Ž .Journal of Mathematical Analysis and Applications 237, 730]739 1999Article ID jmaa.1999.6514, available online at http:rrwww.idealibrary.com on

NOTE

sŽ .The Monotonic Property of L m -Averaging Domainsand Weighted Weak Reverse Holder Inequality¨

Bing Liu

Department of Mathematics, College of St. Scholastica, Duluth, Minnesota 55811

and

Shusen Ding*

Department of Mathematics, Uni ersity of Minnesota, Duluth, Minnesota 55812

Submitted by William F. Ames

Received May 11, 1998

We first prove a local weighted weak reverse Holder inequality for A-harmonic¨sŽ .tensors. Then, we study the monotonic property of newly introduced L m -averag-

ing domains, which can be viewed as an application of the local weighted reversesŽ .Holder inequality in L m -averaging domains. By applying the local result, we also¨

obtain some similar results in other kinds of domains. Q 1999 Academic Press

1. INTRODUCTION

In recent years there have been remarkable advances made in the fieldof A-harmonic tensors. Many interesting results of A-harmonic tensorsand their applications in fields such as potential theory, quasiregular

Ž wmappings, and the theory of elasticity have been found see 1, 8, 10, 12,x. sŽ .15 . Meanwhile, Ding and Nolder introduce the L m -averaging domain

w x s w x5 , which is a generalization of L -averaging domains 14 to weightedaveraging domains, and obtain similar characterizations as in Ls-averagingdomains. There is also a series of studies of the integrability of A-harmonic

Ž w x.tensors in different kinds of domains see 1]4, 6 . In this paper we prove

*Current address: Mathematics Department, Seattle University, Seattle, WA 98122.

7300022-247Xr99 $30.00Copyright Q 1999 by Academic PressAll rights of reproduction in any form reserved.

NOTE 731

a local weighted weak reverse Holder inequality for A-harmonic tensors.¨Then, as applications of this local result, we prove the monotonic property

sŽ .of L m -averaging domains. We first introduce the existing results andw xrelated definitions. We will use notations as in 5, 12 .

Let V be a connected open subset of Rn. Balls in Rn are denoted by BŽ .and s B is a ball with the same center as B and diameter diam s B s

Ž . ns diam B , s ) 0. The n-dimensional Lebesgue measure of a set E : R< < 1 Ž n.is denoted by E . We call w a weight if w g L R and w ) 0 a.e.loc

Ž .Throughout this paper we define a measure m such that dm s w x dx.For an integrable function u, we denote the m average over B by u sB, m

1 H u dm.Bm BŽ .

DEFINITION 1.1. Let s ) 1 be any constant. We say that w satisfies aŽ .reverse Holder inequality and write w g WRH V when there exist con-¨

stants b ) 1 and C ) 0, independent of w, such that

1rb1 1bw dx F C w dx 1.1Ž .H Hž /< < < <B BB s B

for all balls B ; V with s B ; V.

Ž .DEFINITION 1.2. We call w a doubling weight and write w g D V ifŽ . Ž .there exists a constant C such that m 2 B F Cm B for all balls B with

2 B ; V. If this condition holds only for all balls B with 4B ; V, then w isŽ .weak doubling and denote w g WD V . The factor 4 here is for conve-

nience and in fact these domains are independent of this expansion factor;w xsee 13, 14 .

DEFINITION 1.3. We say that a weight w satisfies the A condition,rŽ .where r ) 1, and write w g A V whenr

ry11 11rŽ1yr .sup w dx w dx - `, 1.2Ž .H Hž / ž /< < < <B BB BB

where the supremum is over all balls B ; V. If w satisfies the ArlocŽ . w xcondition for all balls with 2 B ; V, we write w g A V . From 12 wer

have

LEMMA 1.4. If w g A , then there exist constants b and C, independentrof w, such that

5 5 < < Ž1yb .r b 5 5w F C B w 1.3Ž .b , B 1, B

for all balls B ; Rn.

NOTE732

n sŽ .DEFINITION 1.5. We call a proper subdomain V ; R an L m -averag-Ž .ing domain, if for s G 1 and m V - ` there exists a constant C such that

1rs1 s< <u y u dmH B , m0ž /m VŽ . V

1rs1 s< <F C sup u y u dm 1.4Ž .H B , mž /m BŽ . B2 B;V

s Ž .for some ball B ; V and all u g L V; m , where the supremum is over0 locall balls B with 2 B ; V.

DEFINITION 1.6. The quasi-hyperbolic distance between x and y in Vis given by

1k x , y s k x , y ; V s inf ds, 1.5Ž . Ž . Ž .H d z , ­ Vg Ž .g

Ž .where g is any rectifiable curve in V joining x and y, and d z, ­ V is theEuclidean distance between z and ­ V, the boundary of V.

w xThe following Theorems 1.7 and 1.8 are given by Ding and Nolder 5 .

Ž . sŽ .THEOREM 1.7. If w g WRH V and V is an L m -a¨eraging domain,then there exists a constant a such that

1rs1 sk x , x dm F a, 1.6Ž . Ž .H 0ž /m VŽ . V

Ž . Ž Ž Ž . ..where a only depends on n, s, m V , m B x , d x , ­ V r2 and the con-0 0Ž .stant C in 1.4 .

Ž .THEOREM 1.8. Let w g WD V . If

1r21 sk x , x dm F a 1.7Ž . Ž .H 0ž /m VŽ . V

sŽ .for some fixed point x in V and a constant a, then V is an L m -a¨eraging0Ž .domain and inequality 1.4 holds, with constant C depending on n, s, and a.

We give a brief introduction to A-harmonic tensors. Let e , e , . . . , e1 2 ndenote the standard unit basis of Rn. For l s 0, 1, . . . , n, the linear spaceof l-vectors, spanned by the exterior products e s e n e n ??? e corre-I i i i1 2 l

Ž .sponding to all ordered l-tuples I s i , i , . . . , i , 1 F i - ??? - i F n,1 2 l 1 ll lŽ n. I Iis denoted by L s L R . For a s Ý a e g L and b s Ý b e g L, theI I

NOTE 733

² : I Iinner product in L is given by a , b s Ý a b with summation over allŽ .l-tuples I s i , . . . , i and all integers l s 0, 1, . . . , n. We define the1 l

Hodge star operator w: L ª L by the rule w1 s e n e n ??? n e and1 2 n² : Ž . la n wb s b n wa s a , b w1 for all a , b g L , l s 1, 2, . . . , n. The

< < 2 ² : Ž . 0norm of a g L is then a s a , a s w a n wa g L s R. TheHodge star is an isometric isomorphism on L with w: Ll ª Lny l and

Ž . lŽnyl . l lww y1 : L ª L .For 0 - p - ` we denote the weighted L p-norm of a measurable

5 5 Ž < Ž . < p Ž . .1r pfunction f over E by f s H f x w x dx . A differentialp, E, w ElŽ n.l-form v on V is a Schwartz distribution on V with values in L R

Ž . Ž . Ž .which can be written as v x s Ý v x dx s Ý v x dx n dxI I I i i ? ? ? i i i1 2 l 1 2pŽ .n ??? n dx with v g L V, R for all ordered l-tuples I. The norm of vi Il

is defined by

1rp1rp pr2p 25 5 < < < <v s v x dx s v x dx .Ž . Ž .Ýp , V H H Iž /ž / ž /V V I

We consider solutions to the A-harmonic equation

dwA x , dv s 0, 1.8Ž . Ž .lŽ n. lŽ n.here A: V = L R ª L R satisfies the following assumptions:

< Ž . < < < py1 ² Ž . : < < pA x, j F a j , A x, j , j G j for almost every x g V and alllŽ n.j g L R . Here a ) 0 is constant and 1 - p - ` is a fixed exponent

Ž . Ž .associated with 1.8 . A solution to 1.8 is an element of the Sobolev space1 Ž ly1. ² Ž . : 1Ž ly1.W V, L such that H A x, dv , dw s 0 for all w g W V, Lp, loc V p

with compact support. Such differential forms are called A-harmonicw xtensors; see 8, 9, 15 .

w xThe following Theorem 1.9 is due to Nolder 12 .

THEOREM 1.9. Let u be an A-harmonic tensor in V, s ) 1, and 0 - s,t - `; then there exists a constant C, independent of u, such that

5 5 < < Ž tys.r st 5 5u F C B u 1.9Ž .s , B t , s B

for all balls or cubes B with s B ; V.

2. MAIN RESULTS

sŽ .We will show that L m -averaging domains are monotonic in terms ofn tŽ .s; i.e., if a set V ; R is an L m -averaging domain, then it is also an

sŽ .L m -averaging domain for any 1 F s F t - `. We first prove the follow-ing Theorems 2.1, 2.2, and 2.3.

NOTE734

w xAccording to 7 and Lemma 1.4, if w g A , r ) 1, then w is a doublingrweight and also satisfies the weak reverse Holder inequality, i.e., w g¨

Ž . Ž .WD V and w g WRH V . Thus, combining Theorems 1.7 and 1.8, wesŽ .obtain a sufficient and necessary condition of being an L m -averaging

domain.

THEOREM 2.1. Let w g A for r ) 1 and m be a measure defined byrŽ . sŽ .dm s w x dx. Then V is an L m -a¨eraging domain if and only if the

inequality1rs1 s

k x , x dm F aŽ .H 0ž /m VŽ . V

holds for some fixed point x in V and a constant a, which is independent of0Ž .k x, x .0

Ž .Clearly, the A-harmonic equation 1.8 is not affected by adding aclosed form to u. Therefore, any type of estimates about u must bemodulo such forms. Hence, we may set m-average of a form u over B in

Ž .inequality 1.4 ; u s 0 since u is a closed form. Next, we prove theB, m B, m

Ž .following local weighted weak reverse Holder inequality 2.1 for A-¨harmonic tensors on any subdomain of Rn.

THEOREM 2.2. Let u be an A-harmonic tensor in a domain V ; Rn.Assuming that 0 - s, t - `, s ) 1, and w g A for some r ) 1, then thererexists a constant C, independent of u, such that

1rs 1rt1 1s t< < < <u w dx F C u w dx 2.1Ž .H Hž / ž /m B m s BŽ . Ž .B s B

for all balls B with s B ; V.

Ž .Proof. Since w g A for some r ) 1, by definition 1.1 there exists arŽ .constant b ) 1, such that 1.3 holds. For any s ) 0, we can write 1rs s

Ž .b y 1 rbs q 1rbs. Then, using the Holder inequality, there exists a¨constant C , such that1

1rs 1rsss 1r s< < < <u w dx s u w dxŽ .H Hž / ž /B B

Ž . 1rb sby1 rb sb srŽ by1. b< <F u dx w dxH Hž / ž /B B

5 51r s 5 5s w ub , B b srŽ by1. , B

< < Ž1yb .r b s 5 51r s 5 5F C B w u .1, B b srŽ by1. , B1

NOTE 735

Ž . Ž .If we choose k s s r y 1 rt for any t G 1, and let m s str s q tk , thenŽ .m - t. By virtue of 1.9 ,

5 5 < < ŽmŽ by1.yb s.r b sm 5 5u F C B u ,b srŽ by1. , B m , s B2

which yields

1rss wŽ1yb .r b s xqwŽmŽ by1.yb s.r b sm x 1r s< < < < 5 5 5 5u w dx F C B w uH 1, B m , s B3ž /B

1rsyŽ1r m.< < 5 5s C B m B u .Ž .Ž . m , s B3

Therefore

1rs1 s yŽ1r m.< < < < 5 5u w dx F C B u . 2.2Ž .H m , s B3ž /m BŽ . B

Ž .On the other hand, noting that t y m rm s r y 1 and using the Holder¨inequality again, we have

1rmm1r t yŽ1r t .5 5 < <u s u w w dxŽ .m , s B Hž /

s B

Ž .1rt tym rmtŽ .t mtr tym1r t yŽ1r t .< <F u w dx w dxŽ .Ž .H Hž / ž /

s B s B

1r tt 1r ty1< < 5 5s u w dx w . 2.3Ž .H 1rŽ ry1. , s Bž /

s B

Ž .By 1.2 ,

1r t11r t5 5w 1, s BŽ .w 1r ry1 , s B

1r try1Ž .1r ry11s w dx dxH Hž / ž /ž /ws B s B

1r try1Ž .1r ry11 1 1r< <F s B w dx dxH H ž /ž / ž /< < < <s B s B ws B s B

< < rr tF C s B ,4

NOTE736

which gives1r t1 rr t yŽ1r t . rr t yŽ1r t .< < 5 5 < < 5 5F C s B w F C B w . 2.4Ž .1, s B 1, s B4 5

Ž .w 1r ry1 , s B

r 1Ž . Ž . Ž .Combining 2.2 , 2.3 , and 2.4 , and y s 0, we finally obtaint m

1rs 1rt1 1s t< < < <u w dx F C u w dx .H H6ž / ž /m B m s BŽ . Ž .B s B

Thus, Theorem 2.2 is proved.As an application of the local result, Theorem 2.2, we prove the

sŽ .following global result in L m -averaging domains which can also beproved by the generalized Holder inequality.¨

sŽ .THEOREM 2.3. Let u be an A-harmonic tensor in an L m -a¨eragingdomain V ; Rn and 1 F s F t - `; assume that w g A for some r ) 1r

Ž .and a measure m such that dm s w x dx. Then there exists a constant C,independent of u, such that

1rs 1rt1 1s t< < < <u w dx F C u w dx . 2.5Ž .H Hž / ž /m V m VŽ . Ž .V V

1 t 1r tŽ < < .Proof. We may assume that sup H u w dx - `. Choose2 B ; V Bm BŽ .a ball B ; V and a constant C large enough, such that0 1

1r t 1rt1 1t t< < < <sup u w dx F sup u w dxH Hž / ž /m B m BŽ . Ž .B s B2 B;V 2 B;V

1r t1 t< <F C u w dx .H1 ž /m BŽ . s B0 0

Ž . Ž . Ž . Ž .Moreover, since m B s H w x dx F H w x dx s m s B , and by theB s Bvirtue of Theorem 2.2, we have

1rs1rs 1rs1 1 1s s< < < <u w dx s m B u w dxŽ .H H0ž / ž /m V m V m BŽ . Ž . Ž .V V0

1rs 1rs1 11rs s< <F m B sup u w dxŽ .Ž . H0ž /m V m BŽ . Ž . B2 B;V

1rs 1rt1 11rs t< <F m B sup u w dxŽ .Ž . H0ž /m V m s BŽ . Ž . s B2 B;V

1rs 1rt1 C31rs t< <F m B sup u w dxŽ .Ž . H0ž /m V m BŽ . Ž . s B2 B;V

NOTE 737

1r t1rs1 11rs t< <F C m B u w dxŽ .Ž . H4 0ž /m V m BŽ . Ž . s B0 0

1r t1rs1 Ž . Ž .1rs y 1rt t< <F C m B u w dxŽ .Ž . H5 0ž /m VŽ . s B0

1rs 1rt1 Ž . Ž .1rs y 1rt t< <F C m V u w dxŽ .Ž . H6 ž /m VŽ . V

1r t1 t< <s C u w dx .H6 m VŽ . V

Ž .Thus, we yield 2.5 .

Theorem 2.4 is a direct conclusion of Theorem 2.1, noting the definitionŽ .1.6 and the following inequality:

1rs1 sk x , x w dxŽ .H 0m VŽ . V

1r t1 tF k x , x w dx F a for 1 F s F t - `.Ž .H 0ž /m VŽ . V

THEOREM 2.4. Let w g A, r ) 1. Then the fact that V ; Rn is antŽ . sŽ .L m -a¨eraging domain implies that it is an L m -a¨eraging domain, if

1 F s F t - `.

3. APPLICATION

sŽ .We know that a d-John domain is an L m -averaging domain when ww x nsatisfies the A condition 5 . Thus Theorem 2.3 holds if V ; R is ar

d-John domain for s, t such that 1 F s F t F `. In fact, Theorem 2.3 isalso true for any s, t such that 0 - s F t - ` if V is a d-John domain. Weprove it by using Theorem 2.2. For a reference of the d-John domain seew x11 . We give the definition and a lemma we need as follows.

DEFINITION 3.1. We call V, a proper subdomain of Rn, a d-Johndomain, d ) 0, if there exists a point x g V which can be joined with any0

Ž . <other point x g V by a continuous curve g ; V so that d j , ­ V G d x y< Ž .j for each j g g . Here d j , ­ V is the Euclidean distance between j

and ­ V.

NOTE738

LEMMA 3.2. Let V ; Rn be a d-John domain. Then there exists aco¨ering n of V, consisting of open cubes, such that:

Ž . Ž . Ž . ni Ý x x F Nx x , x g R .Q g n s Q V

Ž . Ž .ii There is a distinguished cube Q g n called the central cube0which can be connected with e¨ery cube Q g n by a chain of cubesQ , Q , . . . , Q s Q from n such that for each i s 0, 1, . . . , k y 1, Q ; NQ .0 1 k i

n Ž .There is a cube R ; R this cube does not need to be a member of n suchithat R ; Q l Q , and Q j Q ; NR .i i iq1 i iq1 i

THEOREM 3.3. Let V ; Rn be a d-John domain, u be any A-harmonictensor, 0 - s F t - `, and w g A for some r ) 1. Then there exists arconstant C, independent of u, such that

1rs 1rt1 1s t< < < <u w dx F C u w dx . 3.1Ž .H Hž / ž /m V m VŽ . Ž .V V

Proof. Let V ; Rn be a d-John domain; then by Theorem 2.2 andLemma 3.2, there exists a cover n of V such that

5 5 s < < s < < su s u w dx F u w dxÝs , V , w H HV QQgn

srtŽ .y srt t< <F C m Q m s Q u w dxŽ . Ž .Ý H1 ž /s QQgn

srtŽ .1y srt t< <F C m s Q u w dxŽ .Ý H1 ž /s QQgn

srtŽ .tys rt t< <F C m V u w dxŽ . Ý H1 ž /s QQgn

srtŽ .tys rt t< <F C m V u w dx .Ž . H2 ž /

V

Thus,

1rs 1rtŽ .tys rsts t< < < <u w dx F C m V u w dx .Ž .H H3ž / ž /

V V

t y s 1 1 Ž .Noting that s y , we obtain 3.1 .st s t

Ž .Remark. 1 Our local result, Theorem 2.2, holds in any kind ofdomain.

NOTE 739

Ž . sŽ .2 Here we only apply Theorem 2.2 to L m -averaging domainsand d-John domains, and obtain Theorem 2.3 and 3.3, respectively. Theo-rem 2.2 can be applied to some other kinds of domains, such as the

w x ndomains with Whitney covers. Specifically, from 12 , if V ; R is adomain with Whitney covers, then there is a modified Whitney cover of

� 4cubes W s Q which satisfyi

Q s V ,D ii

and

x F Nx ,Ý VŽ .5r4 Q'QgW

for all x g Rn and some N ) 1. Using Theorem 2.2 and the aboveproperties of the Whitney cover, we can extend the result of Theorem 3.3to any domains with Whitney covers.

Ž .3 Theorem 1.9 is a special case of Theorem 2.2 as w s 1.

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