Table Of ContentLATTICE ACTIONS ON THE PLANE REVISITED
FRANC¸OIS MAUCOURANT AND BARAK WEISS
Abstract. We study the action of a lattice Γ in the group G=SL(2,R) on
the plane. We obtain a formula which simultaneously describes visits of an
0 orbitΓutoeitherafixedball,oranexpandingorcontractingfamilyofannuli.
1
We also discuss the ‘shrinking target problem’. Our results are valid for an
0
explicitly described set of initial points: all u∈R2 in the case of a cocompact
2
lattice,andallusatisfyingcertaindiophantineconditionsincaseΓ=SL(2,Z).
n
The proofs combine the method of Ledrappier with effective equidistribution
a
J resultsforthehorocycleflowonΓ\GduetoBurger,Str¨ombergsson,Forniand
7 Flaminio.
2
]
S
D 1. Introduction, statement of the results
.
h A classical problem in ergodic theory is to understand the distribution of or-
t
a bits for the action of a group on a space. This has been particularly well-studied
m
under the hypotheses that the acting group Γ is amenable and preserves a finite
[ measure. Removing these two assumptions leads to a realm which is not suffi-
1 ciently understood. Our purpose in this note is to describe some features which
v
arise when one studies a non-amenable group acting on a space preserving an
4
2 infinite Radon measure. We will consider the simplest setup with these features.
9 Namely, let Γ be a lattice in G = SL(2,R), that is, a discrete subgroup of finite
4
covolume. It acts on the punctured plane P = R2 (cid:114) {0} by linear transforma-
.
1
tions, preserving Lebesgue measure. It is well-known that this action is ergodic.
0
0 Moreover, when Γ is cocompact all orbits are dense, and when Γ is non-uniform,
1 any orbit is either discrete or dense.
:
v Consider an orbit Γu and an increasing family {Γ : T > 0} of finite sets in
T
i
X Γ. We will refer to Γ u ⊂ P as a ‘cloud’; we wish to understand its distribution
T
r for large values of T. For example one can ask for the frequency of visits to a
a
fixed ball in the plane. One can also consider the behavior of the orbit under
rescaling, i.e. the frequency of visits to a family of balls; if the balls are expanding
with T this corresponds to the ‘large scale’ behavior of the orbit and if the balls
are shrinking this corresponds to the behavior of the orbit ‘at a point.’ The
answers to these questions turn out to depend rather delicately on the choice of
the averaging sets Γ and the initial point u.
T
Fix a norm (cid:107)·(cid:107) on M (R), the space of two by two matrices with real entries.
2
Define for any T > 0 the set
Γ = {γ ∈ Γ : (cid:107)γ(cid:107) ≤ T},
T
1
2 FRANC¸OIS MAUCOURANT AND BARAK WEISS
let f be a compactly supported function on P and u ∈ R2.
Figure 1. ‘Cloud’ for the cocompact lattice SL (D (Z)), u =
1 2,3
(1,0), T = 100.
The asymptotics of the orbit-sum
(cid:88)
S (T) = f(γu)
f,u
γ∈ΓT
(cid:20) (cid:21)
u
were studied in [L, N, GW]. Write u = 1 ∈ R2, and define a norm |·| and
u
2
a ‘product’ (cid:63) on R2 by
(cid:13) (cid:13)
(cid:20) (cid:21)
(cid:13) −u v u v (cid:13)
|v| = max{|v |,|v |}, v(cid:63)u = (cid:13) 2 1 1 1 (cid:13),
1 2 (cid:13) −u v u v (cid:13)
(cid:13) 2 2 1 2 (cid:13)
(cid:20) (cid:21)
v
where v = 1 . Let dx denote the Lebesgue measure on P. It was shown in
v
2
the above-mentioned papers (see particularly [GW, §12.4]) that
(cid:90)
S (T) 2 f(v)
f,u
−→ dv.
T→∞
T µ(Γ\G) v(cid:63)u
P
LATTICE ACTION ON THE PLANE 3
We would like to understand S (T) at a finite time T, i.e. obtain an effective
f,u
error estimate in this asymptotic formula. In particular we would like to be able
to change the function f and the initial point u depending on the time T. Before
stating our results we introduce some notation.
Write suppf = f−1(R(cid:114){0}), and set
R(f)
r(f) = inf |v|, R(f) = sup |v|, v(f) = .
v∈suppf v∈suppf r(f)
The homogeneous space Γ\G carries a finite measure µ invariant under the right
action of G; we will normalize this measure by assuming that its lift to G satisfies
(2.9). Fix θ ∈ (0,1]. We say that a continuous compactly supported function f
on P is θ-H¨older if (cid:107)f(cid:107) < ∞, where
θ
(cid:18)(cid:90) dv (cid:19)1/2 |x|θ|f(x)−f(y)|
(1.1) (cid:107)f(cid:107) = sup|f|+ f(v)2 + sup .
θ |v|2 |x−y|θ
P 0<|x−y|≤|x|/2
Theorem 1.1. For a cocompact lattice Γ in G there are positive constants c
and δ such that for any θ ∈ (0,1], there is a positive constant C such that the
0 θ
following holds. For any u ∈ P and for any θ-H¨older f : P → R, of compact
support, let
(cid:18) (cid:19)
R(f) |u|
(1.2) D = D (u,f) = max ,
0 0
|u| r(f)
and
(cid:18)R(f)(cid:19)−θδ0
(1.3) B = B(u,f) = (logv(f)+1).
|u|
Then for any
(1.4) T > T = cD
0 0
one has
(cid:12) (cid:90) (cid:12)
(cid:12) 2T f(v) (cid:12) R(f) (cid:0) (cid:1)
(1.5) (cid:12)S (T)− dv(cid:12) ≤ C (cid:107)f(cid:107) D +BT1−θδ0 .
f,u θ θ 0
(cid:12) µ(Γ\G) v(cid:63)u (cid:12) |u|
P
Remark 1.2. (1) Our proof shows one may take δ = δ /21, where δ ≤ 1/2
0 Γ Γ
satisfies δ (δ −1) ≤ λ, with λ < 0 the first eigenvalue of the Laplacian
Γ Γ
on Γ\G. Our exponent is not optimal.
(2) The inequality (1.5) behaves as should be expected under rescaling. More
precisely, for any λ > 0, if one replaces f(v) by g(v) = f(λv), and u by
w = λ−1u, then D (u,f) = D (w,g), B(u,f) = B(w,g), T does not
0 0 0
change either, and both sides of (1.5) are unaffected.
4 FRANC¸OIS MAUCOURANT AND BARAK WEISS
The explicit error term in Theorem 1.1 is useful for studying the asymptotic
behavior of an orbit under rescaling. For an ‘expansion coefficient’ ρ > 0 con-
(cid:18) (cid:19)
x
sider the function f (x) = f . Then for a parameter α, f describes a one-
ρ Tα
ρ
parameter family of functions, and we are interested in sampling them with the
cloud Γ u. For instance, if f is the indicator function of an annulus of radius 1
T
then the orbit-sum S (T) describes the number of orbit points in the cloud
fTα,u
contained in the similar annulus of radius Tα. Since the diameter of the cloud
is approximately T, if α > 1 the orbit-sum will vanish for large T, and similarly
for α < −1. However as long as the expansion of the cloud is faster than that of
the support of the expanded function, the cloud equidistributes in the support
of the function, with respect to the same asymptotic density as in Theorem 1.1.
Namely we have:
Corollary 1.3. Given a cocompact lattice Γ in G and −1 < α < 1, 0 < θ ≤ 1,
there is δ > 0 such that for any u ∈ P and any compactly supported θ-H¨older
function f on P there are positive T and C such that for all T > T ,
0 0
(cid:12) (cid:12)
(cid:12) 1 (cid:88) (cid:16)γu(cid:17) 2 (cid:90) f(v) (cid:12)
(1.6) (cid:12) f − dv(cid:12) < CT−δ.
(cid:12)T1+α Tα µ(Γ\G) v(cid:63)u (cid:12)
(cid:12) P (cid:12)
γ∈ΓT
Remark 1.4. (1) Adapting the arguments used in the proof of Corollary 1.3
one can show that for any continuous compactly supported function φ on
φ(T)
P for which → 0 and Tφ(T) → ∞, one has
T
S 2 (cid:90) f(v)
f ,u
φ(T) → dv.
Tφ(T) µ(Γ\G) v(cid:63)u
P
(2) The case α = 1, that is the asymptotic behavior of 1 S was studied in
T2 fT,u
[M]. In this case the asymptotic density is different.
Theorem 1.1 does not hold in the non-uniform case. For example, there are
discrete orbits for Γ in the plane, and these certainly will not satisfy (1.5) if
suppf does not intersect the orbit; consequently, for a fixed T the conclusion of
Corollary 1.3 will also fail for all u sufficiently close to a point with a discrete
orbit. The behavior of the orbit will depend in a subtle way on the diophantine
properties of the slope of the initial vector; to make this precise, we will need a
bit of notation.
Let z ∈ [0,1), and denote z = [0;a ,a ,...] its continued fraction expansion,
1 2
(cid:12) (cid:12)
p ,q its convergents. Let t = −log(cid:12)z − pk(cid:12); the theory of continued fractions
k k k (cid:12) q (cid:12)
k
(see [HW]) tells us that (t ) is an increasing sequence, and that
k k
(1.7) a q2 ≤ etk ≤ (a +2)q2.
k+1 k k+1 k
LATTICE ACTION ON THE PLANE 5
(a) T =100 (b) T =25 (c) T =9
Figure 2. The ‘scaling property’. For Γ = SL(2,Z), parts of the
orbitofthreedifferentpointsareshown, atdifferentscalesandwith
different values of T. Each contains approximately 200 points.
Define, in the case where z is irrational
ˆ
ξ(z,τ ,τ ) = max(a : τ ≤ t , t ≤ τ )
1 2 k 1 k+1 k−1 2
(if the set on the right hand side is empty we set ξˆ(z,τ ,τ ) = eτ2). If z is
1 2
rational, the sequences (a ), (q ) and (t ) are finite. Let k be the length, that is
k k k 0
ˆ
z = p /q . If τ ≤ t , ξ(z,τ ,τ ) is defined by the preceding formula; if not,
k0 k0 2 k0−1 1 2
ξˆ(z,τ ,τ ) = min(cid:0)eτ2,max(cid:0)max{a : τ ≤ t },eτ2/q2 (cid:1)(cid:1).
1 2 k 1 k+1 k0
(cid:20) (cid:21)
u
If u = x ∈ P, denote by z the unique real number in the set [0,1] ∩
u
y
ˆ ˆ
{u /u ,u /u +1,−u /u ,−u /u +1}. We define ξ(u,τ ,τ ) = ξ(z,τ ,τ ).
x y x y y x y x 1 2 1 2
Theorem 1.5. For Γ = SL(2,Z) there are positive constants c and δ such that
0
for any θ ∈ (0,1], there exists C > 0 such that the following holds. For any u ∈ P
and for any compactly supported θ-H¨older map f : P → R, let D and B be as
0
in (1.2) and (1.3), and let
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
T|u| T|u|
ˆ ˆ
ξ = ξ u,log ,log .
f,T,u
R(f) r(f)
Then for any T > cD such that
0
ˆ
|u|ξ
f,T,u
(1.8) T ≥ c
R(f)
one has
(cid:12) (cid:90) (cid:12)
(1.9) (cid:12)(cid:12)S (T)− 2T f(v) dv(cid:12)(cid:12) ≤ C(cid:107)f(cid:107) R(f) (cid:16)D +BT1−θδ0ξˆθδ0 (cid:17).
(cid:12) f,u µ(Γ\G) v(cid:63)u (cid:12) θ |u| 0 f,T,u
P
6 FRANC¸OIS MAUCOURANT AND BARAK WEISS
Remark 1.6. (1) Examining our argument one sees it is possible to take δ =
0
1/48 in Theorem 1.5.
(2) Our arguments prove an analogous result for any lattice Γ in place of
ˆ
SL(2,Z). In this case, the quantity ξ(u,τ ,τ ) is replaced by the supremum
1 2
of the distance between a fixed reference point in Γ\G and Ψ(u/|u|)a ,
s
where s ∈ [τ ,τ ] — see §§7-8. Also, the δ can be taken to be δ /24,
1 2 0 Γ
where δ is defined as explained in Remark 1.4(1).
Γ
For β > 0, say z ∈ R is β-diophantine if there is c > 0 such that |z −p/q| ≥
cq−β for all p,q ∈ Z. Note that quadratic irrationals are 2-diophantine, and by
Roth’s theorem, all algebraic numbers are 2+ε-diophantine for any ε > 0, like
Lebesgue almost any real number. In the following, we extend the definition
to the case β = +∞ with the convention that every number (even rationals) is
∞-diophantine. It is well-known (see Lemma 7.2) that when z is β-diophantine,
ˆ
ξ can be bounded in terms of β. This yields:
z,τ1,τ2
Corollary 1.7. Given β ∈ [2,+∞], θ ∈ (0,1] and − 1 < α < 1, there is δ > 0
β−1
such that for any u ∈ P a vector with a β-diophantine slope and any compactly
supported θ-H¨older function f on P there are positive T and C such that for all
0
T > T ,
0
(cid:12) (cid:12)
(cid:12) 1 (cid:88) (cid:16)γu(cid:17) 2 (cid:90) f(v) (cid:12)
(1.10) (cid:12) f − dv(cid:12) < CT−δ.
(cid:12)T1+α Tα µ(Γ\G) v(cid:63)u (cid:12)
(cid:12) P (cid:12)
γ∈ΓT
Remark 1.8. In a recent and independent work [N2], Nogueira proved a similar
result, with a better estimate of the error term, but in the particular case in which
f is the characteristic function of a square, and the norm ||·|| is the supremum
norm, generalizing his previous results [N]. The method used is completly different
from ours.
Applying Theorem 1.5 to the ‘shrinking target problem’, we obtain:
Corollary 1.9. Let δ be as in Theorem 1.5, and let v,u ∈ P. Then:
0
(1) If u has β-diophantine slope, then there are positive constants C and T
0
such that for all T ≥ T there is γ ∈ Γ such that
0 T
|γu−v| < CT−23δβ0.
(2) If the slope of u is irrational then there is a positive constant C such that
there are infinitely many γ ∈ Γ solving
|γu−v| < C||γ||−δ0.
3
1.1. Notation. Throughout this paper the Vinogradov symbol A (cid:28) B means
that there is a constant C such that A ≤ CB, where A and B are expressions
depending on various quantities and the implicit constant C is independent of
these quantities. In particular, throughout the paper the implicit constant may
LATTICE ACTION ON THE PLANE 7
depend on Γ, on the choice of the norm (cid:107) · (cid:107), on auxilliary functions Ψ, φ, but
not on the function f nor the initial point u. The notation A (cid:16) B means that
A (cid:28) B and B (cid:28) A.
2. The norm estimate
2.1. The setup. Let G act on R2 by matrix multiplication on the left. Define
the following matrices
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
1 s et/2 0 cosθ −sinθ
h = , a = , r = .
s 0 1 t 0 e−t/2 θ sinθ cosθ
(cid:20) (cid:21)
1
The stabilizer of u = is precisely the unipotent subgroup
0 0
H = {h : s ∈ R},
s
so that P is identified with the quotient G/H via the map gH (cid:55)→ gu . Let Γ be
0
a lattice in G, and let τ : G → G/H and π : G → Γ\G be the natural quotient
maps.
G (cid:68)
(cid:123)π(cid:123)(cid:123)(cid:123)(cid:123)(cid:123)(cid:123) (cid:68)(cid:68)(cid:68)(cid:68)(cid:68)τ(cid:68)(cid:68)
(cid:125)(cid:125)(cid:123) (cid:33)(cid:33)
Γ\G G/H
We define a haar measure λ on H by dλ(h ) = ds.
s
2.2. The section. Define Ψ : P → G by
(cid:18)(cid:20) (cid:21)(cid:19) (cid:20) (cid:21)
x x −y/(x2 +y2)
(2.1) Ψ = ,
y y x/(x2 +y2)
The function Ψ is a section in the sense that τ ◦Ψ = Id| , i.e. for all u ∈ P,
G/H
(2.2) Ψ(u)u = u.
0
The following equation is easily verified:
(cid:13) (cid:13)
(cid:20) (cid:21)
(cid:13) 0 1 (cid:13)
(2.3) v(cid:63)u = (cid:13)Ψ(v) Ψ(u)−1(cid:13).
(cid:13) 0 0 (cid:13)
(cid:13) (cid:13)
Note that (2.3) does not depend on the choice of the section Ψ (as might not be
obvious from the formula). It can be also checked that for any t ∈ R and v ∈ P,
we have
(2.4) Ψ(etv) = Ψ(v)a .
2t
Define
(2.5) D = D(u,f) = sup (cid:107)Ψ(x)Ψ(u)−1(cid:107),
x∈suppf
8 FRANC¸OIS MAUCOURANT AND BARAK WEISS
this quantity satisfies
(2.6) D(u,f) (cid:16) D (u,f),
0
where D is as in (1.2). Indeed, let x ∈ suppf, then
0
(cid:13) (cid:13)
(cid:13) (cid:18) x (cid:19) (cid:18) u (cid:19)−1(cid:13)
(cid:107)Ψ(x)Ψ(u)−1(cid:107) = (cid:13)Ψ a Ψ (cid:13)
(cid:13) |x| 2log(|x|/|u|) |u| (cid:13)
(cid:13) (cid:13)
(cid:18) (cid:19)
|x| |u|
(cid:16) (cid:107)a (cid:107) (cid:16) max , .
2log(|x|/|u|)
|u| |x|
2.3. The cocycle. Let u ∈ P and g ∈ G, define c (g) by the following implicit
u
equation:
(cid:20) (cid:21)
1 c (g)
(2.7) u = Ψ(gu)−1gΨ(u).
0 1
This makes sense because the right hand side stabilizes u , so is in H. It is easily
0
checked that c is a cocycle, meaning it satisfies for any g ,g ∈ G and u ∈ P,
1 2
c (g g ) = c (g )+c (g ).
u 1 2 g2u 1 u 2
One also sees that in terms of Iwasawa decomposition, we have
(2.8) g = kan =⇒ ka = Ψ(gu ), n = h where s = c (g).
0 s u0
Therefore we can write haar measure µ on G by the formula
(2.9) dµ(g) = dτ(g)dλ(c (g)).
u0
Note that the normalization of Lebesgue measure on R2 and λ determine a nor-
malization for µ. Changing the section Ψ gives rise to a homologous cocycle, so
that µ is actually independent of Ψ. It follows from (2.8) that
(2.10) c (gh ) = c (g)+s, c (ga ) = e−tc (g),
u0 s u0 u0 t u0
and from (2.2) that
(2.11) c (g) = c (gΨ(u)).
u u0
Lemma 2.1. Let D = D(u,f) be as in (2.5). For any f ∈ C (P), any u ∈ P
c
and any g ∈ G for which gu ∈ suppf we have
(cid:12) (cid:12)
(2.12) (cid:12)(cid:107)g(cid:107)−|c (g)|(gu(cid:63)u)(cid:12) ≤ D.
(cid:12) u (cid:12)
Proof. By (2.7), we have
(cid:18) (cid:20) (cid:21)(cid:19)
0 c (g)
g = Ψ(gu) Id+ u Ψ(u)−1
0 0
(cid:20) (cid:21)
0 1
= Ψ(gu)Ψ(u)−1 +c (g)Ψ(gu) Ψ(u)−1.
u 0 0
LATTICE ACTION ON THE PLANE 9
By (2.3),
(cid:13) (cid:20) (cid:21) (cid:13)
(cid:13) 0 1 (cid:13)
(cid:13)c (g)Ψ(gu) Ψ(u)−1(cid:13) = |c (g)|(gu(cid:63)u).
(cid:13) u 0 0 (cid:13) u
The claim follows. (cid:3)
2.4. Some useful inequalities. Here we state and prove elementary inequal-
ities that will be useful later. The first remark is that the (cid:63)-product is well-
approximated by the product of the norms:
(2.13) v(cid:63)u (cid:16) |v||u|.
Indeed,
(cid:13) (cid:13)
(cid:20) (cid:21)
(cid:13) −u v u v (cid:13)
v(cid:63)u = (cid:13) 2 1 1 1 (cid:13) (cid:16) max{|u v |} = max{|u |}max{|v |}.
(cid:13) −u v u v (cid:13) i j i j
(cid:13) 2 2 1 2 (cid:13) i,j=1,2 i=1,2 j=1,2
The following upper bound will also prove helpful: for any θ-Ho¨lder compactly
supported f : P → R
(cid:12)(cid:90) (cid:12)
(cid:12) f(v) (cid:12) R(f)
(2.14) (cid:12) dv(cid:12) (cid:28) (cid:107)f(cid:107) .
θ
(cid:12) v(cid:63)u (cid:12) |u|
P
This is proved as follows:
(cid:12)(cid:12)(cid:90) f(v) (cid:12)(cid:12) (2.13) (cid:12)(cid:12)(cid:90) f(v) (cid:12)(cid:12) (cid:18)(cid:90) (cid:19)1/2(cid:18)(cid:90) (cid:19)1/2
(cid:12) dv(cid:12) (cid:28) (cid:12) dv(cid:12) (cid:28) |u|−1 f2(v)|v|−2dv dv ,
(cid:12) v(cid:63)u (cid:12) (cid:12) |v||u| (cid:12)
P suppf P suppf
by the Cauchy-Schwarz inequality, and since suppf is in a disk of radius approx-
imately R(f), this implies (2.14).
3. From the plane to the homogeneous space
¯
In this section we pass from a function f : P → R to a function f : Γ\G → R.
This is done in two steps: lifting to a function f(cid:101)using the section Ψ and a bump
¯
function; and summing along Γ-orbits to obtain a function f on Γ\G.
Assume f is compactly supported and non-negative. Fix φ : R → R a non-
(cid:82)
negative C∞ function, vanishing outside [−1,1], such that φ(t)dt = 1. Set
R
(3.1) f(cid:101): G → R, f(cid:101)(g) = f(τ(g))φ(c (g))
u0
(a compactly supported smooth function on G) and
(cid:88)
¯
(3.2) f(x) = f(cid:101)(g)
g∈π−1(x)
(a finite sum for each x). The normalization (2.9) for µ ensures that
(cid:90) (cid:90)
¯
(3.3) f dµ = f(x)dx.
Γ\G R2
10 FRANC¸OIS MAUCOURANT AND BARAK WEISS
ThedistributionofthecloudΓ turnsouttobelinkedwiththenormv (cid:55)→ v(cid:63)u,
T
and for this reason we will have to work with the more precise measures of the
support of f:
r(u)(f) = inf v(cid:63)u (cid:16) r(f)|u|, R(u)(f) = sup v(cid:63)u (cid:16) R(f)|u|,
v∈suppf v∈suppf
R(u)(f)
v(u)(f) = (cid:16) v(f).
r(u)(f)
Lemma 3.1. Let u ∈ P, u(cid:101) = Ψ(u), γ ∈ Γ, f ∈ Cc(P) and let f(cid:101)be as in (3.1).
Let D = D(u,f) be as in (2.5).
Then
(cid:90) 1+(T+D)/r
r ≤ r(u)(f), (cid:107)γ(cid:107) ≤ T =⇒ f(cid:101)(γu(cid:101)hs)ds = f(γu),
−(1+(T+D)/r)
(3.4)
(cid:90) (T−D)/R−1
R(u)(f) ≤ R, (cid:107)γ(cid:107) ≥ T =⇒ f(cid:101)(γu(cid:101)hs)ds = 0.
−((T−D)/R−1)
(cid:82)
Proof. Since φ(s)ds = 1 and suppφ ⊂ [−1,1], for the first claim it suffices to
show that for (cid:107)γ(cid:107) ≤ T and γu ∈ suppf we have
[−1,1] ⊂ {c (γuh ) : |s| ≤ 1+(T +D)/r},
u0 (cid:101) s
or by (2.10), that
T +D
|c (γu)| ≤ .
u0 (cid:101) r
This follows from (2.11) and (2.12). The proof of the second claim is similar. (cid:3)
¯
We will need to control the H¨older norm of f in terms of that of f. For
θ ∈ (0,1], define a Ho¨lder norm on compactly supported function f on G or Γ\G:
|f(x)−f(y)|
||f|| = sup .
θ dist(x,y)θ
dist(x,y)≤1
Here by dist we denote a left-invariant Riemannian metric on G, or the corre-
sponding metric induced on Γ\G.
Lemma 3.2. For any σ > 1 and any θ ∈ (0,1] there is a constant c = c > 0
σ,θ
and a compact set K ⊂ Γ\G such that for any θ-H¨older compactly supported
σ
function f with
suppf ⊂ A = {w ∈ P : σ−1 ≤ |w| ≤ σ},
σ
¯
we have suppf ⊂ K and
σ
¯
(3.5) (cid:107)f(cid:107) ≤ c(cid:107)f(cid:107) .
θ θ
