PDF Lindelo¨f theorems for monotone Sobolev functions on uniform domains

34 (2004), 413–422

Lindelo¨f theorems for monotone Sobolev functions on uniform domains

Toshihide Futamura

(Received September 8, 2003) (Revised January 28, 2004)

Abstract. This paper deals with Lindelo¨f type theorems for monotone Sobolev functions on a uniform domain.

1. Introduction

A continuous function u on an open set Din the n-dimensional Euclidean space Rⁿ, nb2, is called monotone in the sense of Lebesgue (see [6]) if the equalities

max

G

u¼max

qG u and min

G

u¼min

qG u

hold whenever G is a domain with compact closure GHD. If u is a monotone Sobolev function on D and p>n1, then

juðxÞ uðx⁰ÞjaCðn;pÞr^1n=p ð

Bðz;rÞ

j‘uðyÞj^pdy

!1=p

ð1:1Þ

whenever x;x⁰ABðz;r=2Þ with Bðz;rÞHD, where Bðz;rÞ is the open ball centered at zwith radiusrandCðn;pÞis a positive constant depending only on n and p (see [11, Chapter 8] and [13, Section 16]). Using this inequality (1.1), we proved Lindelo¨f theorems for monotone Sobolev functions on the half space ofRⁿ in [1]. For related results, see Koskela-Manfredi-Villamor [5], Manfredi- Villamor [7, 8] and Mizuta [10]. In this paper we will generalize this result to a uniform domain in a metric space.

Let X be a metric space with a metric d and m be a Borel measure on X which is positive and finite on balls. We denote by Bðx;rÞ the open ball centered at xAX with radius r>0 and set lB¼Bðx;lrÞ for each ball B¼Bðx;rÞ and l>0. A domain D in X with qD0 q is a uniform domain if there exists a constant Ab1 such that each pair of points x;yAD can be joined by a curve g in D for which

2000 Mathematics Subject Classification. Primary 31B25 (46E35)

Key words and phrases. monotone Sobolev functions, Lindelo¨f theorem, uniform domain, Hausdor¤ measures

lðgÞaAdðx;yÞ;

ð1:2Þ

dDðzÞbA¹minflðgðx;zÞÞ;lðgðy;zÞÞg for all z2g;

ð1:3Þ

where lðgÞ;dDðzÞ and gðx;zÞ denote the length of g, the distance from z to qD and the subarc ofgconnecting xand z, respectively (see [9] and [12]). Here a curve means simple curve.

Our first aim in this paper is to deal with Lindelo¨f type theorems for functions u on a uniform domain D for which there exist a nonnegative Borel function gAL_loc^p ðD;mÞ, p>1, constants M>0 and 0<la1 such that

juðxÞ uðx⁰ÞjaMr ð

Bðz;rÞ

gðyÞ^pdmðyÞ

!1=p

ð1:4Þ

whenever x;x⁰ABðz;lrÞ with Bðz;rÞHD and ð

D

gðyÞ^pdDðyÞ^admðyÞ<y ð1:5Þ

for some real number a. Here we used the standard notation uF ¼

ð

F

u dm¼ 1 mðFÞ

ð

F

u dm

for a measurable set F with 0<mðFÞ<y. For this purpose we assume that there exists a constant C₁b1 such that

mð2BÞaC₁mðBÞ ð1:6Þ

for all ballsB. We further assume that there exist constantsQb1 and C₂ >0 such that

mðBÞ

mðB0ÞbC₂ diamB diamB0

Q

ð1:7Þ

for all balls B and B0 with BHB0. For xAqD and a>1, consider the set G_Dðx;aÞ ¼ fxAD:dðx;xÞ<ad_DðxÞg:

A function u defined onD is said to have a nontangential limit L at xAqD if for every a>1, lim_z!x;zAGDðx;aÞuðzÞ ¼L. The main result of this paper is the following theorem.

Theorem 1. Let D be a uniform domain in X. Let u be a function on D with gb0 satisfying (1.4) and (1.5). Suppose p>Qþa1 and set

E¼ xAqD:lim sup

r!0

r^pa mðBðx;rÞÞ

ð

Bðx;rÞVD

gðyÞ^pdDðyÞ^admðyÞ>0

( )

:

If xAqDnE and there exists a curve g in D tending to x along which u has a finite limit L, then u has a nontangential limit L at x.

Remark 1. If g satisfies (1.5), then H^QþapðEÞ ¼0, where H^s denotes the s-dimensional Hausdor¤ measure.

Corollary1. Let u be a monotone Sobolev function on a uniform domain D in Rⁿ satisfying

ð

D

j‘uðyÞj^pd_DðyÞ^ady<y; where p>maxfn1;n1þag. Set

E⁰¼ xAqD:lim sup

r!0

r^pan ð

Bðx;rÞVD

j‘uðyÞj^pd_DðyÞ^ady>0

( )

:

If xAqDnE⁰ and there exists a curve g in D tending to x along which u has a finite limit L, then u has a nontangential limit L at x:

2. Proof of Theorem 1

Throughout this paper, let M denote various constants independent of the variables in question.

For a proof of Theorem 1, we need the following Lemmas.

Lemma1. Let D be a uniform domain. Then, for eachxAqD, there exists a curve g_x in D ending at x such that

dDðzÞbA¹₁ lðg_xðx;zÞÞ for all zAg_x; ð2:1Þ

where A₁¼2⁵A³.

Proof. Fix xAqD. For each j su‰ciently large (say jbj₀), take a point wjADVqBðx;2^jÞ. Further, take a curve g_j in D joining wj1 and wjþ1 satisfying (1.2) and (1.3), and take a point z_jAg_jVqBðx;2^jÞ. Since lðg_jðwjþ1;zjÞÞb2^j1 and lðg_jðwj1;zjÞÞb2^j, we have by (1.3)

dDðzjÞbA¹2^j1:

Let gg^_j be a curve in D joining z_j and z_jþ1 satisfying (1.2) and (1.3). Then lð^gg_jÞaA2^jþ1 and dDðzÞbA²2^j3 for all zAgg^_j. Set

^gg_x¼^gg_j0þ^gg_j0þ1þ^gg_j0þ2þ

Then it is not di‰cult to construct a simple curve g_x from ^gg_x satisfying (2.1)

with A1¼2⁵A³. r

For each tAR, consider the function k_tðr1;r₂Þ ¼

ðr2

r1

t^{ð1tQÞ=ðp1Þ}dt

11=p

for 0ar1<r2.

Lemma 2 (cf. [2, Lemma 3]). Let u be a function on D with gb0 satisfying (1.4) and tAR. Then

juðxÞ uðyÞjaMk_tðdDðxÞ;8A max

zAg d_DðzÞÞ r^Q mðBðw;rÞÞ

ð

BðgÞ

gðzÞ^pd_DðzÞ^tdmðzÞ

!1=p

whenever x and y can be joined by a rectifiable curve g in D such that d_DðzÞbA¹lðgðx;zÞÞ for all zAg

ð2:2Þ

and BðgÞ ¼6

zAgBðz;dDðzÞ=2ÞHBðw;rÞ, where M is a positive constant inde- pendent of x;y;g and Bðw;rÞ.

Proof. Let g be a curve in D joining x and y satisfying (2.2) and BðgÞHBðw;rÞ. We can take a finite chain of balls B₀;B₁;. . .;B_N with the following properties:

( i ) B_j¼Bðzj;d_DðzjÞ=2Þ with z_jAg, z₀¼x and yAlB_N; ( ii ) lBjVlBjþ10 q for all 0aj<N;

(iii) For small t>0, the number of z_j such that t<d_DðzjÞa2t is bounded by ð2AþlÞ=l;

(iv) P

jw_BjaC3, where C3 is a positive constant depending only on C1

and l;

see [1, Proof of Theorem 1] and [2, Lemma 2.2].

Pickxjþ1AlB_jVlBjþ1 for 0aj<N: setx₀ ¼x andxNþ1¼y. By (1.4), we see that

juðxjÞ uðxjþ1ÞjaMdDðzjÞ ð

Bj

gðzÞ^pdmðzÞ

!1=p

for 0ajaN. Then we have by (1.7), Ho¨lder’s inequality and (iv) juðxÞ uðyÞj

aMmðBðw;rÞÞ^1=pX^N

j¼0

d_DðzjÞ^1t=p mðBðw;rÞÞ mðBjÞ

1=p ð

Bj

gðzÞ^pd_DðzÞ^tdmðzÞ

!1=p

aMmðBðw;rÞÞ^1=pX^N

j¼0

dDðzjÞ^1t=p r d_DðzjÞ Q=p ð

Bj

gðzÞ^pdDðzÞ^tdmðzÞ

!1=p

aMr^Q=pmðBðw;rÞÞ^1=p X^N

j¼0

d_DðzjÞ^{ðptQÞ=ðp1Þ}

!11=p ð

BðgÞ

gðzÞ^pd_DðzÞ^tdmðzÞ

!1=p

:

Further, since ð2AÞ¹dDðxÞadDðzjÞamaxzAgdDðzÞ, we see from (iii) that X^N

j¼0

dDðzjÞ^{ðptQÞ=ðp1Þ}aM kt dDðxÞ;8A max

zAg dDðzÞ

p=ðp1Þ

:

Thus the proof is completed. r

A sequence fxjg is called regular at x if x_j!x and dðx;xjþ1Þadðx;x_jÞacdðx;xjþ1Þ for some constant c>1.

Lemma 3 (cf. [1, Lemma 1]). Let u;g;D and E be as in Theorem 1. Suppose there exists a regular sequence fxjg at xAqDnE such that xj Ag_x and limj!yuðxjÞ ¼L, where g_x is as in Lemma 1. Then u has a nontangential limit L at x.

Proof. Set rj¼dðx;xjÞ. Since fxjg is regular at x, there exists a constant c>1 such that r_jþ1ar_jacr_jþ1. Fix xAG_Dðx;aÞVBðx;r₁Þ. Then there exists an integer j such that rjadðx;xÞ<rj1. Let g be a curve in D joining x and x_j with (1.2) and (1.3), and take yAg such that lðgðx;yÞÞ ¼lðgðxj;yÞÞ; Set g₁¼gðx;yÞ and g₂¼gðxj;yÞ. Then g_i satisfies (2.2) for i¼1;2 anddðx;zÞac₁r_j for all zAg, wherec₁¼ ðcþ1ÞAþ1. Since dDðxÞba¹rj, dDðxjÞbA¹₁ rj and Bðg_iÞHBðx;2c1rjÞVD, we see from Lemma 2 with t¼a that

juðxÞ uðxjÞjajuðxÞ uðyÞj þ juðyÞ uðxjÞj

aMkaða¹rj;8Ac1rjÞ ð2c1rjÞ^Q mðBðx;2c₁r_jÞÞ

ð

Bðg₁Þ

gðzÞ^pdDðzÞ^admðzÞ

!1=p

þMkaðA¹₁ rj;8Ac1rjÞ ð2c1rjÞ^Q mðBðx;2c₁r_jÞÞ

ð

Bðg₂Þ

gðzÞ^pdDðzÞ^admðzÞ

!1=p

aM r_j^pa mðBðx;2c1rjÞÞ

ð

Bðx;2c1rjÞVD

gðzÞ^pdDðzÞ^admðzÞ

!1=p

:

Since xBE, this implies that u has a nontangential limit L at x. r Now we can prove Theorem 1.

Proof ofTheorem 1. Suppose uðzÞtends toL as z!x along g. Let g_x be as in Lemma 1. For r>0 su‰ciently small, take x₁ðrÞAgVqBðx;rÞ and

x2ðrÞAg_xVqBðx;rÞ. Thenx1ðrÞandx2ðrÞcan be connected by a curveg₀ inD with (1.2) and (1.3). Set g₁¼g₀ðx1ðrÞ;yðrÞÞ and g₂¼g₀ðx2ðrÞ;yðrÞÞ with a point yðrÞAg₀ such that lðg₁Þ ¼lðg₂Þ. Then

dDðzÞbA¹lðg_iðxiðrÞ;zÞÞ for all zAg_i; i¼1;2:

Note that d_Dðx2ðrÞÞbA¹₁ r, dðx;zÞac₂r for all zAg₀ and jrdðx;zÞjadðz;x1ðrÞÞac2dDðzÞ

for all zABðg₁Þ, where c2¼2Aþ1. By Lemma 2 with t¼a, we see that

juðx2ðrÞÞ uðyðrÞÞjaMkaðA¹₁ r;8Ac2rÞ ð2c2rÞ^Q mðBðx;2c₂rÞÞ

ð

Bðg₂Þ

gðzÞ^pdDðzÞ^adm

!1=p

aM r^pa mðBðx;2c₂rÞÞ

ð

Bðx;2c2rÞVD

gðzÞ^pdDðzÞ^adm

!1=p

:

Since p>Qþa1 by our assumption, there exists b>0 such that Qþap<b<1. We have by Lemma 2 with t¼ab

juðx1ðrÞÞ uðyðrÞÞj

aMkabð0;8Ac2rÞ ð2c2rÞ^Q mðBðx;2c₂rÞÞ

ð

Bðg₁Þ

gðzÞ^pdDðzÞ^abdmðzÞ

!1=p

aM r^paþb mðBðx;2c₂rÞÞ

ð

Bðx;c2rÞVD

gðzÞ^pdDðzÞ^ajrdðx;zÞj^bdmðzÞ

!1=p

:

Hence we have

juðx1ðrÞÞ uðx2ðrÞÞj^p ð2:3Þ

aM r^paþb mðBðx;2c₂rÞÞ

ð

Bðx;2c2rÞVD

gðzÞ^pdDðzÞ^ajrdðx;zÞj^bdmðzÞ:

Moreover, since 0<b<1, we see that ð2^j

2^j1

jrdðx;zÞj^bdraM2^jð1bÞ: ð2:4Þ

Hence it follows from (2.3) and (2.4) that

inf

2^j1ara2^jjuðx1ðrÞÞ uðx2ðrÞÞj^p aM

ð2^j 2^j1

r^paþb mðBðx;2c₂rÞÞ

ð

Bðx;2c2rÞVD

gðzÞ^pd_DðzÞ^ajrdðx;zÞj^bdmðzÞ

!dr r

aM 2^jðpaþb1Þ mðBðx;c₂2^jÞÞ

ð

Bðx;c22^jþ1ÞVD

gðzÞ^pd_DðzÞ^a ð2^j

2^j1

jrdðx;zÞj^bdr

! dmðzÞ

aM 2^jðpaÞ mðBðx;c22^jÞÞ

ð

Bðx;c22^jþ1ÞVD

gðzÞ^pdDðzÞ^admðzÞ:

Since xBE, we can find a sequence frjg such that 2^j1<rja2^j and

j!limyuðx2ðrjÞÞ ¼L:

Thus u has a nontangential limit L at x by Lemma 3. r 3. Aq weights

Let w be a Muckenhoupt Aq weight, that is, a nonnegative measurable functions on Rⁿ satisfying

sup ð

B

wðxÞdx

ð

B

wðxÞ^1=ð1qÞdx

q1

<y; ð3:1Þ

where the supremum is taken over all balls B in Rⁿ (see [4]). Let u be a monotone function on a uniform domain D in Rⁿ in the sense of Lebesgue which satisfies

ð

D

j‘uðxÞj^pwðxÞdx<y: ð3:2Þ

Suppose 1aq< p=ðn1Þ. Since p₁¼p=q>n1, then juðxÞ uðx⁰ÞjaMr^1p1=n

ð

Bðz;rÞ

j‘uðyÞj^p1dy

!1=p1

whenever x;x⁰ABðz;r=2Þ with Bðz;rÞHD.

Hence we derive the following extension of a result by Manfredi-Villamor [8] to a uniform domain (see also [1]).

Corollary2. Let 1aq<p=ðn1Þand w be a Muckenhoupt A_q weight.

Suppose u is a monotone function on a uniform domain D in Rⁿ satisfying (3.2). Set

E₁¼ xAqD:lim sup

r!0

r^p wðBðx;rÞÞ

ð

Bðx;rÞVD

j‘uðyÞj^pwðyÞdy>0

( )

;

where wðBðx;rÞÞ ¼Ð

Bðx;rÞwðyÞdy. If xAqDnE1 and there exists a curve g in D tending tox along which u has a finite limit L, then u has a nontangential limit L at x.

Proof. Set

E₂ ¼ xAqD:lim sup

r!0

r^p1n ð

Bðx;rÞVD

j‘uðyÞj^p1dy>0

( )

;

where p1¼p=q. Using Ho¨lder inequality and (3.1), we see that E2HE1. Thus Corollary 2 follows from Theorem 1 with p andmreplaced by p₁ and the

n-dimensional Lebesgue measure. r

4. Generalizations of Lindelo¨f theorems

In this section, we give a generalization of Theorem 1 in case X¼Rⁿ. Let mbe an integer such that 1am<n. We say thatG is anm-approach set at x with l₁ >1 and l₂>0, if there exist a sequence of positive numbers frjg tending to zero and a sequence of contraction maps P_j fromRⁿ toR^m such that rjþ1arjal1rjþ1 and

H^mðPjðGVðBðx;rjÞnBðx;rjþ1ÞÞÞÞbl2r_j^m: ð4:1Þ

Theorem2. Let D be a uniform domain inRⁿ. Let u be a function on D with gb0 satisfying (1.4) and (1.5). Suppose p>Qþam and set

E¼ xAqD:lim sup

r!0

r^pa mðBðx;rÞÞ

ð

Bðx;rÞVD

gðyÞ^pdDðyÞ^admðyÞ>0

( )

:

If xAqDnE and there exists an m-approach setGHD at x along which u has a finite limit L at x, then u has a nontangential limit L at x.

Proof. Letr_j;P_j;l₁ andl₂ be retained from the definition of m-approach set G at x, and set

G_j¼GVðBðx;r_jÞnBðx;r_jþ1ÞÞ:

For oAPjðGjÞ, take x1ðoÞAGj and set r¼ jxx1ðoÞj. Let g_x be as in Lemma 1 and take x₂ðoÞAg_xVqBðx;rÞ. By our assumption, we can take b>0 such thatQþap<b<m. SincejPjðzÞ ojajzx1ðoÞj, in view of the estimate (2.3) in the proof of Theorem 1, we obtain

juðx1ðoÞÞ uðx2ðoÞÞj^p aM r^paþb

mðBðx;2c2rÞÞ ð

Bðx;2c2rÞVD

gðzÞ^pd_DðzÞ^ajPjðzÞ oj^bdmðzÞ:

Further, since PjðGjÞHBðPjðxÞ;rjÞðHR^mÞ and 0<b <m, we see that ð

PjðGjÞ

jPjðzÞ oj^bdH^mðoÞa ð

BðPjðxÞ;rjÞ

jPjðzÞ oj^bdH^mðoÞaMr_j^mb: Hence we have by (4.1)

oAinfPjðGjÞjuðx1ðoÞÞ uðx2ðoÞÞj^paM r_j^pa mðBðx;2c₂l¹₁ r_jÞÞ

ð

Bðx;2c2rjÞVD

gðzÞ^pdDðzÞ^admðzÞ:

From xBE, we can find a sequence fojg such that ojAPjðGjÞ and

j!limy uðx2ðojÞÞ ¼L:

Since fx2ðojÞg is regular atx, we can show that u has a nontangential limit L

at x by Lemma 3. r

Acknowledgement

The author would like to express his deep gratitude to Professor Yoshihiro Mizuta for his valuable advice and encouragement. The author also thanks Professors Fumi-Yuki Maeda and Tetsu Shimomura for their kind comments and suggestions.

References

[ 1 ] T. Futamura and Y. Mizuta, Lindelo¨f theorems for monotone Sobolev functions, Ann.

Acad. Sci. Fenn. Math. 28 (2003), 271–277.

[ 2 ] T. Futamura and Y. Mizuta, Boundary behavior of monotone Sobolev functions on John domains in a metric space, preprint.

[ 3 ] P. Hajłasz and P. Koskela, Sobolev met Poincare´, Mem. Amer. Math. Soc.145(2000), no.

688.

[ 4 ] J. Heinonen, T. Kilpela¨inen and O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Univ. Press, 1993.

[ 5 ] P. Koskela, J. J. Manfredi and E. Villamor, Regularity theory and traces of A-harmonic functions, Trans. Amer. Math. Soc. 348 (1996), 755–766.

[ 6 ] H. Lebesgue, Sur le proble`me de Dirichlet, Rend. Cir. Mat. Palermo24(1907), 371–402.

[ 7 ] J. J. Manfredi and E. Villamor, Traces of monotone Sobolev functions, J. Geom. Anal.6 (1996), 433–444.

[ 8 ] J. J. Manfredi and E. Villamor, Traces of monotone Sobolev functions in weighted Sobolev spaces, Illinois J. Math. 45 (2001), 403–422.

[ 9 ] O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn.

Ser. A. I. Math. 4 (1979), 383–401.

[10] Y. Mizuta, Tangential limits of monotone Sobolev functions, Ann. Acad. Sci. Fenn. Ser.

A. I. Math. 20 (1995), 315–326.

[11] Y. Mizuta, Potential theory in Euclidean spaces, Gakko¯tosyo, Tokyo, 1996.

[12] J. Va¨isa¨la¨, Uniform domains, Toˆhoku Math. J. 40 (1988), 101–118.

[13] M. Vuorinen, Conformal geometry and quasiregular mappings, Lectures Notes in Math.

1319, Springer, 1988.

Toshihide Futamura Department of Mathematics Graduate School of Science

Hiroshima University Higasi-Hiroshima 739-8526, Japan Current address: Department of Mathematics

Daido Institute of Technology Nagoya 457-8530, Japan E-mail address: [email protected]