DECOMPOSITION OF BV FUNCTIONS IN CARNOT-CARATHÉODORY SPACES

2003,23B( 4) :433-439

.At'~cIIlS'cittrtia

1t~f4J1~tIl DECOMPOSITION OF BV FUNCTIONS IN CARNOT-CARATHEODORY SPACES 1 Song Yingqinfl,2 ( *-i!f.7k)

Yang Xiaopinfl ( -1h:t-t )

Liu Zhenghai3

(

~IJ#." )

1.School of Science, Nanjing University of Science and Technology, Nanjing 2100g4, China 2.Department of Mathematics, Hunan University of City, Yiyang 413000, China 3.Department of Mathematics, Changsha University of Electrical Power, Changsha 410077, China

Abstract

The aim of this paper is to get the decomposition of distributional derivatives

of functions with bounded variation in the framework of Carnot-Caratheodory spaces (CC spaces in brievity) in which the vector fields are of Carnot type. For this purpose the approximate continuity of BV functions is discussed first, then approximate differentials of L 1 functions are defined in the case that vector fields are of Carnot type and finally the decomposition Xu = Vu' L" + X'u is proved, where u E BVx(n) and Vu denotes the approximate differential of u. Key words BV function, C-C space, Radon measure, vector field, approximate differential 2000 MR Subject Classification

1

26A45, 49J45

Introduction and Preliminary

Functions of bounded variation (BV functions briefly) in the Euclidean space (R n , I· I ) have been extensively studied (see [1][5]). It is well known that if u E BV(D) where D c R" is an open set, then the Radon measure Du can be decomposed into three parts, i.e. Du = D'i u. + Di u + DC u, where D'tu, Diu, DCu are called the absolute continuity part, jump part and Cantor part respectively. Furthemore, D'n: = 'Vu· u-.os« = (u+ - u-)vuHn-ILSu, where 'Vu stands for the approximate differential of u(see for instance Ambrosio's recent bookl'J). A problem is naturally addressed, that is, if u belongs to the space BVx (D) which consists of functions of bounded variation in the C-C space induced by vector fields {X I, ... , X m}, can we expect a similar decomposition for XU? In general, it is very hard to give an answer to this question because it is very diffficult to define the approximate differential of u E BVx(D) for general vector fields X = {Xl,"', X m } . This paper partially gives a positive answer to the above question only in the case that vector fields X are of Carnot type. We would like to mention that vector fields of Heisenberg groups (more general Carnot groups) are of Carnot type. Our ideas partly stem from Monti and Cassano's result(Theorem 3.2[2]). See also Proposition 4 below. 1

Received April 9, 2001. This work is supported by NNSF of China (19771048)

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As a kind of metric spaces, C-C spaces (Rn, d) which contain the Euclidean space (Rn, 1·1) and Riemann spaces as special cases are recently accepted much attention(see[2][3][4]). Because of the closed relation between BV functions and partial differential equations arising from families of noncommutating vector fields, it becomes an interesting direction to study BV functions in the framework of C-C spaces. Given a family X = {Xl,'" ,Xm} of Lipschitz vector fields Xj(x) =

n

L

i=l

aij(x)oi (j =

1 " ... m) with a·· E Lip(R") 'J' " = 1 ... m , i = 1,"', n, we call a subunit the Lipschitz 1.) continuous curve, : [0, T] -+ R n such that m

i'(t) =

L h (t)X (r(t)) j

(1)

j

j=l

and

m

L

j=l

h;(t) ::; 1 for a.e. t E [0, T] with hI,"', h m measurable coefficients. Then one can

define the C-C distance between the points x, y E R n as d(x,y):= inf{T

~

0, there exists a subunit path

, : [0,T] -+ H" such that ,(a) = x and ,(T) = y}.

(2)

If the above set is empty, put d(x,y) = +00. If d is finite on H" i.e, d(x,y) < 00 for every x, y E H"; it turns out to be a metric on H" and the metric space (Rn, d) is called the C-C space (see for example[2j[3]). Obviously both Euclidean space and Riemann spaces are C-C spaces, so

are Heisenberg groups (see[4][7][8]). The C-C metric ball is denoted by B(x,r) = Br(x) := {y : Rn,d(y,x) < r}, while the Euclidean ball is denoted by BI'I(x,r) = BI'I(x):= {y E Rn,IY -xl < r ]. UE will stand for the average udx, where unless otherwise stated, lEI, dx denotes the Lebesgue measure. By 1 n we denote, throughout this paper, the bounded open set in R": It is convenient to assume that ([2][3][7][8]) (Hj ) the metric d is finite and the identity map I d : (R n, d) -+ (Rn, I . I) is a homeomorphism.

11 IE

(H 2) (the doubling condition) let n c R n , there exists a constant 0 x E n and some ro > 0, the following is true:

> a such that for every

provided that a < r < ro. (H 3) (Rn, d) is complete and is a length space. As for (H 2 ) , for the best constant 0, Q = log20 is called the local homogeneous dimension of n relatively to the vector fields {Xl,' .. ,Xm } . Remark 1 The metric measure space (Rn,d,Ln) is Ahlfors regular (see [6]), i.e. there exist positive constants aI, a2 such that

Remark 2 An interesting condition under which the above hypotheses (Hj ) and (H 2 ) hold is the so-called Chow-Hormander condition: rank Lie(Xl,"', Xm)(x) = n

for every

x E R"

(3)

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where Lie(X, "', X m) is the Lie algebra generated by {Xl,"', X m} E COO (R n, Rn). Namely, if f! E R" is a bounded open set and kEN is the minimum length of the commutators necessary to guarantee (3) relatively to 0, then there is a constant c> 0 such that

1 1 -Ix - yl ::; d(x,y) ::; clx - ylk" c

(4)

for all x, yEO (see [2]). A subunit path, : [0, T] -+ R" is called a geodesic if d(,(O), ,(T)) = T. The following proposition has been proved (see for instance Theorem 1.10[7J). Proposition 3 Let (Rn,d) be X-connected and complete. For every x,y E R" there exists a Lipschitz continuous subunit, : [O,d(x,y)] -+ Rn such that ,(0) = x,,(d(x,y)) = y, and t = d(,(t), x). Let m < n, vector fields Xl," . , X m E COO(Rn, R n) are called "of Carnot type" if they satisfy the Chow-H6rmander condition and are of the form X j = OJ + 2: aij (x )Oi (j = 1, ... , m),

i>j

where aij(x) E coo(Rn). Concerning vector fields of Carnot type, we have (see also Theorem 3.2[2]) the following result: Proposition 4 Let Xl, ... , X m E Coo (Rn , Rn) be of Carnot type and let f E Lip (Rn , d). Then for a.e. x E R" lim f(y) - f(x) -

2:7=1 Xjf(x)(Yj -

Xj) =

d(x,y)

y-tx

o.

(5)

We denote by Xj the operator formally adjoint to X j in L2(Rn), which for all cp,7j; E C(f(Rn) satisfies

r cp(x)XpjJ(x)dx = iRnr 7j;(x)Xjcp(x)dx iRn

and define the X -gradient and X -divergence

Xf For a function

f

:=

(XIi,"" Xmf), divx(cp)

:=

L XjCpj.

E LLoc(O), its distributional derivative along the vector field X j is defined by

(Xjf,if».=

k

fXjif>dx for every if> E Co(O).

(6)

Throughout the paper, if f is a nonsmooth function, then Xjf will be meant in the distributional sense. The space of functions with bounded variation with respect to X has been introduced in [3]. Precisely, given an open set 0 eRn, the space BVx(O) is defined as the set of functions fELl (0) such that IIXfll(f!) :=

SUP{k fdivxcpdx,cp

E CJ(O,R m) and Icp(x) I::; I}

< 00.

(7)

Now we can state the main results of this paper. Theorem 5 at x. Then

Let x E O,u E BVx(Br(x)). Assume that u has the approximate limit u(x)

r

iBr(x)

r

lu(y) - u(x)1 d < IXul(Btr(x)) dt ' d(x,y) y- io tQ

(8)

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where Q is the local homogeneous dimension of O. Theorem 6 Let X = {Xl, ... , X m} be the vector fields of Carnot type, then any function u E BVx (0) is approximately differentiable at Ln- a.e. points of O. Moreover, the approximate differential V'u is just the density of the absolutely continuous part of Xu with respect to L" i.e. Xu = V'uLn + xsu.

2

Approximate Continuity and Differentiability

In this section we extend some results about approximate continuity and differentiability of BV functions in the case of Euclidean spaces to the case of C-C spaces, The ideas of approximate , limits come from [1)[10]. Definition 7 (approximate limit) Let u E Ltoc(O), Bp(x) be a C-C ball centred at x E O. We say u has an approximate limit at x E 0 if there exists z E R such that lim IB p.j.o

~ )1 1rBp(x) lu(y) -

zldy = O.

p x

(9)

The set 8" of points where this property does not hold is called the approximate discontinuity set. For any x E 0\8", the real value z, uniquely determinated by (9), is called the approximate limit of u at x and denoted by u(x). Remark 8 Owing to (Hj ), our definition is equivalent to Ambrosio's one and in general u(x) is different from usual Lebesgue value. Remark 9 In the above definition, Bp(x) can be replaced by any set Q which contains the point x and is of positive Lebesgue measure, while p can be replaced by diam(Q). Proposition 10 (properties of approximate limits) Let u belong to LLc(O). (a) 8" is a Ln-negligible Borel set and u(x) : 0\8" --+ R 1 is a Borel function coinciding with u L'
(b) if x E 0\8", the mollified function Jgu = JRn u(x + eh)J(h)dh converges to u(x) as e -I- 0, where J(h) is the standard mollifer with compact support on B 1 (0). Proof (a) is trivial, in reference to Propsition 3.64[1]. (b)

IJgu - u(x)1

~ 1IJ1100

~ 1IJ1l00

r

lB 1 (O)

r

lBl'I(O)

r

= 1\11100 lB1'1

lu(x

+ eh) -

lu(x

+ eh) -

u(x)ldh u(x)ldh

cnlu(y) - u(x)ldy,

c.

where the second inequality is derived from the inequality ~Iy-xl < d(y,x) < ely-xli- (hence B 1(0) C Bl"(O)). By Remark 9, the expression on the right side tends to 0 as e -I- O. In Euclidean spaces, the approximate differential of a function u E LLc(O) has been well defined. See for instance[1][51. Now we try to define the approximate differential of u E LLc(O) in the framework of C-C spaces. It is not easy to do this because a C-C space is only a metric space which may fail to be a Banach space and one can not expect a linear structure. In general,

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XjU can be only regarded as directional derivatives whose linear combination may not be the limit of the quotient of difference even if U is smooth, i.e, ,

lim

r-'tO

u(y) - u(x) -

1

B(x,r)

m

l: Xju(x)(Yj

j=l d(y, x)

- Xj) dy = 0

(10)

may not hold. So we can' ~ define approximate differentials in general but in the case that vector fields are of Carnot type. From now on we always make such a assumption for vector fields. Definition 11 (approximate differentiability) Let vector fields Xl, ... ,Xm be of Carnot type and U E LLc (0), x E O\Su' We say that U is approximately differentiable at x in the framework of C-C spaces if there exists a Rm-valued operator V'u(x) = (ll(X),"', lm(x)) such that m u(y) - u(x) - '" LJ I,J (y.J - x J.) lim

P--+O

1

IB(x, p)I

1

j=l d(x, y)

B(x,p)

(11)

dy = O.

U is approximately differentiable at x, V'u(x) determinated uniquely by u is called the approximate differential of u at x. For any ~ = (6, ... , ~n) E Rn, V'u( x) acts on ~ as follows:

If

m

V'u(x)(~) = 2:Zj~j.

(12)

j=l

Remark 12 By Proposition 4, V'u(x) is reduced to Xu Lipschitz function in (Rn, d).

= (Xlu,"', Xmu)

when u is a

Proposition 13 Let u E Ltoc(O). The set D u c O\Su of points where u is approximately differentiable belongs to the Borel algebra 8(0). Proof Denote by D = {ld a countable dense set in the space of R'": An argument analogous to Proposition 3.17[lJ shows that D u can be represented by

nu

{x E O\Su : limsupp- Q-

p=l k=O

p.j.O

1

J

ju(y) - u(x)
~}

(13)

P

Bp(x)

and therefore is a Borel set, where (y - x)m := (YI - Xl,'" ,Ym - x m).

3

Proof of Main Results Let us start from stating a lemma. (see appendixl'"). Lemma 14 Let J(h) be the standard mollifier with compact support BI(O), x E O\Su,

i,« =

JRn

u(x + ch)J(h)dh for u E LLc(O), Y =

(a) Y(Jeu) = Je(Yu) _

where Jeu

+ i,«,

n

l: bkfh, bk E Lip(Rn , 1·1).

k=l

= J u(x + ch)Ke(x, h)dh and Ke(x, h) = Rn

n

:

l:

k=l

Then

at [(bk(x + c) - bk(x)]J(h).

(b) limE--+o IIJeull£l(O) = O. Proof of Theorem 5 It is not restrictive to assume x = O. Temporarily assume u to be smooth and p E (0,1). By Proposition 3, for arbitrary y E Br(O)\{O} there exists a geodesic

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Vo1.23 Ser.B

=

=

=

ry: [O,T] -+ H" joining 0 and y such that ry(O) 0, ry(T) ry(d(O,y)) y, and t d(ry(t) , 0). For 0 < p < 1, take ~ E ry such that d(~,O) = pT (equivalently ~ = ry(pT)). Our aim is to estimate u(y) - u(~). Defining a smooth function f(t) := u(ry(tT)) for p ::; t ::; 1, one can get u(y) -

u(~)

= u(ry(T)) -

=

II /

u(ry(pT)) = f(l) - f(p)

(14)

o«. 1" d(O, y)dt.

(15)

(t)dt =

II

So,

t' Dsi- i'dt.

u(y) - u(~) = d(y, 0) Noticing by (1) that

Jp

m

i'(tT)

=L

j=l

h j (t )X j (ry(tT )), Ih(t)1 ::; 1

and Xju(x) = Du - Xj(x), we can conclude that

lu(y) - u(~)1 ::; d(y,O)

r (ry(tT ))ldt . J IX u 1

p

Integrating the above inequality and using Fubini's Theorem, we get

(16) with change of variables ry(td(y, 0)) = T, the right to the above expression can be estimated by for some constant C,

C

rr 1

Jp

rQIXu(T)ldTdt,

JB'rCO)

hence for u E C 1(0) nBVx(O), IXul(O) =

In IXu(x)ldx we have,

r lu(y) - uWI dy::; Jr rQIXul(Btr)dt. Je; d(y,O) 1

p

Letting p .j.. 0, we obtain the desired result. Now we consider the case that u is not necessarily smooth. We have

(17) By Lemma 14, we get

IXJcul(Btr)

=! =!

B'r

B'r

d~~)XjJcu)2)!dx j=l

C£(J1Xju+ ilu)2)!dx j=l

-+!

Bi;

d:(Xju)2)!dx j=l

=

IXul(B tr),

as c.j.. O. Proof of Theorem 6 Let Xu = XUu + X8 U be the Lebesgue decomposition (see P134, Theorem e[l1J) of Xu in absolutely continuous and singular part with respect to L'', and

Song et al: DECOMPOSITION OF BV FUNCTIONS

No.4

439

suppose that v E L I (fl., Rm) is the density of xau with respect to Ln. We shall prove that u is approximately differentiable at any x E fI.\(Su USv) such that IXSul(Bp(xo)) = O(pQ) as p.(. O. By Proposition 10(a), Ln- a.e. point of fI. has these properties. To prove the approximate differentiability at any such point x we apply Theorem 5 to BV function w(y) = u(y) - u(x) (ii(x), (y - x)m). We get r

-Q

J

e.;

lu(y) - u(x) - (v(x), (y - x)m)1 d < IXwj(Btr(x)) y sup d(y,x) - tE(O,I) (tr)Q

(18)

Since Xl,'" ,Xm are of Carnot type, simple computation shows that Xw(y)

= Xu(y) -

ii(x)L n

= (v(y)

- ii(x))L n

+ XSu(y).

Our choice of x yields that

IXwl(Bp(x))

=

r

lBp(x)

Iv(y) - v(x)ldx

+ IXSul(Bp(x)) = o(pQ).

(19)

Letting r .(. 0 in (18), we obtain the approximate differentiability of u at z and uniqueness of approximate differential from the equality \7u(x) = ii(x). References

2 3 4 5 6 7 8 9 10 11

Ambrosio L, Fusco N, Pallara D. Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, 2000 Monti R, Cassano F S. Surface measures in Carnot-Caratheodry spaces. Calc Var, 2000 Garofalo N, Nhieu D M. Isoperimetric and Sobolev Inequalities for Carnot-Caratheodory spaces and the existence of minimal surfaces. Comm Pure Appl Math, 1996,49: 1081-1144 Capogna L, Lin F H. Legendrian energy minimizers. Part I: Heisenberg group target. Calc Var, 2000 Evans L C, Gariepy R F. Measure Theory and Fine Properties of Functions. Boca Raton: CRC Press, 1992 Gromov M. Carnot-Caratheodory spaces seen from within, in sub-Riemannian geometry. In: Bellaiche A, Risler J, eds. Progress in Mathematics 144. Basel: Birkhauser, 1996 Gromov M. Structures metriques pour les varietes Rimannienners. CEDIC, Paris, 1981 Ambrosio 1. Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces. Adv Math, 2001,159: 51-67 Miranda M. JR Functions of bounded variation on "good" metric measure spaces (Preprint) Cheeger J. Differentiability of Lipschitz functions on metric measure spaces. GAFA Geom Funct Anal, 1999,9: 428-517 Halmos P R. Measure Theory. Springer-Verlag, 1989