M(|x_{0}-y|+s)^{1-n}(|x_{0}-y|+r)^{-1}s^{n}$.

Consequently, it follows that
$J_{3}$$=$$\int_{\{y\in D_{0}:|x_{0}-y|\geqq 3s/2,y_{n}>r/4\}}g(y)y_{n}^{-\gamma/p}I_{3}(y)dy$$\leqq$$Ms^{n} \int_{\{y\in D_{0}:|y-x_{0}|\geqq 3s/2,y_{n}>r/4\}}g(y)y_{n}^{-\gamma/p}(|x_{0}-y|+s)^{1-n}(|x_{0}-y|+r)^{-1}dy$.

Setting $t=|x_{0}-y|$ and $|(x_{0})_{n}-y_{n}|=t\cos\theta$ , we note that$(t+r)\cos\theta\leqq|(x_{0})_{n}-y_{n}|+(x_{0})_{n}\leqq 3y_{n}<3(r+t)$

when $y_{n}>r/4.$ Using H\"older’s inequality and applying the polar coordinates about
we have$J_{3}$$\leqq$$\leqq$$Ms^{n}( \int_{3s/2}^{\infty}(t+s)^{p’(1-n)}(t+r)^{p’(-\gamma/p-1)}t^{n-1}dt)^{1/p’}(\int_{D_{0}}g(y)^{p}dy)^{1/p}$$Ms^{n}s^{1-n/p}r^{-1-\gamma/p}( \int_{D_{0}}g(y)^{p}dy)^{1/p}$$x_{0}$,

104
since $p<n.$ Therefore we obtain$|B| \leqq Mr^{-\beta-\gamma/p}s^{(p-n)/p}s^{n}(\int_{D_{0}}g(y)^{p}dy)1/p$

Hence it follows from the definition of$C_{k_{\beta.\gamma},p}$that$r^{\beta p+\gamma_{S}n-p}\leqq MC_{k_{\beta,\gamma},p}(B;D_{0})$,as required.
To obtain the converse inequality, note that for $x\in B$$\int_{B}x_{n}^{1-\beta}|x-y|^{1-n}|\overline{x}-y|^{-1}y_{n}^{-\gamma/p}dy$$\geqq$$\geqq$$Mr^{-\beta-\gamma/p} \int_{B}|x-y|^{1-n}dy$$Mr^{-\beta-\gamma/p_{S}}$,so that$C_{k_{\beta,\gamma},p}(B;D_{0})$$\leqq$$Mr^{(\beta+\gamma/p)p_{S}-p} \int_{B}dy=Mr^{\beta p+\gamma_{S}n-p}$.

Thus the proof is completed.
Let $0\leqq\beta\leqq 1and-1<\gamma<p-1$ . Set$2^{j}\leqq|xj|<2^{j+1}$ and $0<rj\leqq 2^{j+1}.$ If$\mathrm{p}_{\mathrm{R}\mathrm{O}\mathrm{P}\mathrm{O}\mathrm{S}\mathrm{I}\mathrm{T}\mathrm{I}\mathrm{O}\mathrm{N}}3.$with $xj\in\partial D,$$\sum_{j=1}^{\infty}(\frac{r_{j}}{2^{j}})^{n+\gamma-(1-\beta)p}<\infty$

conversely,$V= \bigcup_{j=1}^{\infty}B(x_{j}, r_{j})\cap D$$V\mathrm{j}s(k\beta,\gamma’ p)$$\mathrm{j}f\gamma>(1-\beta)p$and (9) holds, then-thin at infinity, then;

(9)

$V\mathrm{j}s(k\beta,\gamma’ p)$

PROOF. First we show that if $B_{+}=B(x_{0}, r)\cap D$ with, then-thjn at infinity.
$x_{0}\in\partial D,$$1\leqq|x_{0}|<2$and$0<r\leqq 2$$r^{n+\gamma-(1-\beta)p}\leqq MC_{k_{\beta,\gamma\prime}p}(B_{+}; D_{0})$

Let

$g$

be anonnegative measurable function such that.

$g=\mathrm{O}$(10)outside$D_{0}$

and$\int_{D}k_{\beta,\gamma}(x, y)g(y)dy\geqq 1$

for every $x\in B_{+}$ . Then we have by Fubini’s theorem$|B_{+}|$$\leqq$$\int_{B_{+}}(\int_{D_{0}}k_{\beta,\gamma}(x, y)g(y)dy)dx$$=$$\int_{D_{0}}g(y)y_{n}^{-\gamma/p}(\int_{B}x_{n}^{1-\beta}|x-y|^{1-n}|\overline{x}-y|^{-1}dx)+dy$.

105Here we see that if $|x_{0}-y|>2r$ , then$\int_{B}x_{n}^{1-\beta}|x-y|^{1-n}|\overline{x}-y|^{-1}dx\leqq M|x_{0}-y|^{-n}r^{1-\beta+n}+$

and that if $|x_{0}-y|\leqq 2r$ , then$\int_{B_{+}}x_{n}^{1-\beta}|x-y|^{1-n}|\overline{x}-y|^{-1}dx$$\leqq$$\leqq$$Mr^{1-\beta} \int_{B_{+}}|x-y|^{1-n}(|x-y|+y_{n})^{-1}dx$$Mr^{1-\beta}\log(4r/y_{n})$.

Then we have by H\"older’s inequality$J_{1}$$=$$r^{1-\beta} \int_{\{y\in D_{0}:|x\mathit{0}-y|\leqq 2r\}}g(y)y_{n}^{-\gamma/p}\log(4r/y_{n})dy$$\leqq$$r^{1-\beta}( \int_{\{y\in D_{0}:|x_{0}-y|\leqq 2r\}}\{\log(4r/y_{n})\}^{p’}y_{n}^{-\gamma p’/p}dy)^{1/p’}(\int_{D_{0}}g(y)^{p}dy)^{1/p}$$\leqq$$Mr^{1-\beta-\gamma/p+n/p’}( \int_{D_{0}}g(y)^{p}dy)^{1/p}$$=$$r^{1-\beta+n} \int_{\{y\in D_{0}:|x0-y|>2r\}}g(y)y_{n}^{-\gamma/p}|x_{0}-y|^{-n}dy$$\leqq$$r^{1-\beta+n}( \int_{\{y\in D_{0}:|x0-y|>2r\}}y_{n}^{-\gamma p’/p}|x_{0}-y|^{-p’n}dy)^{1/p’}(\int_{D_{0}}g(y)^{p}dy)^{1/p}$$\leqq$$Mr^{1-\beta-\gamma/p+n/p’}( \int_{D_{0}}g(y)^{p}dy)^{1/p}$

and$J_{2}$

Therefore we have$|B_{+}|| \leqq Mr^{1-\beta-\gamma/p+n/p’}(\int_{D_{0}}g(y)^{p}dy)^{1/p}$

so that it follows from the definition of$C_{k_{\beta,\gamma},p}$

,

that

$r^{n+\gamma-(1-\beta)p}\leqq MC_{k_{\beta,\gamma\prime}p}(B_{+}; D_{0})$,as required.

To obtain the converse inequality, note that for $x\in B_{+}$$\int_{B_{+}}x_{n}^{1-\beta}|x-y|^{1-n}|\overline{x}-y|^{-1}y_{n}^{-\gamma/p}dy$$\geqq$$\geqq$$\geqq$$\int_{B\cap B(x,x_{n}/2)}x_{n}^{1-\beta}|x-y|^{1-n}|\overline{x}-y|^{-1}y_{n}^{-\gamma/p}dy+$$Mx_{n}^{1-\beta-1-\gamma/p} \int_{B\cap B(x,x_{n}/2)}|x-y|^{1-n}dy+$$Mx_{n}^{1-\beta-\gamma/p}\geqq Mr^{1-\beta-\gamma/p}$,

106since $1-\beta<\gamma/p$ . Hence it follows from the definition of$C_{k_{\beta,\gamma},p}(B_{+}; D_{0})$$\leqq$that$C_{k_{\beta,\gamma},p}$$Mr^{-(1-\beta)p+\gamma} \int_{B_{+}}dy=Mr^{n+\gamma-(1-\beta)p}$.

Thus the proof is completed.

For anondecreasing function
apositive constant $M$ , we set$\varphi$on$\mathrm{R}^{1}$such that $0<\varphi(2t)\leqq M\varphi(t)$ for$T_{\varphi}=\{x=(x’, x_{n});0<x_{n}<\varphi(|x’|)\}$$t>\mathrm{O}$

with

.

PROPOSITION 4(cf. Aikawa [1, Proposition 5.1]). Let $0<\beta\leqq 1$ and $p(1-\beta)-1<$
Assume further that$\gamma<p-1$ .

$\lim\underline{\varphi(r)}=0$$rarrow\infty$Then$T_{\varphi}$is$(k\beta,\gamma’ p)$

.

(11)

$r$

-thin at inBnity if and only if$\int_{1}^{\infty}(\frac{\varphi(t)}{t})^{p(-1+\beta)+\gamma+1}\frac{dt}{t}<\infty$

.

(12)

For example, $\varphi(r)=r[\log(1+r)]^{-\delta}$ satisfies (12), when $\delta\{p(-1+\beta)+\gamma+1\}>1$ .

5Limits of monotone functions
Finally we consider the limits at infinity for monotone BLD functions. Acontinuousfunction is called monotone on in the sense of Lebesgue (see [5]) if for every relatively
compact open subset of ,$u$$\mathrm{D}$$G$$\mathrm{D}$$G \cup\partial G\partial G\mathrm{m}\mathrm{a}\mathrm{x}u=\max u$and$\min_{G\cup\partial G}u=\min_{\partial G}u$.

For examples and fundamental properties of monotone functions, see [12] and [16].
Among them the following result is only needed for monotone functions.

LEMMA 1. If is amonotone $BLD$ function on $B(x, 2r)$ and $p>n-1$ , then$u$$|u(z)-u(x)|^{p} \leqq Mr^{p-n}\int_{B(x,2r)}|\nabla u(y)|^{p}dy$

for every $z\in B(x, r)$ .

(13)

107
THEOREM 5. Let $p>n-1,$ $-1<\gamma<p-1$ and $n+\gamma-p\geqq 0$ . Iffunction on $D$ satisfying (1) then there exist anumber such that$u$is a monotone$A$$\rangle$$\lim_{|x|arrow\infty,x\in D}x_{n}^{(n+\gamma-p)/p}[u(x)-A]=0$

in case $n+\gamma-p>\mathrm{O}$ and$\lim_{|x|arrow\infty,x\in D}(\max\{\log(1/x_{n}), \log|x|\})^{-1/p’}[u(x)-A]=0$

in case $n+\gamma-p=0$ .
denote the distance
PROOF. For $x\in D$ , let $r=|x|,$ $C(x)=(0, \ldots, 0, r)$ andcovering $\{Bj\}$ ,takeafinite.
We, that is,from the boundaryof, such that$\rho_{\mathrm{D}}(x)$$\rho_{\mathrm{D}}(x)=x_{n}$$\partial D$$x\in \mathrm{D}$$B_{j}=B(X_{J}, 4^{-1}\rho_{\mathrm{D}}(X_{j}))$(i) $X_{1}=x$ and $X_{N+1}=C(x)$ ;(ii) $r/2<|z|<2r$ for $z \in A(r)=\bigcup_{j}2B_{j}$ , where $2B_{j}=B(Xj, 2^{-1}\rho \mathrm{D}(Xj))$ ;(iii)$B_{j}\cap B_{j+1}\neq\emptyset$(iv)$\sum_{j}\chi_{2B_{j}}$for each ;$j$is bounded, where$\chi_{A}$denotes the characteristic function of$A$.

By the monotonicity of , we see that$u$$|u(y)-u(X_{j})| \leqq M\rho_{\mathrm{D}}(X_{j})^{(p-n)/p}\int_{2B_{j}}|\nabla u(z)|^{p}dz$

for $y\in B_{j}$ . First suppose $n+\gamma-p>0$ . Using Theorem 1, we can find anumber
such that $C_{1}(x)\in B_{N+1}$ and

$A$

and

$C_{1}(x)$$\lim_{|x|arrow\infty}|x|^{(n+\gamma-p)/p}[u(C_{1}(x))-A]=0$.

Then we have by H\"older’s inequality$|u(x)-A|$$\leqq$$|u(X_{1})-u(X_{2})|+|u(X_{2})-u(X_{3})|+\cdots+|u(X_{N})-u(X_{N+1})|$$+|u(X_{N+1})-u(C_{1}(x))|+|u(C_{1}(x))-A|$$\leqq$$\leqq$$M \sum_{j}\rho_{\mathrm{D}}(X_{j})^{(p-n-\gamma)/p}(\int_{2B_{j}}|\nabla u(z)|^{p}\rho_{\mathrm{D}}(z)^{\gamma}dz)^{1/p}+|u(C_{1}(x))-A|$$M( \sum_{j}\rho_{\mathrm{D}}(X_{j})^{p’(p-n-\gamma)/p})^{1/p’}(\int_{A(r)}|\nabla u(z)|^{p}\rho_{\mathrm{D}}(z)^{\gamma}dz)^{1/p}$$+|u(C_{1}(x))-A|$$\leqq$$Mx_{n}^{(p-n-\gamma)/p}( \int_{\mathrm{D}-B(0,r/2)}|\nabla u(z)|^{p}\rho_{\mathrm{D}}(z)^{\gamma}dz)^{1/p}+|u(C_{1}(x))-A|$,

108
which proves$\lim_{|x|arrow\infty}x_{n}^{(n+\gamma-p)/p}[u(x)-A]=0$,

as required.
The case $n+\gamma-p=\mathrm{O}$ can be treated similarly.

References[1] H. Aikawa, On the behavior at infinity of non-negative superharmonic functions inahalf space, Hiroshima Math. J. 11 (1981), 425-441.
[2] J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier 5(1955), 305-370.

[3] M. Ess\’en and H.L.Jackson, On the covering properties of certain exceptional sets
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[4] T. Kurokawa and Y. Mizuta, On the order at infinity of Riesz potentials, Hiroshima
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[5] H. Lebesgue, Sur le probl\’eme de Dirichlet, Rend. Circ. Mat. Palermo 24 (1907),

371-402.
[6] J. Lelong-Ferrand, Etude au voisinage de la fronti\‘ere des fonctions surharmoniques
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[7] N. G. Meyers, Atheory of capacities for potentials in Lebesgue classes, Math.
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[8] I. Miyamoto and H. Yoshida, Two criterions of Wiener type for minimally thin sets
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[9] Y. Mizuta, On the behavior at infinity of Green potentials in ahalf space, Hiroshima
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[10] Y. Mizuta, On semi-fine limits of potentials, Analysis 2(1982), 115-139.

[11] Y. Mizuta, Continuity properties of potentials and Beppo-Levi-Deny functions,
Hiroshima Math. J. 23 (1993), 79-153.

[12] Y. Mizuta, Potential theory in Euclidean spaces, Gakk\={o}tosho, Tokyo, 1996.
[13] Y. Mizuta, Integral representations, differentiability properties and limits at infinity
for Beppo Levi functions, Potential Analysis 6 (1997), 237-276.
[14] Y. Mizuta and T. Shimomura, Minimally fine limits at infinity for -precice functions, to appear in J. Math. Soc. Japan.
$p$

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[15] M. Ohtsuka, Extremal length and precise functions in 3-space, Lecture Notes at
Hiroshima University, 1972.
[16] M. Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in
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[17] W. P. Ziemer, Extremal length as acapacity, Michigan Math. J. 17 (1970), 117-128.

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