Uniformly spread measures and vector fields

0.15 MB·English

by  Mikhail Sodin

Uniformly spread measures and vector fields
Mikhail Sodin∗ and Boris Tsirelson∗
8
0
0 Abstract
2
We show that two different ideas of uniform spreading of locally
n
finite measures in the d-dimensional Euclidean space are equivalent.
a
J The first idea is formulated in terms of finite distance transportations

6 to the Lebesguemeasure, while the second idea is formulated in terms
1
ofvectorfieldsconnectingagivenmeasurewiththeLebesguemeasure.

]
A
C
1 Introduction
.
h
t
a
This text aims to disentangle and make explicit some ideas implicit in our
m
work [9]. It can be read independently of [9].
[

1 Given a locally finite non-negative measure ν on the Euclidean space Rd,
v we are interested to know how evenly is the measure ν spread over Rd? First,
5
we consider counting measures for discrete subsets X ⊂ Rd: ν = δ
0 X x∈X x
5 where δ is a unit measure sitting at x. Following Laczkovich [7, 8]P, we say
x
2 that the set X (and the measure ν ) are uniformly spread in Rd if there
. X
1 exists a bijection S: Zd → X such that sup{|S(z) − z|: z ∈ Zd} < ∞.
0
8 Equivalently, there exists a measurable map T: Rd → X called the marriage
0
between the d-dimensional Lebesgue measure m and ν (a.k.a. “matching”,
: d X
v “allocation”) that pushes forward the Lebesgue measure m to ν and such
i d X
X that sup{|T(x)−x|: x ∈ Rd} < ∞.
r ToextendthenotionofuniformspreadingtoarbitrarymeasuresonRd,we
a
use the idea of the mass transfer that goes back to G. Monge and L. V. Kan-
torovich[5,ChapterVIII,§4]. Letν andν belocallyfinitepositivemeasures
1 2
on Rd. We call a positive locally finite measure γ on Rd×Rd a transportation
from ν to ν , if γ has marginals ν and ν , that is
1 2 1 2
ϕ(x)dγ(x,y) = ϕ(x)dν (x),
ZZ Z 1
Rd×Rd Rd
∗Supported by the Israel Science Foundation of the Israel Academy of Sciences and
Humanities
1
and
ϕ(y)dγ(x,y) = ϕ(y)dν (y)
ZZ Z 2
Rd×Rd Rd
for all continuous functions ϕ: Rd → R1 with a compact support. Note that
if there exists a map τ: Rd → Rd that pushes forward ν to ν , then the
1 2
corresponding transportation γ is defined as follows:
τ
ψ(x,y)dγ (x,y) = ψ(x,τ(x))dν (x)
ZZ τ Z 1
Rd×Rd Rd
for an arbitrary continuous function ψ: Rd ×Rd → R1 with a compact sup-
port.
The better γ is concentrated near the diagonal of Rd × Rd, the closer
the measures ν and ν must be to each other. We shall measure such a
1 2
concentration in the L∞-norm and set
Tra(ν1,ν2) = inf||x−y||L∞(γ) = inf sup{|x−y|: x,y ∈ spt(γ)} ∈ [0,∞],
γ γ
where the infimum is taken over all transportations γ, and ‘spt’ denotes the
closed support. Clearly, Tra(ν ,ν )+Tra(ν ,ν ) ≥ Tra(ν ,ν ). By Tra(ν) =
1 2 2 3 1 3
Tra(ν,m ) we denote the transportation distance between the measure ν and
d
the Lebesgue measure m . If ν = ν with a discrete set X ∈ Rd, then
d X
const·Tra(ν) ≤ inf sup |S(x)−x| ≤ Const·Tra(ν)
S x∈Zd
where the infimum is taken over all bijections S: Zd → X. This follows, for
instance, from the locally finite marriage lemma discussed two paragraphs
below. Throughout the paper, ‘Const’ and ‘const’ mean positive constants
that depend only on the dimension d. The values of these constants can be
changed at each occurrence.
There exists a dual definition of the transportation distance Tra(ν ,ν ).
1 2
The distance Di(ν ,ν ) is defined as the infimum of r ∈ (0,∞) such that
1 2
ν (B) ≤ ν (B ), and ν (B) ≤ ν (B ), (1.1)
1 2 +r 2 1 +r
for each bounded Borel set B ⊂ Rd. Here, B is the closed r-neighbourhood
+r
of B (actually, for our purposes, we could take open neighbourhoods as well).
The distance Di ranges from 0 to +∞, the both ends are included. We define
the discrepancy of the measure ν as D(ν) = Di(ν,m ). The following duality
d
is classical.
Theorem 1.2. Tra(ν ,ν ) = Di(ν ,ν ). In particular, Tra(ν) = D(ν).
1 2 1 2
2
For finite measures ν and ν , it follows from a result of Strassen [10,
1 2
Theorem 11] and Sudakov [11]. If the measures ν and ν are counting
1 2
measures of discrete sets X and X , then it follows from a locally finite
1 2
versionofthemarriagelemmaduetoM.HallandR.Rado,seeLaczkovich[8].
Note that the locally finite marriage lemma asserts existence of a bijection
between the sets X and X which is more than a transportation from ν to
1 2 1
ν . Theorem 1.2 is also mentioned in Gromov [3, Section 31], though the
2 2
exposition there is quite sketchy. For the reader’s convenience, we recall the
proof in Appendix.
A different idea of connecting the measures ν and m comes from the
d
potential theory. We say that a locally integrable vector field v connects the
measures ν and m if divv = ν −m (in the weak sense), that is
d d
hv(x),∇ϕ(x)idm (x) = − ϕ(x)d(ν −m )(x)
Z d Z d
Rd Rd
for all smooth compactly supported functions ϕ: Rd → R1. It is easy to see
that such a field always exists. For instance, we can take v = ∇h, where h is
a solution to the Poisson equation ∆h = ν −m in Rd which exists due to a
d
subharmonic version ofWeierstrass’ representation theorem[4, Theorem4.1].
Let B(x;r) be a ball of radius r centered at x, and rB = B(0;r). Set
χr = md(1rB)1lrB where1lrB istheindicatorfunctionoftheballrB. Wemeasure
the size of the field v as follows.
Definition 1.3. For a locally integrable vector field v on Rd, we set
Ra(v) = inf {r +kv∗χ k } and Ra(v) = inf {r +k|v|∗χ k } ,
r ∞ r ∞
r>0 r>0
f
where ∗ denotes the convolution.
Evidently, Ra(v) ≤ Ra(v) ≤ kvk . Note that the multiplicative group
∞
R acts by scaling on the measures and vector fields: ν (B) = ν(tB), v (x) =
+ f t t
t−1v(tx). Theseactionsare‘coordinated’: ifdivv = ν−m ,thendivv = ν −
d t t
m , and they are respected by our definitions of Tra, Ra and Ra: Tra(ν ) =
d t
t−1Tra(ν), Ra(v ) = t−1Ra(v) and Ra(v ) = t−1Ra(v).
t t f
Theorem 1.4. Let ν be a non-negaftive locally fifnite measure on Rd. Then
const·infRa(v) ≤ Tra(ν) ≤ Const·infRa(v),
v v
f
where the infimum is taken over all vector fields v connecting the measures
ν and m .
d
3
This is the main result of this note. In the proof of the upper bound we
use duality and actually prove that D(ν) ≤ Const·Ra(v). For this reason,
our technique gives no idea how transportations γ may look like in the case
when Tra(ν) is finite.
Corollary 1.5. Let u be a C2-function on Rd such that ∆u = ν−m . Then
d
Tra(ν) ≤ Const kuk .
∞
p
One can juxtapose this corollary with classical discrepancy estimates due
to Erd˝os and Tur´an and Ganelius. In [1] Ganelius proved that if ν is a prob-
ability measure on the unit circumference T ⊂ C, and m is the normalized
Lebesgue measure on T, then
sup|ν(I)−m(I)| ≤ Const supUν ,
I q T
where the supremum is taken over all arcs I ⊂ T, and
Uν(z) = log|z −ζ|dν(ζ)
Z
is the logarithmic potential of the measure ν. Since Um vanishes on T, we
can rewrite this as
sup|ν(I)−m(I)| ≤ Const supUν−m.
I q T
Notethesupremum ontheright-handside, not thesupremum oftheabsolute
value as in our result.
Proof of Corollary 1.5. Consider the convolution u = u∗χ . We have
r r
∇u = u∗∇χ , and ∆u = div∇u = ν ∗χ −m .
r r r r r d
Noting that ∇χ is a finite vector measure of total variation
r
k∇χ k = k∇χ k ·r−1 = Const·r−1,
r 1 1 1
we have
Const
Ra(∇u ) ≤ k∇u k ≤ kuk ·k∇χ k = ·kuk ,
r r ∞ ∞ r 1 r ∞
and
Const
Tra(ν) ≤ Tra(ν,ν∗χ )+Tra(ν∗χ ) ≤ r+Const·Ra(∇u ) ≤ r+ ·kuk .
r r r ∞
r
Choosing r = kuk , we get the result.
∞
p
4
This corollary immediately yields a seemingly more general result (cf. [9,
Theorem 4.3]).
Corollary 1.6. Let u be a locally integrable function in Rd such that ∆u =
ν −m weakly. Then
d
Tra(ν) ≤ Const·inf r + ku∗χ k . (1.7)
r ∞
r>0(cid:8) p (cid:9)
Proof of Corollary 1.6. Denote by χ the 3-rd convolution power of χ and
r r
put u = u∗χ . Then u is a C2-function and ∆u = ν ∗χ −m . Since the
r r r r r d
function χ is supported by the balle3rB, we have Tra(ν) ≤ 3r+Tra(ν∗χ ).
r r
e e
Corollary 1.5 applied to the smoothed potential u yields Tra(ν ∗ χ ) ≤
r r
e e
Const ku k . At last, note that ku k ≤ ku∗χ k ·kχ ∗χ k = ku∗χ k
r ∞ r ∞ r ∞ r r 1 r ∞
e
complepting the argument.
2 Proof of Theorem 1.4
2.1 The lower bound
Here, we construct a vector field v that connects the measure ν with the
Lebesgue measure m and such that Ra(v) ≤ Const·Tra(ν).
d
Let r > Tra(ν). For any x,y ∈ Rd such that |x−y| ≤ r, there exists a
f
vector field v concentrated on theballB x+y;r such that divv = δ −δ
x,y 2 x,y x y
(as usual, δ is a point measure at x of th(cid:0)e unit(cid:1)mass), and
x
|v (ξ)|dm (ξ) ≤ Const·r.
Z x,y d
Rd
(In order to see that such a field v exists, first, consider a special case r = 1;
then the general case follows by rescaling.)
Now, we take
v = v dγ(x,y)
ZZ x,y
Rd×Rd
where the transportation γ connects the measures ν and m, and is concen-
trated on the set {(x,y) : |x−y| ≤ r}. Then
divv = (δ −δ )dγ(x,y) = ν −m ,
ZZ x y d
Rd×Rd
and for every z ∈ Rd
|v(ξ)|dm (ξ) ≤ dγ(x,y) |v (ξ)|dm (ξ)
Z d ZZ Z x,y d
B(z;r) Rd×Rd B(z;r)
≤ Const·r · dγ(x,y),
ZZ
5
where thelatter integralistaken over such (x,y)thatB x+y;r ∩B z;r 6= ∅,
2
which implies |y −z| ≤ 5r. Thus, (cid:0) (cid:1) (cid:0) (cid:1)
2
Z |v(ξ)|dmd(ξ) ≤ Const·r·ZZ 1lB(z;5r/2)(y)dγ(x,y)
B(z;r) Rd×Rd
= Const·r· dm (y) ≤ Const·rd+1,
Z d
B(z;5r/2)
that is, Ra(v) ≤ Const·r, q.e.d.
Notefthat in the argument given above, the Lebesgue measure m can be
d
replaced with any measure µ satisfying µ ≤ Constm . The other inequality
d
Tra(ν) ≤ Const·Ra(v) does not permit such a replacement. Indeed, if η is
x
a normalized volume within the unit ball centered at x, then for |x−y| ≥ 2
we have Tra(η ,η ) ≥ const|x−y|, whereas it is easy to construct a vector
x y
field v connecting the measures η and η with ||v|| ≤ Const. Just take
x y ∞
v = (∇E) ∗(η −η ), E being a fundamental solution for the Laplacian in
x y
Rd.
2.2 The upper bound
In what follows, by a unit cube we mean Q = d [a ,a + 1], a ∈ Z,
i=1 i i i
1 ≤ i ≤ d. The proof of the upper bound relies onQthe following.
Lemma 2.1 (Laczkovich). Suppose that for any set U ⊂ Rd which is a finite
union of the unit cubes, we have
|ν(U)−m (U)| ≤ ρm (∂U) (2.2)
d d−1
with ρ ≥ 1. Then D(ν) ≤ Constρ.
In [8], Laczkovich proved this lemma for the counting measure ν of a
X
discrete set X ⊂ Rd. For the reader’s convenience, will recall the proof of
this lemma in A-2.
Now, the upper bound in Theorem 1.4 will readily follow from the diver-
gence theorem. We need to show that D(ν) ≤ ConstRa(v). A simple scaling
argument shows that it suffices to consider only the case Ra(v) = 1. Then
thereexists r ≤ 2such thatkv∗χ k ≤ 2. Notethatdiv(v∗χ ) = ν∗χ +m .
r ∞ r r d
let U ⊂ Rd be a finite union of the unit cubes. Then denoting by n the
6
outward unit normal to U, we have
|(ν ∗χ )(U)−m (U)| = div(v ∗χ )dm
r d (cid:12)Z r d(cid:12)
(cid:12) U (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
= hv ∗χ ,nidm ≤ kv ∗χ k m (∂U) ≤ 2m (∂U),
(cid:12)Z r d−1(cid:12) r ∞ d−1 d−1
(cid:12) ∂U (cid:12)
(cid:12) (cid:12)
whence, by L(cid:12)aczkovich’s lemma, D(cid:12) (ν ∗ χ ) ≤ Const, and finally, D(ν) ≤
r
r +D(ν ∗χ ) ≤ Const.
r
Appendix
A-1 Transportation supported by a given set
Here, we shall prove a somewhat more general result than Theorem 1.2. Let
F ⊂ Rd ×Rd be a closed symmetric set such that
F ∩(Rd ×B) is bounded whenever B is bounded. (A-1.1)
For U ⊂ Rd, set U = {x ∈ Rd: ∃y ∈ U (x,y) ∈ F}. If C ⊂ Rd is a
+F
compact set, then the set C is compact as well.
+F
Definition A-1.2.
(i) Tra(F) is a set of all pairs (ν ,ν ) of locally finite positive measures ν ,
1 2 1
ν on Rd such that there exists a transportation γ with spt(γ) ⊂ F.
2
(ii) Di(F) is a set of all pairs (ν ,ν ) of locally finite positive measures ν ,
1 2 1
ν on Rd such that
2
ν (C) ≤ ν (C ) and ν (C) ≤ ν (C )
1 2 +F 2 1 +F
for any compact subset C ⊂ Rd.
Theorem A-1.3. For any closed symmetric set F ⊂ Rd × Rd satisfying
(A-1.1), Tra(F) = Di(F).
See also Kellerer [6, Corollary 2.18 and Proposition 3.3] for a wide class
of non-closed sets F.
Theorem 1.2 follows immediately from Theorem A-1.3: just take a closed
symmetric set F = {(x,y) ∈ Rd ×Rd: |x−y| ≤ r}. Then
r
(ν ,ν ) ∈ Tra(F ) ⇐⇒ Tra(ν ,ν ) ≤ r
1 2 r 1 2
and
(ν ,ν ) ∈ Di(F ) ⇐⇒ Di(ν ,ν ) ≤ r.
1 2 r 1 2
7
Proof of Theorem A-1.3. The inclusion Tra(F) ⊂ Di(F) is rather obvious:
ν (C) = γ(C ×Rd) = γ(C ×C ) ≤ γ(Rd ×C ) = ν (C ),
1 +F +F 2 +F
and the same for the other inequality.
The proof of the opposite inclusion Di(F) ⊂ Tra(F) is based on duality.
Consider a linear space C (Rd) of continuous functions with compact support
0
in Rd endowed with standard convergence: f → f in C (Rd) if there is a ball
n 0
B such thatspt(f ) ⊂ B foralln, andthesequence f converges uniformlyto
n n
f. The dual space of continuous linear functionals M(Rd) consists of signed
measures of locally finite variation on Rd with a usual pairing ν(f) = f dν.
If a linear functional ν on C (Rd) is positive (e.g. ν(f) ≥ 0 whenevRer the
0
function f is non-negative), then it is continuous and is represented by a
non-negative locally finite measure. The same facts are true for the linear
space C (F) of continuous functions with a compact support in F, and its
0
dual space M(F).
Consider a mapping π: M(F) → M(Rd)⊕M(Rd) acting as πγ = (ν ,ν ),
1 2
where ν and ν are the marginals of the measure γ. The mapping π is well-
1 2
defined due to our assumption (A-1.1). The conjugate mapping π′: C (Rd)⊕
0
C (Rd) → C(F) is π′(f,g)(x,y) = f(x)+g(y) for (x,y) ∈ F. Assume, that
0
(ν ,ν ) ∈ Di(F). We need to show that the pair (ν ,ν ) belongs to the image
1 2 1 2
of the cone of positive measures M (F) under π; in other words, that there
+
exists γ ∈ M (F) such that
+
′
γ(π (f,g)) = (ν ,ν )(f,g) = f dν + gdν . (A-1.4)
1 2 Z 1 Z 2
We shall check below that condition (ν ,ν ) ∈ Di(F) ensures that the RHS
1 2
of (A-1.4) defines a positive linear functional on a linear subspace L =
π′(C (Rd) × C (Rd)) of C (F). The linear space C (F) is subordinated to
0 0 0 0
its linear subspace L; i.e. for any ϕ ∈ C (F) there are functions f,g in
0
C (Rd) such that
0
|ϕ(x,y)| ≤ f(x)+g(y), (x,y) ∈ F .
Then by the classical M. Riesz’ extension theorem (see e.g. [2, Chapter II,
§6, Theorem 3]) we can extend this linear functional to a positive linear
functional on the whole space C (F).
0
It remains to check that the linear functional is well-defined and positive.
Assume that it does not hold; i.e. there is a pair of functions f,g ∈ C (Rd)
0
such that
f(x)+g(y) ≥ 0, (x,y) ∈ F ,
8
however,
f dν + gdν < 0.
Z 1 Z 2
Replacing g by −g, we get a pair of functions such that
f(x) ≥ g(y), (x,y) ∈ F , (A-1.5)
and
f dν < gdν . (A-1.6)
Z 1 Z 2
Then, by virtue of (A-1.5),
{y: g(y) ≥ t} ⊂ {x: f(x) ≥ t},
+F
{x: f(x) ≤ t} ⊂ {y: g(y) ≤ t}.
+F
Using, at last, condition (ν ,ν ) ∈ Di(F), we get
1 2
ν {y: g(y) ≥ t} ≤ ν {x: f(x) ≥ t} , t > 0,
2 1
(cid:0) (cid:1) (cid:0) (cid:1)
ν {y: g(y) ≤ t} ≥ ν {x: f(x) ≤ t} , t < 0.
2 1
(cid:0) (cid:1) (cid:0) (cid:1)
Then
gdν ≤ f dν
Z 2 Z 1
which contradicts (A-1.6) and completes the proof of the theorem.
A-2 Proof of lemma of Laczkovich
We check that, for any bounded Borel set V ⊂ Rd,
ν(V) ≤ m (V ), (A-2.1)
d +Cρ
m (V) ≤ ν(V ). (A-2.2)
d +Cρ
Take M = [2ρd]+1 and denote by Q the collection of all cubes of edge
M
length M,
d
Q = [a M,(a +1)M]
i i
Y
i=1
with a ∈ Z, 1 ≤ i ≤ d. Given a bounded Borel set V, consider the cubes
i
Q , ..., Q from Q that intersect the set V, and denote by Q′ = 3Q the
1 n M i i
cube concentric with Q of thrice bigger size, 1 ≤ i ≤ n. Set
i
n n
′
A = Q , B = Q .
i i
[ [
i=1 i=1
We’ll need a simple geometric claim.
9
Claim A-2.3.
2d 2d
m (∂A) ≤ m (B \A), m (∂B) ≤ m (B \A).
d−1 d d−1 d
M M
Proof of Claim A-2.3. First, we consider the boundary of the set A: ∂A =
r F where F is a face of some cube Q . By P we denote the cube
j=1 j j ij j
Sobtained by reflection of Q in F ; clearly, for all j, P ⊂ B \A. Each cube
ij j j
can be listed at most 2d times in the list of cubes P , ..., P (since every P
1 r j
cannot have more than 2d neighbours among the cubes Q , ..., Q ). Thus,
1 n
r
2dm (B \A) ≥ m (P ) = rMd = M ·rMd−1 = Mm (∂A).
d d j d−1
X
j=1
Thisgivesusthefirstinequality. Toestimatem (∂B), wenotethatB\A =
d−1
s R where R , ..., R are different cubes from the collection Q , and
j=1 j 1 s M
Sthat ∂B ⊂ s ∂R . Whence,
j=1 j
S
s
2d 2d
m (∂B) ≤ m (∂R ) ≤ s·2dMd−1 = sMd = m (B \A)
d−1 d−1 j M M d
X
j=1
proving the claim.
Now, we readily finish the proof of the lemma. We choose a constant C
(depending on the dimension d) so big that B ⊂ V . Then
+Cρ
ν(V) ≤ ν(A) ≤ m (A)+ρm (∂A)
d d−1
2dρ
≤ m (A)+ m (B \A) ≤ m (B) ≤ m (V ),
d d d d +Cρ
M
whence (A-2.1); and
2dρ
ν(V ) ≥ ν(B) ≥ m (B)−ρm (∂B) ≥ m (B)− m (B \A)
+Cρ d d−1 d M d
≥ m (B)−m (B \A) = m (A) ≥ m (V),
d d d d
whence (A-2.2).
References
[1] T. Ganelius, Some applications of a lemma on Fourier series. Acad.
Serbe Sci. Publ. Inst. Math. 11 (1957), 9–18.
10