数理解析研究所講究録1116 巻 1999 年 51-68

51

Hyperplane integrals of BLD and monotone BLD functions
広島大総合科学部
明石工業高専1
Let

水田義弘 (Yoshihiro Mizuta)
下村哲 (Tetsu Shimomura)

Introduction$\mathrm{D}$denote the half space$\mathrm{D}=\{x=(x’, x_{n})\in \mathrm{R}^{n-1}\cross \mathrm{R}^{1} :x_{n}>0\}$and set$\mathrm{H}=\partial \mathrm{D}$we sometimes identify$H_{q}(u)$over$\mathrm{H}$$x’\in \mathrm{R}^{n-1}$

with

;

$(x’, \mathrm{O})\in \mathrm{H}$. We define the$q\mathrm{t}\mathrm{h}$hyperplane integralby$H_{q}(u)=( \int_{\mathrm{H}}|u(X’)|^{q}dx’)^{1}/q$

for a measurable function on H.
Our main aim in this note is to study the existence of limits of
where
$u$

$H_{q}(U_{r})$at $r=0$ ,$U_{r}(x)’=u(x’, r)- \sum_{k=0}\frac{r^{k}}{k!}[(\partial/\partial X_{n}m-1)k]u(X’, 0)$

for quasicontinuous Sobolev functions

$u$

on

$\mathrm{D}$, where the vertical limits$( \partial/\partial X_{n})^{k}u(X’, 0)=\lim_{x_{n^{arrow}}0}(\partial/\partial X_{n})^{k}u(x’, Xn)$exist for almost all
and $0\leqq k\leqq m-1$ (see [15, Theorem 2.4]). Moreprecisely, we show (in Theorem 3.1 below) that$x’=(x^{l}, 0)\in \mathrm{H}$$\lim_{rarrow 0}r^{-\omega}H_{q}(U_{r})=0$

for some $\omega>0$ .
Consider the Dirichlet problem for polyharmonic operator$\Delta^{m}u(x)$$=0$with the boundary conditions$(\partial/\partial x_{n})^{k}u(X’, \mathrm{o})$$=$$f_{k}(x’)$$(k=0,1, \ldots, m-1)$ .

52We show (in Corollary 3.1 below) that if $1<p\leqq q<\infty,$ $n/p-(n-1)/q<1$ andforis a solution of the Dirichlet problem with
$0\leqq k\leqq m-1$ , then$f_{k}(x^{l})=(\partial/\partial X_{n})^{k}u(x’, 0)$$u\in W^{m,p}(\mathrm{D})$$\lim_{rarrow 0}r^{n/p-(}n-1)/q-mH(q)U_{r}=0$,

where $U_{r}(x’)=u(x’, r)- \sum_{k=0}\frac{r^{k}}{k!}fm-1k(X’)$ .

To prove our results, we apply the integral representation in $[12, 15]$ . For this purdefined bypose, we are concerned with $K$-potentials$U_{K}f$$U_{K}f(x)= \int K(x-y)f(y)dy$

for functions

$f$

on

$\mathrm{R}^{n}$satisfying the weighted$\int_{\mathrm{R}^{n}}|f(y)|^{p}|yn|\beta dy<\infty$$L^{p}$

condition :$-1<\beta<p-1$ .

,for a multiIn connection with our integral representation, $K(x)$ is of the form
index with length . Our basic fact is stated as follows (see Theorem 2.1 below):$x^{\lambda}|x|^{-}n$$m$$\lambda$$\lim_{rarrow 0}r^{n/p-(n-}1)/q-mH(q)u_{r}=0$

where$u_{r}(x’)=U_{K}f(X’, r)-m \sum_{k=0}^{-}\frac{r^{k}}{k!}[(\partial/\partial Xn)^{k}U_{K}f](1x)/$,.

In Section 4, we give growth estimates of higher differences of Sobolev functions.
In the final section, we study the existence of limits of hyperplane integrals formonotone BLD functions on satisfying
$u$

(1.1)

$\mathrm{D}$$\int_{\mathrm{D}}|\nabla u(X)|^{p}x^{\beta}dxn<\infty$$p>n-1$ ,,denotes the gradient, $1<p<\infty$ and $-1<\beta<p-1$ ; see Section 5 forwhere
the definition of monotone functions. We here note that harmonic functions are monotone, $A$-harmonic functions and hence coordinate functions of quasiregular mappings aremonotone (see [4] and [20]), and thus the class of monotone functions is considerably
wide.
For related results, see Gardiner [2], Stoll [24, 25, 26], the first author [11, 12, 16]
and the authors $[17, 18]$ .
$\nabla$2Hyperplane integrals of potentials

For a multi-index$\lambda$

and $\ell>0$ , set$K(X)= \frac{x^{\lambda}}{|x|^{\ell}}$.

53

We define the $K$-potential$U_{K}f$by$U_{K}f(x)= \int_{\mathrm{R}^{n}}K(x-y)f(y)dy$

for a measurable function
(2.1)

$f$

on

$\mathrm{R}^{n}$satisfying$\int_{\mathrm{R}^{n}}(1+|y|)^{|\lambda|\ell}-|f(y)|dy<\infty$

and
(2.2)

$\int_{\mathrm{R}^{n}}|f(y)|^{p}|y_{n}|\beta dy<\infty,$$y=(y_{1}, \cdots, y_{n})$.

In particular, $K$ is the Riesz -kernel whenand $\ell=n-\alpha$ . In this case,is$\alpha=|\lambda|-l+n$
written aswith. Note here that (2.1) is equivalent to the condition$\lambda=0$$\alpha$$U_{K}f$$U_{\alpha}f$

that

(2.3)

$U_{\alpha}|f|\not\equiv\infty$.

Throughout this paper, let $M$ denote various constants independent of the variables

in question.
For a nonnegative integer $m$ , consider$K_{m}(x, y)=K(x-y)- \sum_{=k0}\frac{X_{n}k}{k!}[(\partial/\partial_{X_{n})^{k}](_{X}-}K/y)m$where$x=(x’, x_{n})\in \mathrm{R}^{n-1}\cross \mathrm{R}$; we sometimes identify

LEMMA 2.1 ([19, Lemma 2.1]). Let
$m+1$

.

$m$

$x’$

,

with $(x’, 0)$ .

be a nonnegative in teger such that$|\lambda|-\ell<$(1) $If|x’-y|\geqq x_{n}/2>0$ and $|x-y|\geqq x_{n}/2>0$ , then$|K_{m}(x, y)|\leqq MX_{n}|X’-y|^{||-}m+1\lambda\ell-m-1$.

(2) $If|x-y|<x_{n}/2$ , then $|K_{m}(x, y)|\leqq M(x_{n}^{|\lambda|-\ell}+|x-y|^{1}\lambda|-\ell)$ ,(3) $If|x’-y|<x_{n}/2$ , then $|K_{m}(x, y)|\leqq M(x_{n}^{|\lambda|-\ell}+x_{n}^{m}|x’-y||\lambda|-\ell-m)$ .

For a point
and radius .

$x\in \mathrm{R}^{n}$

and $r>0$ , we denote by $B(x, r)$ the open ball with center at

$x$

$r$

$be$

LEMMA 2.2 (cf. [16, Lemma 3.2]). Let $\beta>-1,$ $q>0$ and $|\lambda|-\ell+n/q>0$ . Leta nonnegative integer such th at$m<| \lambda|-\ell+\frac{n+\beta}{q}<m+1$

.

$m$

54

Then
$( \int|K_{m}(x, y)|^{q}|yn|^{\beta}dy)1/qn\leqq Mx_{n}^{1}-\ell(+\lambda|+\beta)/q$

for all $x=(x’, x_{n})\in \mathrm{D}$ .
PROOF. For fixed$x\in \mathrm{D}$, consider the sets

$E_{1}=B(x, x_{n}/2)$ ,$E_{2}=B(x’, x_{n}/2)$,$E_{3}=\mathrm{D}-(E_{1}\cup E_{2})$.

Since $|\lambda|-l+(n+\beta)/q-m-1<0$ , applying the polar coordinates about
by Lemma 2.1(1)$( \int_{E_{3}}|K_{m}(X, y)|q|yn|\beta dy)^{1}/q$$x’$, we have$\leqq$$Mx_{n}^{m+1}( \int_{E_{3}}|X’-y|(|\lambda|-\ell-m-1)q|yn|^{\beta}dy)^{1}/q$$\leqq$$Mx_{n}^{m+1}( \int_{x_{\mathfrak{n}}/2}^{\infty}r-m-)q+\beta n-r1dr)(|\lambda|-\ell 11/q$$=$$Mx_{n}^{|\lambda|+}-\ell(n+\beta)/q$.

Similarly, since $|\lambda|-\ell+n/q>0$ , we have by Lemma 2.1(2)$( \int_{E_{1}}|K_{m}(_{X}, y)|q|y_{n}|^{\beta}dy)1/q$$\leqq$$=$$Mx_{n}^{\beta/q}( \int_{E_{1}}(x_{n}^{1}-+|\lambda|\ell-yX|^{||-\ell}\lambda)qdy)1/q$$Mx_{n}^{|\lambda|+}-\ell(n+\beta)/q$.

Finally, since $|\lambda|-\ell+(n+\beta)/q-m>0$ , we obtain by Lemma 2.1(3)$( \int_{E_{2}}|K_{m}(x, y)|q|_{Jn}\tau|^{\beta}dy)1/q$$\leqq$$M( \int_{E_{2}}(x_{n}^{|\lambda|m}-\ell+x_{n}|_{X}/-y||\lambda|-l-m)^{q}|yn|^{\beta}dy)1/q$$\leqq$$Mx_{n}^{|\lambda|+}- \ell(n+\beta)/q+MX_{n}^{m}(\int_{0}^{x_{n}/2}r^{(1}-m\lambda|\ell_{-})q+\beta dr^{n-1}r\mathrm{I}^{1/q}$$=$$Mx_{n}^{|\lambda|}-l+(n+\beta)/q$.

The required inequality now follows.
LEMMA 2.3 ( . $.[16$ , Lemma 3.4]). Let $q>0$ and $m$ be a nonnegative in teger suchthat$m<| \lambda|-\ell+\frac{n-1}{q}<m+1$ .
$\mathrm{c}\mathrm{f}$If $x=(x’, x_{n})\in D$ and $y=(y’, y_{n})\in R^{n}$ , then$( \int_{\mathrm{R}^{n-1}}|K(mx, y)|qdx)’\leqq M_{X}m+1(_{X+|y}nnn|)|\lambda|1/q)-p_{-m}-1+(n-1/q$.

55

PROOF. Let $x=(x’, x_{n})\in \mathrm{D}$ and. If$|\lambda|-\ell-m-1+(n-1)/q<0$ , we have by Lemma 2.1(1)$y=(y^{l}, y_{n})\in \mathrm{R}^{n}$($\int_{\mathrm{R}^{n-1}}|K_{m}(X, y)|qdx^{\prime)^{1}}/q$$|y_{n}|\geqq 2x_{n}$, then, since$\leqq$$Mx_{n}^{m+1}( \int_{\mathrm{R}^{n-1}}|x’-y|^{(}|\lambda|-\ell-m-1)qdX’)^{1}/q$$=$$Mx_{n}^{m+1}( \int_{0}^{\infty}(r^{2}+y_{n^{2}})(|\lambda|-l-m-1)q/2r-dn2r)^{1/q}$$=$$M_{X_{n}^{m}|}+1y_{n}|^{|\lambda|+}-\ell_{-}m-1(n-1)/q$.

If $|y_{n}|<2x_{n}$ , then we have as in the proof of Lemma 2.1$( \int_{\mathrm{R}^{n-1}}|K(xm’ y)|^{q}dX’)^{1/}q\leqq M(\int_{\{x’:}(x’,xn)\in E_{1}\})^{1/}(x_{n}-+||\lambda|\ell-y|^{1}x\lambda|-\ell)qdX’q$$+M( \int_{\{x’:}(x’,xn)\in E2\})^{1/}(x^{|\lambda|-}+x_{n}^{m}|n\ell X’-y|^{1}\lambda|-l-m)^{q}d_{X}/q$$+Mx_{n}^{m+1}( \int_{\{x’:}(xxn)’,\in E_{3}\}||x-/y-\ell-m-1qd(|\lambda|)X^{\prime)^{1}}/q$$\leqq$$M_{X_{n}+}| \lambda|-\ell+(n-1)/qM(\int_{B(y’,x/2)}n|_{X}/-y/|(|\lambda|-l)qdX)^{1}//q$$+Mx_{n}^{m}( \int_{B(yx_{n}/2)}’,)^{1}|X-/y’|^{(|\lambda}|-\ell-m)qdx//q$$+Mx_{n}^{m+1}( \int_{\mathrm{R}^{n-1}}(x_{n}+|_{X’}-y|’)(|\lambda|-\ell-m-1)qdx)^{1}’/q$$=$$Mx_{n}^{|\lambda|\ell+(}-n-1)/q$.

Therefore the required inequality now follows.
LEMMA 2.4 (cf. [1, Theorem 13.5], [15, Section 6.5]). Let $\alpha=|\lambda|-\ell+n,$ $\alpha p>1$ ,$\alpha p>1+\beta \mathrm{a}nd-1<\beta<p-1$ . If
is a measurable function onand (2.3), thenhas thproperty, in particular, $U_{K}f(XX_{n}/,)$ is a bsolutely
continuo us on for almost every. Moreover, in case $m$ is a positive integer
such that $(\alpha-m)p>1$ and $(\alpha-m)p>1+\beta$ ,$f$$U_{K}f$$\mathrm{e}(\mathcal{A}CL)$$\mathrm{R}^{n}\mathrm{s}at\mathrm{j}\mathrm{s}6\Gamma ing(\mathit{2}.\mathit{2})$$\cdot$$x’\in \mathrm{R}^{n-1}$$\mathrm{R}$$( \partial/\partial X_{n})^{m}U_{K}f(X’, X_{n})=\int(\partial/\partial_{X_{n})^{m_{K(x}}}-y)f(y)dy$

is a bsolu tely contin us on$\mathrm{u}o$$\mathrm{R}$for almost every$x’\in \mathrm{R}^{n-1}$.

THEOREM 2.1 (cf. [11, Theorem 2.1] and [16, Theorem 2.1]). Let $\alpha=|\lambda|-\ell+n$
sa tisfy $m+1/p<\alpha<m+n$ . Let $1<p\leqq q<\infty,$ $\beta<p-1$ and$\frac{n-\alpha p}{p(n-\alpha)}<\frac{n-1}{q(n-\alpha+m)}$

when $n-\alpha>0$ .

56

Fur ther suppose $m<\omega<m+1$ , where $\omega=(n-1)/q-(n-\alpha p+\beta)/P$. If
sa tisfying (2.2) and (2.3), thennonnegati measurable function on

$f$

is a

$\mathrm{R}^{n}$$\mathrm{r}^{r}e$,$\lim_{rarrow 0}r^{-\omega}H_{q}(u_{\Gamma})=0$

where$u_{r}(x’)=U_{K}f(x’, r)- \sum_{0k=}\frac{r^{k}}{k!}[m(\partial/\partial Xn)kUKf](x’, \mathrm{o})$PROOF. Under the assumptions onsuch that$p,$$\alpha,$$\beta,$$q$.

and $m$ in Theorem 2.1, we can take$(\delta, \gamma)$$\beta<\gamma<p(n-\alpha+m+1)\delta+\beta-p(n-1)/q$ ,

(2.4)

(2.5)

$p(n-\alpha+m+1)\delta+(\alpha-m-1)p-n<\gamma<p(n-\alpha+m)\delta+(\alpha-m)_{P}-n$ ,(2.6)$\beta<\gamma<p-1$

(2.7)

,

$0<\delta<1$ ,$\delta p(n-\alpha)>n-\alpha p$

and
(2.8)

Set

$\frac{n-1}{q(n-\alpha+m+1)}<\delta<\frac{n-1}{q(n-\alpha+m)}$$a=(1-\delta)p\mathrm{a}\mathrm{n}\mathrm{d}/b=-\gamma p’/p$.

. Then, by (2.6) and (2.7),$b>-1$

and$\alpha-n+\frac{n}{a}>0$.

Further, (2.5) implies$m< \alpha-n+\frac{n+b}{a}<m+1$ .

We first note from Lemma 2.4 that$u_{x_{n}}(x’)$$U_{K}f(x)- \sum_{=k0}\frac{x_{n}^{k}}{k!}[m(\partial/\partial Xn)kU_{K}f](x/, 0)$$=$$\int K_{m}(x, y)f(y)dy$.

Using H\"older’s inequality and Lemma 2.2, we have$|u_{x_{n}}(_{X’})|$$\leqq$$( \int|K_{m}(x, y)|a|y_{n}|bdy)^{()/}1-\delta a(\int|K_{m}(x, y)|\delta pf(y)^{p}|y_{n}|^{\gamma}dy)1/p$$\leqq$$Mx_{n}^{(\alpha}-n)(1- \delta)+n/p’-\gamma/p(\int|K_{m}(x, y)|\delta pf(y)^{p}|y_{n}|\gamma dy)1/p$

57

In view of Minkowski’s inequality for integral we have$H_{q}(u_{x_{n}})$$Mx_{n}^{()(}\alpha-n1-\delta)+n/p’-\gamma/p$$\leqq$$\cross\{\int(\int_{R}n-1|K_{m}(_{X}, y)|\delta qdX/)p/qf(y)^{p}|yn|^{\gamma}dy\}^{1}/p$

Here, noting (2.8), we have by Lemma 2.3$( \int_{\mathrm{R}^{n-1}}|K_{m}(x, y)|\delta qd_{X}/)p/qn\leqq M[Xm+1(x_{n}+|yn|)\alpha-n-m-1+(n-1)/\delta q]\delta p$.

Consequently$H_{q}(u_{r})$$\leqq$$Mr^{(\alpha-n)}(1-\delta)+n/p^{J}-\gamma/p+(m+1)\delta$$\cross\{\int[(r+|y_{n}|)\alpha-n-m-1+(n-1)/\delta q]^{\delta}p|yn|\gamma-\beta f(y)^{p}|yn|^{\beta}dy\}1/p$

Consider the function$k(r, y_{n})$$=r^{p[(}-n-)/q]r^{p}n-\alpha p+\beta)/p(1[(\alpha-n)(1-\delta)+n/p’-\gamma/P+(m+1)\delta]$$\cross[(r+|y_{n}|)\alpha-n-m-1+(n-1)/\delta q]^{\delta}p|yn|^{\gamma-}\beta$.

Then$r^{-\omega}H_{q}(u_{r}) \leqq M\{\int k(r, y_{n})f(y)p|y_{n}|\beta dy\}1/p$,

where $\omega=(n-1)/q-(n-\alpha p+\beta)/p$. It follows from (2.4) that$r^{-\omega}r^{(-n}\alpha)(1-\delta)+n/p’-\gamma/\mathrm{P}+(m+1)\delta(n=r-\alpha+m+1)\delta+(\beta-\gamma)/p-(n-1)/qarrow 0$

as$rarrow \mathrm{O}$. If $r<|y_{n}|$ , then$k(r, y_{n})\leqq M(r/|y_{n}|)^{(}n-\alpha+m+1)\delta p+(\beta-\gamma\rangle$if$|y_{n}|\leqq r$$-p(n-1)/q\leqq M$ ;, then.

$k(r, y_{n})\leqq M(|y_{n}|/r)^{\gamma-\beta}\leqq M$

Hence Lebesgue’s dominated convergence theorem implies that$\lim_{rarrow 0}r^{-\omega}H_{q}(u_{r})=0$

Now the proof of Theorem 2.1 is completed.

.

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Sobolev functions

3

For an open set$G\subset \mathrm{R}^{n}$, we denote by $BL_{m}(L_{loc}^{p}(c))$ the Beppo Levi space$BL_{m}(L_{l}^{p_{OC}}(c))=\{u\in L_{loC}^{p}(G):D\lambda L^{p}(u\in loc)G(|\lambda|=m)\}$satisfying(see [15]). In view of [15], each$u\in BL_{m}(L_{\iota oc}^{p}(\mathrm{D}))$(3.1)$\int_{\mathrm{D}}|\nabla_{m}u(X)|pX\beta ndX<\infty$

has an $(m,p)$ -quasicontinuous representative , where $| \nabla_{m}u(X)|=(\sum_{|\mu|=m}|D^{\mu}u(X)|^{2})1/2$ ,is given by$1<p<\infty$ and-l $<\beta<p-1$ . Moreover,$\tilde{u}$$\tilde{u}$$\tilde{u}(x)=\sum_{=|\lambda|m}a_{\lambda}\int\tilde{K}_{\lambda,m}(x, y)D^{\lambda}\overline{u}(y)dy+P(X)$

where

is an extention of$\overline{u}$$K_{\lambda}(x)=x^{\lambda}|x|^{-}n$

$u$

to

$\mathrm{R}^{n},$$P(x)$,

is a polynomial of degree at most $m-1$ ,and$\tilde{K}_{\lambda,m}(x, y)=$Note further from Lemma 2.4 that for each$x’\in \mathrm{R}^{n-1}$$k$with $0\leqq k\leqq m-1$ and for almost every,$( \partial/\partial_{X_{n}})^{k}\int\tilde{K}_{\lambda,m}(x, y)D^{\lambda}\overline{u}(y)dy=\int(\partial/\partial X_{n})^{k}\tilde{K}_{\lambda},m(X, y)D^{\lambda}\overline{u}(y)dy$

holds forSince

$x_{n}\in \mathrm{R}$, where $x=(X’, X_{n})$ .

$Q(x)- \sum_{k=0}^{m-1}\frac{r^{k}}{k!}[(\partial/\partial_{X_{n})^{k}Q]}(x’)=0$

for any polynomial$Q$of degree at most

$m-1$ , we have$U(x)$$\equiv$$=$

for$\mathrm{q}.\mathrm{e}$.

$x\in \mathrm{D}$$\tilde{u}(x)-\sum_{k=0}^{m-1}\frac{X_{n}k}{k!}(\partial/\partial Xn)k\tilde{u}(X’)$$\sum_{|\lambda|=m}a_{\lambda}\int K_{\lambda,m}(x, y)D^{\lambda}\overline{u}(y)dy=\tilde{u}(X)-P(X)$, where $K_{\lambda,m}(x, y)=K_{\lambda}(X-y)-m- \sum_{k=0}1\frac{x_{n^{k}}}{k!}[(\partial/\partial X_{n})^{k}K\lambda](x-/y)$.

Theorem 2.1 now gives the following result.

THEOREM 3.1 ($\mathrm{c}\mathrm{f},$$[19$, Theorem 3.1]). Let $1<p\leqq q<\infty$ ,$\frac{n-mp}{p(n-m)}<\frac{1}{q}$when $n-m>0$

59

and
$\frac{n-p+\beta}{p(n-1)}<\frac{1}{q}<\frac{n+\beta}{p(n-1)}$

If$\mathrm{D}$$u\in BL_{m}(L^{p}(loC\mathrm{D}))$, then.

satisfying (3.1) for-l $<\beta<p-1$ is $(m,p)$ -quasicontin uous on$\lim_{rarrow 0}r^{(n-mp+}\beta)/p-(n-1)/qH(qU_{r})=0$

where$U_{r}(x’)=u(x’, r)-m-1 \sum\frac{r^{k}}{k!}[(k=0\partial/\partial_{X_{n}})ku](x’, 0)$,.

Consider the Dirichlet problem for polyharmonic operator:$\Delta^{m}u(X)=0$

with the boundary conditions$(\partial/\partial x_{n})^{k}u(X’, \mathrm{o})=f_{k}(X’)$$(k=0,1, \ldots, m-1)$.

We denote by $W^{m,p}(G)$ the Sobolev space$W^{m,p}(G)=\{u\in L^{p}(G) : D^{\lambda}u\in L^{p}(G)(|\lambda|\leqq m)\}$(see Stein [23, Chapter 6]). If
is a solution of the Dirichlet problem,
then the vertical limit
exists for almost every
and$0\leqq k\leqq m-1$ (see [12], [14]).
We also see that every function in
can be extended to a function in(see Stein [23]). Hence Theorem 3.1 gives the following result.
$u\in W^{m,p}(\mathrm{D})$$(\partial/\partial x_{n})ku(x’, 0)$$x’=(X’, \mathrm{o})\in\partial \mathrm{D}$$W^{m,p}(\mathrm{D})$$W^{m,p}(\mathrm{R}^{n})$COROLLARY 3.1. Let $1<p\leqq q<\infty$ and$(0<) \frac{n}{p}-\frac{n-1}{q}<1$.

is a solu tion of the Dirichlet problem with, then$Ifu\in W^{m,p}(\mathrm{D})$$0\leqq k\leqq m-1$$\lim_{rarrow 0}r^{n/p-(n-}1)/q-mH_{q}(U_{r})=0$

where

4

$U_{r}(x’)=u(x’, r)- \sum_{k=0}\frac{r^{k}}{k!}fkm-1\wedge(x’)$$f_{k}(x’)=(\partial/\partial x_{n})^{k}u(X’, 0)$,.

Higher differences

For $r>0$ and a function , we define the first difference$u$$\Delta_{r}u(t)=\Delta r1u(t)=u(t+r)-u(t)$

for

60

and the m-th difference$\Delta_{r}^{m}u(t)=\Delta^{m-1}r(\Delta_{r}u(\cdot))(t)$.

It is easy to see that$\Delta_{r}^{m}u(t)=\sum^{m}k=0(-1)^{m-k}u(t+kr)$.

As in Section 2, we consider$K(X)= \frac{x^{\lambda}}{|x|^{\ell}}$and define$u_{r}(_{X}/)= \Delta_{r}m_{UK}f(X\cdot)/,(0)=\sum^{m}k=0(-1)^{m-k}U_{K}f(_{X’,kr})$.

THEOREM 4.1. Let $\alpha=|\lambda|-\ell+n,$ $1<p\leqq q<\infty,$ $\beta<p-1$ and$\frac{n-\alpha p}{p}<\frac{n-1}{q}$(when $n-\alpha>0$ ).

rther ppose $0<\omega<m$ , where $\omega=(n-1)/q-(n-\alpha p+\beta)/P$ . If is a nonnegative
satisfying (2.2) and (2.3), thenme as urable function on$F\mathrm{u}$$f$$s\mathrm{u}$$\mathrm{R}^{n}$$\lim_{rarrow 0}r^{-\omega}H_{q}(u_{r})=0$

where$u_{r}(x’)=\Delta_{r}^{m_{U_{K}}}f(x\cdot)/,(\mathrm{o})$,.

To prove this, we have only to prepare the following two lemmas instead of Lemmas
2.2 and 2.3.
LEMMA 4.1 ([19, Lemma 4.1]). Let $\beta>-1,$ $q>0$ and $|\lambda|-\ell+n/q>0$ . Let $m$ bea positi integer such th at$\mathrm{r}^{r}\mathrm{e}$$0<| \lambda|-\ell+\frac{n+\beta}{q}<m$.

Then$( \int|K_{m}^{*}(x, y)|^{q}|yn|\beta dy)1/qn\leqq Mxn|\lambda|-\ell+(+\beta)/q$

for all $x=(x’, x_{n})\in \mathrm{D}$ , where $K_{m}^{*}(x, y.)=\Delta_{x}mn..K(x-ly’, \cdot-y_{n})(0)$ for $x=(x’, x_{n})\in \mathrm{D}$.
and$y=(y’, y_{n})\in \mathrm{R}^{n}$

61

LEMMA 4.2 ([19, Lemma 4.2]). Let $q>0$ and$m$be a positive integer such that$0<| \lambda|-l+\frac{n-1}{q}<m$

If $x=(X’, X_{n})\in \mathrm{D}$ and$y=(y’, y_{n})\in \mathrm{R}^{n}$.

, then$( \int_{\mathrm{R}^{n-1}}|K_{m}*(x, y)|qdx’)^{1}/q|\leqq M_{X}m(nxn+|y_{n}|)\lambda|-\mathit{1}-m+(n-1)/q$.

THEOREM 4.2. Let $1<p\leqq q<\infty$ and$(0<) \frac{n}{p}-\frac{n-1}{q}<m$If$u\in BL_{m}(L^{p}(\mathrm{R}^{n}))$is $(m,p)$ -quasicontinuous on$\mathrm{R}^{n}$.

, then$\lim_{rarrow 0}r^{n/p-(n-}1)/q-mH_{q}(U_{r})=0$

where$U_{r}(x’)=\Delta_{r}^{m}u(X’, \cdot)(\mathrm{o})$In fact, since$\Delta_{r}^{m}Q=0$$U(x)$

where

5

,

for $r>0$ .

for any polynomial$Q$of degree at most $m-1$ , we have$\equiv$$\triangle_{x_{n}}^{m}u(x’, \cdot)(\mathrm{o})=||\sum_{\lambda=m}a_{\lambda}\int K_{\lambda,m}^{*}(x, y)D\lambda u(y)dy$$K_{\lambda,m}^{*}(x, y)=\triangle_{x_{n}}^{m}K_{\lambda(\cdot-y}X’-y’,n)(0)$with$K_{\lambda}(x)=x^{\lambda}|x|^{-}n$,.

Monotone functions

We say that a continuous function
Lebesgue ([6]), if both$u$$\mathrm{m}_{\frac{\mathrm{a}}{D}}\mathrm{x}u(x)=\max_{D}\partial u(x)$is monotone in an open set

and

$G$

, in the sense of$\mathrm{m}_{\frac{\mathrm{i}}{D}}\mathrm{n}u(x)=\min_{\partial D}u(x)$hold for every relatively compact open set $D$ with the closure.
The class of monotone functions is considerably wide. We give some examples ofmonotone functions.
$\overline{D}\subset G$

EXAMPLE 1. Let $f(r)$ be a non-increasing (or non-decreasing) continuous function. If we defineon$(0, \infty)$$u(x)=f(|x-\xi|)$

for $x\in G$ and$\xi\in\partial G$, then$u$

is monotone in

$G$

.

62

EXAMPLE 2. Harmonic functions on an open set are monotone in . More generally, solutions of elliptic partial differential equations of second order may be monotone(see Gilbarg-Rudinger [3]).
$G$

$G$

EXAMPLE 3. Weak solutions for variational problems may be monotone; in particular, weak solutions of the p–Laplacian are monotone. Moreover, if is a quasi-regularmapping on , then the coordinate functions of are monotone in . For these facts,
see Heinonen-Kilpel\"ainen-Martio[4], Reshetnyak [20], Serrin [21] and Vuorinen [27], [28].
$f$

$G$

$G$

$f$

A key result for monotone BLD functions is the following fact.
$u$

LEMMA 5.1 (cf. [5, Lemma 7.1], [7, Remark, p.9], $[28$ , Sect. 16]). Let $p>n-1$ . If
is a monotone $BLD$ fun tion on $B(x_{0},2r)$ , then$\mathrm{c}$whenever$|u(x)-u(y)|^{p} \leqq Mr^{p-n}\int_{B(x\mathrm{o},r)}2|\nabla u(Z)|^{p}dz$$x,$$y\in B(x_{0}, r)$.

This lemma is shown by the application of Sobolev’s inequality on the sphericalsurfaces, so that the restriction $p>n-1$ is needed; for a proof of Lemma 5.1, see forexample [5, Lemma 7.1] or [15, Theorem 5.2, Chap.8].

Hyperplane integrals of monotone functions

6

We define the$q\mathrm{t}\mathrm{h}$integral $H_{q,N}(u)$ over $\{x’ : |x’|<N\}$ by$H_{q,N}(u)=( \int_{\{x’:}|x’|<N\}||u(_{X’)}qdX’)1/q$

for a measurable function$u$

on $\{x’ : |x’|<N\}$ .

THEOREM 6.1 (cf. [17, Theorem 2]). Let(1.1). If $n-1<p<n+\beta,$ $p\leqq q<\infty$ and$u$be a monot one fun ction on$\frac{1}{q}<\frac{n-p+\beta}{p(n-1)}$$\mathrm{D}$satisfying,

then$\lim_{rarrow 0}r^{(n-p+}\beta)/p-(n-1)/qH_{q},N(u_{r})=0$,

where $u_{r}(x’)=u(x’, r)$ for $r>0$ .
be a monotone function on
If $|s-t|\leqq r<t/2$ , then Lemma 5.1 yields
PROOF. Let$u$$|H_{q,N}(u_{\mathit{8}})-H_{q,N}(u_{t})|$$\mathrm{D}$satisfying (1.1) with $n-1<p<n+\beta$ .

$\leqq$$( \int_{\{||\}}x’:x’<N|u_{\epsilon}(_{X’})-u_{t}(x’)|^{q}dx’)^{1}/q$$\leqq$$Mr^{()/_{\mathrm{P}}}p-n( \int_{\{x’:}|x|’<N\}(\int_{B((i}x’,),2r)\nabla|u(z)|^{p}dz)^{q}/pdx’)^{1}/q$,

63

so that Minkowski’s inequality for integral yields$|H_{q_{)}N}(u_{s})-H_{q,N}(u_{t})| \leqq Mr^{(p-h)/}(p2r)^{(1)}n-/q(\int_{\{z=(}zz_{n})/,<;t-2rzn<t+2r\}|\nabla u(Z)|^{p}dz)1/p$

Let $r_{j}=2^{-j-1},$ $t_{j}=r_{j-1}$ and $A_{j}=\{z=(z’, z_{n}) : r_{j}<z_{n}<3r_{j}\}$ for $j=1,2,$$\ldots$$\omega=(n-p+\beta)/p-(n-1)/q>0$ .

Then we find$|H_{q,N}(ut_{j})-H_{q,N}(ur)| \leqq Mr_{j}^{-\omega_{1}}+(\int_{A_{j}}|\nabla u(z)|pz_{n}d\beta)z1/p$

for $t_{j}-r_{j+1}<r\leqq t_{j}$ ,$|H_{q,N}(u_{t}r_{j}+1) \mathrm{j}^{-}-H_{q,N}(ur)|\leqq Mr_{j}^{-\omega}+2(\int_{A_{j}}|\nabla u(z)|pz_{n}d\beta)z1/.p$

for $t_{j}-r_{j1^{-r}}+j+2<r\leqq t_{j}-r_{j+1}$ and$|H_{q,N}(u_{r})-H_{q,N()|}ut_{j+}1 \leqq Mr_{j2}^{-\omega}+(\int_{A_{j+1}}|\nabla u(Z)|^{p_{Z_{n}d}}\beta z)^{1}/p$

for $t_{j+1}<r\leqq t_{j}-r_{j1^{-r}}+j+2$ . Collecting these results, we have$|H_{q,N}(u_{t_{\mathrm{j}}})-Hq,N(u_{r})|$$\leqq$$Mr_{j}^{-\omega}( \int_{A_{j}}|\nabla u(z)|p\beta Z_{n}d_{Z})^{1/}\mathrm{P}$$+Mr_{j+1}^{-\omega}( \int_{A_{j+1}}|\nabla u(Z)|^{p_{Z^{\beta}}}ndZ)^{1}/\mathrm{p}$for$t_{j+1}<r\leqq t_{\mathrm{j}}$. Hence it follows that$|H_{q,N}(u_{t}j)-Hq,N(u_{t})j+m| \leqq Mj\sum_{l=j}^{+m}r_{\ell}-\omega(\int_{A_{\ell}}|\nabla u(z)|^{p}z_{n}d\beta Z)^{1}/p$

Since$A_{l}\cap A_{k}=\emptyset$

when $P\geqq k+2$ , H\"older’s inequality gives$|H_{q,N}(u_{t}j)-Hq,N(ut_{j+m})|$$\leqq$$M(_{\ell=}^{j+m} \sum_{\mathrm{j}}r^{-}p\omega)\ell\prime 1/p’(^{j}=\sum_{\ell j}^{+m}\int_{A}\ell|\nabla u(Z)|^{p}zd\beta)nz1/p$$\leqq$$Mr_{j+m}^{-\omega}( \int_{\{z=(z’,z_{n})z_{n}}:rj+m<<3rj\}|\nabla u(Z)|^{p_{Z^{\beta}}}ndZ)^{1}/p$

. Set

64

More generally, if $0<r\leqq t_{j}$ , then we take $m$ such that $t_{j+m}<r\leqq t_{j+m-1}$ , and establish$|H_{q,N}(u_{t_{j}})-Hq,N(u_{r})| \leqq Mr^{-\omega}(\int_{\{z=()3r_{j}}Z’,Zn:0<z_{n}<\}|\nabla u(z)|pz_{n}d\beta)z1/p$,

which implies that$\lim_{rarrow}\sup_{0}r^{\omega}Hq,N(u_{r})\leqq M(\int_{\{z=(Z}\prime_{z_{n}},):0<z_{n}<3r_{j}\}|\nabla u(z)|^{p\beta}ZdnZ)^{1}/p$

for all . Therefore it follows that$j$$\lim_{rarrow 0}r^{\omega}H_{q,N}(ur)=0$,

as required.
is bounded.
In case $1/q=(n-p+\beta)/p(n-1)>0$ , one might expect that$0\leqq\beta<p-1$
without assuming
In fact, we can show that this is true only in casethe monotonicity. We refer the reader to the result by Yamashita [29] who showedaffirmatively the case $p=2$ and $\beta=1$ for harmonic functions. In the hyperplane case,
we refer to [16, Theorem 2.2]. The case $\beta=p-1$ remains open.
$H_{q,N}(u_{r})$For Sobolev functions, we have a weak limit result as follows.

THEOREM 6.2 (cf. [16, Theorem 2.1]). $L\mathrm{e}t-1<\beta<p-1,1<p\leqq q<\infty$ and$\frac{n-p-1}{p(n-1)}<\frac{1}{q}<\frac{n-p+\beta}{p(n-1)}$

If is a $(1, p)$ -quasicon tin
such th at$u$$\mathrm{u}o$

us func tion on$\mathrm{D}$.

satisfying (1.1), then there exists a number$A$$\lim_{rarrow}\inf_{0}r^{(+}n-p\beta)/p-(n-1)/qH_{q}(u_{r}-A)=0$,

where $u_{r}(x)’=u(x’, r)$ for $r>0$ .
By this together with Theorem 6.1, we can prove the follwing result.
be a monotone function on$p<n+\beta,$ $-1<\beta<p-1,$ $p\leqq q<\infty$ and

COROLLARY 6,1. Let$u$$\frac{1}{q}<\frac{n-p+\beta}{p(n-1)}$

then there exists a number$A$$\mathrm{D}$satisfying (1.1). If $n-1<$,

such th at$\lim_{rarrow 0}r^{(+}n-p\beta)/p-(n-1\rangle/qHq(ur-A)=0$

where $u_{r}(x’)=u(X’, r)$ for $r>0$ .

,65

THEOREM 6.3 (cf. [18, Theorem 1]). Let be a monotone fun ction on(1.1) with $n-1<p\leqq n+\beta$ . If $p\leqq q<\infty$ and$u$$\frac{1}{q}>\frac{n-p+\beta}{p(n-1)}$$\mathrm{D}$satisfying,

then$\lim_{rarrow 0}r^{(+}n-p\beta)/p-(n-1)/qH_{q}(Ur)=0$,

where $U_{r}(x’)=u(x’, r)-u(x’, 0)$ for $r>0$ .
be a monotone function on$|s-t|\leqq
r<t/2$, then Lemma 5.1 gives
If

PROOF. Let$u$$|H_{q}(u_{s}-u_{t})|$$=$$\leqq$$\mathrm{D}$satisfying (1.1) with $n-1<p\leqq n+\beta$ .

$( \int_{\mathrm{R}^{n-1}}|u_{s}(X’)-u_{t}(X/)|^{q}dx’)^{1}/q$$Mr^{(p-n)}/p( \int_{\mathrm{R}^{n-1}}(\int_{B((x^{J}},t),2r)u|\nabla(z)|^{p}dz)q/pXd/)1/q$,

so that Minkowski’s inequality for integral yields$|H_{q}(u_{S}-ut)| \leqq Mr^{(-}pn)/p(2r)^{(}n-1)/q(\int_{\{=}z(Z’,z_{n}):t-2r<z_{n}<t+\mathrm{z}_{r}\}|\nabla u(Z)|^{p}dz)1/p$

Let $r_{j}=2^{-j-1},$ $t_{j}=r_{j-1}$ and $A_{j}=\{z=(z’, z_{n}) : r_{j}<z_{n}<3r_{j}\}$ for $j=1,2,$
simplicity, set$\omega=(n-p+\beta)/p-(n-1)/q<0$.

Then we find$|H_{q}(u_{t_{j}}-u_{r})| \leqq Mr^{-\omega}j+1(\int_{A_{j}}|\nabla u(Z)|^{p_{Z^{\beta}}}ndZ)1/p$

for $t_{j}-r_{j+1}<r\leqq t_{j}$ ,$|S_{q}(u_{r}-u_{s})| \leqq Mr^{-\omega_{2}}j+(\int_{A_{\mathrm{j}}}|\nabla u(z)|pdzz_{n}^{\beta})1/p$

for $t_{j}-r_{j1^{-r}}+j+2<r<s\leqq t_{j}-r_{j+1}$ , and$|S_{q}(u_{S^{-u)}}t_{j+}1| \leqq Mr_{j}^{-\omega}+2(\int_{A_{\mathrm{j}+1}}|\nabla u(Z)|^{\mathrm{P}}Z\beta d)^{1}nZ/p$$\ldots$

. For

66

for $t_{j+1}<s\leqq\iota_{j}-r_{j+1}-r_{j2}+\cdot$ Collecting these results, we have$|H_{q}(u_{t_{j}}-ur)|$$\leqq$$Mr_{j}^{-\omega}( \int_{A_{j}}|\nabla u(z)|p_{Zd}\beta zn)1/p$$+Mr_{j+}^{-\omega_{1}}( \int_{A_{j+1}}|\nabla u(_{Z)}|p\beta dz)^{1}z_{n}/p$

for $t_{j+1}<r\leqq t_{j}$ . Hence it follows that$|H_{q}(u_{r}-u_{t_{j+m}})|$

4

for $t_{j+m}<r\leqq t_{j}$ . Since$\leqq$$A_{\ell}\cap A_{k}=\emptyset$$|H_{q}(u_{r}-ut_{j}+m)|$$\leqq$$\leqq$$M \sum_{\ell}^{m}j+=jr_{l}^{-\omega}(\int_{A}\ell|\nabla u(Z)|^{p}Z^{\beta}dZ)n1/p$

for $\ell\geqq k+2$ , H\"older’s inequality gives$M(_{\ell=j}^{jm} \sum^{+}r\ell-p’\omega)^{1/}p’(=\sum_{lj}^{j+m}\int_{A_{\ell}}|\nabla u(z)|p_{Zd}\beta zn)1/p$$Mr_{j}^{-\omega}( \int_{\{z=(z’},zn):r_{j+}m<zn<3r_{j}\}\nabla|u(z)|pz_{n}^{\beta}dz)1/p$

for $t_{j+m}<r\leqq t_{j}$ , where $1/p+1/p’=1$ . Now, letting $marrow\infty$ , we establish$|H_{q}(U_{r})| \leqq Mr^{-\omega}(\int_{\{Z=()}Z’,zn:0<z_{n}<3rj\}||\nabla u(z)pndZ^{\beta}Z)1/p$

for $t_{j+1}<r\leqq t_{j}$ , which implies that$\lim_{rarrow 0}r^{\omega}H_{q}(U_{r})=0$,

as required.
As to the condition on , compare Theorem 6.3 with Theorem 3.1.
$q$

67

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Y.Mizuta
The Division of Mathematicaland Information Sciences
Faculty of Integrated Arts and Sciences
Hiroshima University
Higashi-Hiroshima 739-8521
Japan
mizuta@mis.hiroshima-u.ac.jp

T.Shimomura
General Studies
Akashi National College of Technology
Nishioka Uozumi 674-8501
Japan
shimo@akashi.ac.jp