Nets oî subgroups in locally compact groupsAbstract. The approximation of the integral of a function / in a locally compact

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE X X (1978)

J

L. B

* (Princeton)

Nets oî subgroups in locally compact groups

Abstract. The approximation of the integral of a function / in a locally compact group by average functions / # defined by subgroups H of the group is studied in some detail, with other related questions and a few applications.

0. Introduction. An old well-known result, due to Kolmogorov, states th a t given a function f e L 1 (T) = и ( [ 0 , 1 ) ) , the functions

/« 0*0 = — У / р + — I, n J-J \ П n = 1 > 2 , 3 , . . . , i

converge in L 1 to I = J f(x)dx (see [ 6 .]; УП.4 for a related result). More 0

precisely, if cop denotes the modulus of continuity in L P1 one finds (see [4]

or [5])

(0 .1 ) !l/„ ~i\\r < <op if-, - i - j (1 < P s; с о ).

On the other hand, Jessen proved later (see [2]) th a t ( 0 . 2 ) / 2n (*)->! (a.e.).

This type of results also holds if we replace the torus T — [0 , 1 ) by the real line B, defining for each / e L l (B)

f r(x) = r j £ f ( x + hr) (r > 0 )

h e Z

and making r->0. The convergence in L 1 or L p is local in this case.

Our aim is to give a treatm ent of these questions in the general setting of locally compact groups. Besults of the type (0.1) are Theorems 2,3 and 4 below, and Corollary 3, while Corollary 4 provides the natural extension of Jessen’s result (0.2). All this is studied^'n Section 2 . Pre-

* Dedicated to Prof. Luis Vigil (Univ. of Zaragoza, Spain) who introduced

me to this problem.

viously, we give the definition and basic facts of fundamental domains for quotient groups in Section 1 . In the definition of f n and f r above, the

f 1 n - 1)

subgroups H n = <0, — ---> and H r = rZ give rise to funda-

[ n n j

mental domains ( 0 , 1 jn) in T and ( 0 , r) in B. Similarly for T n (or Т ш) and B n.

If / e L l {T) and ektn(x) = (f(x)e~2nikx)n one easily verifies П

(0.3) f(x ) = 2 скш+1(х)еык*.

k=—n

Thus, (0.1) and (0.2) seem to lead (only formally) to convergence of Fourier series. However, this approach only gives estimates like

\\sn - f l i p < (const) cop 1 ogn

which for p = 1 or p — oo means the Dini-Lipschitz test and its integral analogue (see [ 6 ]; I I .6 and TV. Example 7).

Elementary inversion formulae like (0.3) are also valid for T n and B n. The extension to other groups is given in Section 3, Theorem 7.

Finally, the fundamental domains associated to an increasing sequence of subgroups behave in some sense like the rectangles in B n, and Section 4 is devoted to the study of the corresponding maximal function.

The notation to be used is as follows : G will denote a locally compact group with identity e and left H aar measure m. A normal closed sub­

group of G will always be written H (or Hj), m H will be a left H aar measure for H and, given a function / defined on G

/ я И = j f{ x t) d m H(t) H

will be defined whenever the right-hand side exists (a.e.). The function f(n{x)) = / н ( ж) is then well defined on G/Н, and there is a left H aar

measure m on GjH such th a t Weil’s identity holds:

(0.4) J fd m — J fd m (f e L x((r))

G G/H

(see [3]; III.4.5). We shall assume m, mH and m adjusted in this way.

If the subgroup is щ we write fj and fa, and similarly m j, щ . Moreover, any compact group will be assumed to have total measure 1 .

If f e IP (G) and V is a neighbourhood of e, the modulus of con­

tinuity is defined in an obvious way

V) = SUP Wf(™)-f(x)\\p-

veV

Finally, when G is Abelian, G will be its dual group, with H aar measure m* related to.m so th a t Plancherel’s theorem has the form: ||/||2 = ||/||2-.

1. Fundamental domains. An open subset V of G is called a fu n ­ damental domain for the quotient group G/H if these two conditions hold:

(1.1) V V ~ l r\E = {(e},

( 1 . 2 ) the complement of VH in G is a locally null set.

Observe th a t (1.1) means th a t the restriction to V of the projection 7i : G-+E/H must be one-to-one. I t is clear th a t no fundamental domain can exist for G/H unless H is discrete. On the other hand, we have an existence result for discrete H.

T

1. I f H is discrete, each open subset W of G such that W H = G, contains a fundamental domain V for G/H.

L

1 . Let be the family of all neighbourhoods U of e in G which satisfy

(a) TJV~l rsH = {e},

(b) the boundary of tz {TJ) has measure zero in GjH.

Then % is a basis of neighbourhoods of e in G.

P r o o f . We can start with a neighbourhood U0 of e such th a t I7 0 Z7^1n n H = {e}. Let g be a continuous function on GjH with compact support and

O < 0 < 1 ; g(e) = 1 ; supp(^) с n( TJ0) . For each r, with 0 < r < 1, define

Ür = {x e G /H : r < g ( x ) } .

The boundary of V r is contained in g~x{r). But (J g~l (r) is contained

0 < r < l

in supp(gf), so th a t a t most countably many of the sets g~l {r) have positive measure. Choose r such th a t ^-1(r ) has measure zero. Then U = U0r\

r\7i~l {Vr) is a neighbourhood of e which satisfies (a) and (b). Q.E.D.

Before we prove the theorem, observe th a t we can reduce it to the case of G being <r-compact, because we can consider in any case a ^-com­

pact open subgroup 8 of G, and a fundamental domain for S / H n S would be also a fundamental domain for G[H.

Another observation is th a t (1.2) is equivalent to (1.2') The complement of ti {V) in G/H is locally null.

The equivalence follows from Weil’s identity (0.4).

P r o o f of such th a t

T h e o r e m 1. For each x e G / H , let x e W, TJx eW be x = n(x), xUx c W .

As G/H is assumed to be cr-compact, choose a sequence {TJj} of open sub­

sets of G satisfying conditions (a) and (b) of Lemma 1, and such th a t

Now, define

G IE = TJj c W .

= * № ) 4 U « W } , * = 2 , 3 , . . . i-i

oo

Then V = (J Vk is open in GjH, and its complement is contained in the union of the boundaries of л( Uk), which is locally null (actually, it has measure zero) due to Lemma 1. Finally, let

Vk ^ n - 1( t h)r\Uk, Tc = 1 , 2 , 3 , ' . . . ,

OO

v = u v k- fc=1

The sets Vk are pairwise disjoint and the restrictions of л to each of them are one-to-one. Thus, V satisfies (1.1). Condition (1.2') has already been 'proved because л(У) = V, and

V <= U TJk cr W.

*=i Q.E.D.

The measure mH on H will be the counting measure multiplied by some constant c > 0, namely: c = mH({e}). Then, for any f e L x(G)

HV teH tv / ^ f { t x ) d m { x )

V teH “ f f H(a>)dm{x).

G

Using Weil’s identity and the fact th a t every g e L X(GIH) has the form g = f for some f e L x(G), we get

C

1. Let V be a fundamental domain for GjH. Then j g dm

GjH

1

G f ^{(go л) dm.}

Furthermore, the measure of all fundamental domains is the same:

m( V) — oo if Gj H is not compact,

m( V) = e if G/ H is compact,

where, in the second case, we mnst choose c = ты {(е}) so th a t the total measure of G jE is 1.

A trivial remark is th a t if У is a fundamental domain for GjE, so are F -1, у V and Vy, for any y e G .

Assume now th a t G is Abelian, H is discrete and G /E is compact.

The same properties are true for G, E ° and G jE °, where E ° = [sc eG:

S(E ) = { 1 }). Then the dual measures of m, mH and m are precisely m*, (m*)~ and (m*)Eo (see [3]; V.5.4), where mH and (т*)но are adjusted so th a t both quotient groups have total measure 1. Combining this with the proceeding corollary, we find

C

2 . Let G be Abelian, E discrete and G IE compact. I f V and W are fundamental domains for G IE and G jE 0, respectively, then

m(V) m*(W) = 1 .

2. Nets of subgroups. In this section we study the approximation of the integral of a function / (or the function itself) by functions of type /я? when the subgroup E becomes very large (or very small).

T

2 . Let V be a relatively compact open neighbourhood of e such that Y E = G. Let f e L 1n L p (G), with integral I — J fdm . Then

( 2 . 1 ) I f 1 < oo: ( / I f - I f d m \ ' b < <»,(/; V).

GIH

(2.2) I f 1 < ^ < oo, and G is compact

( / \fH- H pdm)Ilr < « , ( / ; V).

a

(2.3) I f p =

and supp(/) = К compact

SUP |/я (ж )- 1 К w (FA )to00(/; F ).

х е в

P ro o f. As G/Е is compact and therefore unimodular 1 = f f ( ÿ & ) d m ( ÿ ) = f dm{y) j f(yœt)dmH{t)

GIH GIH H

for any œ eG. Thus

/ я И - I = f <*»»(£) j (f(xt)-f(ynt))dmH(t).

G/H H

We can assume th a t the representative y e G chosen for each y e Gj E belongs to our neighbourhood V. Then (2.1) and (2.2) are proved from the proceeding equality by application of Minkowski’s inequality and Weil’s identity, while (2.3) follows immediately if we observe th a t | f(sct) —

~-f{yxt) I = 0 when yæt $ VK. Q.E.D.

The condition “V H = G” can be obviously replaced by the slightly weaker “G W H is locally nnll”.

Observe also th a t for compact G (2.3) means

(2.4) snp \fH( x ) - I \ < <»«,(/; V) for any f eL°°(G).

xeG

In the rest of this section we shall consider a directed set (J , -<) and, associated with it, a net of snbgronps of onr gronp G. We say th a t {Hs} is a dense net of subgroups (D.N.S.) if, given any neighbour­

hood U of e, there is some j 0 e J snch th a t

UHj = G whenever j 0 -< j .

C

3. Let {JSj} be a D .N.S. in G, and let f e L 1n L p(G) with I = f f d m (1 < p < oo). Then

( 2 . 1 ') lim J ifj —l\pdrhj = 0 . 3 G/Hj

( 2 . 2 ') I f G is compact: limfj = I in DP{G).

i *

(2.3') I f f is continuous with compact support:

lim fj{x) — I (uniformly).

j

In (2.1') the integration m ust be performed over different groups GIHj for each index j. Nothing more can be said in general, b u t when the subgroups are discrete, the use of fundamental domains enables us to obtain convergence over compact subsets of the original group G.

More precisely

T

3. Let {Hj} be a D .N .S. with each discrete, and let f e L 1n L p (G), 1 < p < oo, with I — J f dm. Then

lim// = I in Lf0C(G).

j

P ro o f. Given a compact subset К of G such th a t m{K) > 0 , let us take an open set U with these conditions

K a U, m( U) < 2 m(K).

Now, let Ж be a neighbourhood of e such th a t K W ~ l W c TJ and con­

sider any j Ej satisfying WHj = G. For these j ’s (which are all following some j 0) we can choose fundamental domains Vj of G /H j, all of them contained in TP. Fix any of these H i and assume, with no loss of generality, th a t К c VjHj. Then

K a l J t i V j , f e Hj ,

i = l ^{UV j}

^TJ.

The sets f Vj , i = 1, 2 , r, are pairwise disjoint, and therefore Г

f If s - I f dm < V j \ f j - i f dm - r f i f , - I f dm

К i = l t t V j V j

=

rm(Vj) J I fj

\pdrhj < m( U)

jfj

drhj .

G j H j G / H j

The theorem follows now from (2.1'). Q.E.D.

If every Hj is discrete and we associate to each of them a fundamental domain Vi which is a neighbourhood of e, a review of the preceeding proof shows th a t we actually have the estimate

(2.5) l i ms up— -- ( f \fj - I \ pdm)llP < m( K) llp jej cQp (f', Vj) \ J !

for any compact K c= G.

Now, we shall consider monotone nets of subgroups. We say th a t {Hj} is increasing (decreasing) in case th a t Hj с B... (Hj. a Ej) whenever 3 <3'-

T

4 . Let

be increasing, the group G compact, and U

i

dense in G. I f f e L p (G), 1 < p < oo, and I — f f dm, then lim J \ f j — I\pdm = 0 .

i о

P ro o f. Let g

be the a-algebra of measurable sets satm ated modulo

H j ,

th a t is

t$j — {E cz G : E measurable and EHj = E } . For every E e we have

f f j d m = f{f %E)jdm = f f %Edm = f f d m (x).

E G O S

On the other hand, fj is ^-m easurable, because it is constant over each cogroup of Hj. Thus

fj =

where ê stands for conditional expectation. As c when j -< j ', we find th a t {fj}jeJ is an inverse martingale, so it is convergent in IE

lim J \ fj — g\pdm = 0 i a

p) The equality f g g = fg holds when II is compact (even if О is not) because the modular function of 6? is 1 on H.

15 — Roczniki FTM Prace Mat. XX.Î

for some g e l f (G) which will satisfy, for every t e [J Hj g{xt) = g(x) a.e. j

The set of all t for which the above relation holds is known to be a closed subgroup (see [3]; III.6.5 and YIII.2.6). In this case, it will be the whole G, so th a t g = const (a.e.) and th a t constant can only be I. Q.E.D.

C

4. Let H x c= H 2 c= ... c H n a ... be such that each G/Hn is compact and U H n is dense in G. For every f e L 1 (G)

П

lim f n(x) = f f d m (a.e.).

n Q

P ro o f. If G is compact, the same argument of Theorem 4 gives the convergence (a.e.). For the general case, consider the increasing se­

quence H nj Hx (n = 1 , 2 , . . . ) in the compact group GjHly and apply Weil’s identity. Q.E.D.

T

5. I f {Hj}jeJ is decreasing, each Hj is compact and P | Hj = ( ф

then for every f e l f (G), 1 < p < oo

lim f If j - f f d m = 0 . i в

Moreover, i f is a sequence (i.e., ( J , -<) = (N , <)) lim fj(os) —f(x) a.e.

3

P ro o f. For the same reasons as in Theorem 4, {fj} is now an ordinary martingale, and there exist g e L p(G) such th a t

Kmfj = g in L P(G)

3

and there is also convergence (a.e.) when the martingale is a sequence.

But, given any compact set K c G, we have

J g dm = lim J f r dm = J fd m (j e J )

K H j j ' K H j K H j

because K H j e gy whenever j < j ' . If we take now the limit in j j g dm = J f d m

K K

because (~}KHj = К (for any œ e n K 3 n the sets (x 1K ) n H j have the

3 3

finite intersection property). As К was arbitrary, we have shown th a t g = f a.e. Q.E.D.

B e m a rk s . If {Hj}jeJ is increasing (decreasing) and we denote Я = ( J Hj (H = C\Hj)y then we can replace in Theorem 4 and Corollary 4

j i

I by f H (in Theorem 5 / by f H). This can be seen by slightly modifying the proceeding proofs or by an appropriate pass to the quotient gronp.

The results for convergence (a.e.) are not valid for nets of subgroups in general, because the almost sure convergence of martingales is no longer true for martingales which are not sequences. On the other hand, for a monotone sequence {Hn}neN of compact subgroups, the almost sure convergence is derived from the fact of being the maximal operator

= sup \fn\

П

of weak type (1.1) and of type ( p, p) for p > 1. This, however, is not true under the assumptions of Corollary 4.

Under the hypotesis of Theorem 3, the condition fj exists for every j and fj->I locally in measure

provides a definition of the integral for a class of functions estrictly wider than L 1 (except in some trivial cases). See [4] for the study of th e case when G is compact. When G = T = [0,1) this more general integral is very close to the integrals В and B x, suggested by Denjoy and studied by Boks [ 1 ] (see also [ 6 ]; V II. 4).

3. Inversion formulae. Under the hypotesis of Theorem 5, if we also assume th a t G is Abelian, the orthogonal subgroups ^ are an increasing net with \ J H) = G, and each of them is open. Con- versely, if we have this situation for G, the hypotesis of Theorem 5 holds i for G. On the other hand, an easy computation shows:

f f(&) <#, x)dm*(x) = f]{oo) (a.e.)

ttO

provided th a t the integral is absolutely convergent (the equality results from the ordinary inversion formula for the case of an integrable Fourier transform). Thus we get

Th e o r e m; 6 .

Let G be a locally compact Abelian group, and {Sj}jeJ an increasing net of open subgroups in G such that : [J 8j — G. Let f e L p(G)

(1 < P < 2 ) be such that j

(3.1) J |/|d m * < oo for all j e J . Sj

Then we have

(3.2) lim f f ( x ) ( x , xydm*(x) =/(a?) i Sj

with convergence in I f . In cctse that {Sj} is a sequence, (3.2) also holds (a.e.).

R e m a rk s . (3.2) is also true with convergence in I f whenever / e L r,

2 <

r

Examples of groups G with the structure required by Theorem 6 are (i) G = T 01 (Fourier series of infinitely many variables).

(ii) G = Q when Q has discrete topologie {G is then a quotient group of the Bohr group R).

(iii) G = /7 Qï with all G{ compact Abelian groups and I infinite.

i e l

(iv) G = (p-adic numbers}.

( v )

G = dual of (jp-adic numbers}.

Condition (3.1) is in any (non-trivial) case strictly weaker th a n / e l 1;

otherwise the theorem will be of no interest. Observe th a t (3.1) holds automatically when the subgroups Sj are also compact; this is the case in examples (iv), (v) and in example (iii) when all Gt are finite.

Our next theorem, though simple, can be regarded as an elementary inversion formula, in view of the convergence results stated in the pro­

ceeding section.

T

7. Assume that G is Abelian, H a discrete subgroup of G, and G/H compact. Let f be defined on G and such that f H exists, that is

J £ l/(® 0 I < « > л.е.

teH K

Then, for every fundamental domain W of the quotient group G/H0 (3.3) j {fx~1)H{x)^x,xydm*(x) = f(x) a.e.

w

P ro o f. According to Corollaries 1 and 2, the left-hand side of the identity to be proved can be written

m*(W) j ^ f { x t ) f t , x 1}dm*(x)

W t e H

and the identity follows from th e following fact m (TP) J* <t, x}dm*(x) = 0

w

for all t e H \{ e }

which is proved by considering each t e H as a character on the compact group G/H0 and using the properties of fundamental domains. Q.E.D.

If we apply (3.3) to each Hi of a net under conditions like those of Theorems 3 and 4 or Corollary 4, and then pass formally to the limit in j , we are led to an ordinary inversion formula for the Fourier transform.

However, we did not attem pt here to make this procedure rigorous in

the general setting of locally compact groups.

4. A maximal function. We shall assume now th a t our locally com­

pact group G has a sequence of discrete subgroups E x c: E z a ... cz E n c ... cz G

such th a t each GIEn is compact and { J H n is dense in G (then {E n} is П

a D.IST.S.). Let us denote

К = order (En+1IE n) which is a finite number for each n.

L

2. There is a sequence of open sets Vx=> V t =>...=> ...

such that Vn is a fundamental domain for Gj En, and each Vn is (except for a set of measure zero) the disjoint union of Jcn translates of Vn+l by ele­

ments of E n+l.

P r o o f . Take as V x any fundamental domain for GIEX. Let W be a fundamental domain for the quotient group of the group G/ Ex over its discrete subgroup E ZI EX. If тс: G-^G/EX is the natural projection, we know th a t V x is mapped bijectively onto n ( V x), so th at, defining

we find th a t V z is a fundamental domain for Gj Ez and к

У г — и Ъ У * ,

i - 1

where h = kx and {^, is a complete system of representatives of E z modulo E x such th a t

GI EX = U t,W . /=i

Similarly we define Vz from V z and so on. Q.E.D.

We know th a t Vn V~l does not contain any element of E n except e, but the number of elements it contains of a cogroup of E n is finite and bounded for all cogroups

cn = sup # ( V n V - ' n y E J = sup # ( {t

E n : yV nc\tVn Ф&}) < oo.

yeG yeG

I t can be verified th a t cn does not depend on the choice of the fundamental domain Vn.

T

8. Suppose the following two conditions hold supkn = к < oo,

П

supCn = C< oo.

(4.1)

(4.2)

I f the sets {Fn} are chosen as in Lemma 2 and we denote by £% the family of all sets yV n (y eG, n — 1 , 2 ,. ..) , then the maximal operator

Mf(x) = sup — f \f\dm xelteSt m{R) J Jti

is of weak type (1 .1 ) and of type ( p, p) for 1 < p < oo.

P r o o f . For each n — 1, 2, ... we define an operator FJ(oo) = m ^ y y J ^{fd m} (when x e tV n, t e Hn)

tvn

(in the definition we neglect the locally null set N — complement of (~)HnVn). Let 23n denote the <r-algebra generated by the sets tV n (t e H n).

n

Then E n is the operator of conditional expectation with respect to 53n, and

SBj C ® 2 C ... с ! 8 и C ...

because of the way of selecting the fundamental domains Vn . Thus, {Enf } neN is a martingale and, given a > 0 we write

E *f = supF7n( |/|) , A = { x e G: E*f{x) > a}.

П

We know th a t E* is of weak type ( 1 , 1 ) and E nf(x)->f(x) a.e. because the conditional expectation with respect to the <r-algebra generated by U23n is the identity operator. So

G

^ a for almost all x ф A .

Assume for simplicity th a t f \ f \ d m < am{ Vx) (otherwise, we can define in an appropriate way new (7-algebras < $>_j c= 33 _7-+1 c= ... c 33 0 <= 53 x).

Then A is the disjoint union of the sets {E{ ( \f |) < a < E n ( \f |), i = 1 , 2 , . . .

. . . , n —1}, each of which has the form F V n for some finite subset F of S n, so th a t there is a disjoint family of sets tV n {t e H n), n = 1 , 2 , ...

whose union is A and such th a t

a < — i —— f \ f \ d m < h a . m>{Vn) J 1 У П

The constant Jc appears because m ( V n) > ~ m i V ^ f ) due to (4.1). On

к

the other hand, the definition of cn implies Vn V -1 c I J t j V - 1 ( t j e H n),

i=i

\JsjV» (sj e Hn)

3 = 1

(except, possibly, for some set of measure zero). Both together and (4.2) give

mi VnVn' Vn) < c2m ( V n).

Thus, if we define A* by replacing each t Vn <= A by tVn V~xVn, all we have to prove is

Mf(x) ф (const) a whenever x ф A*.

Let x $ A * and x e y V m. Then 1

™(Vm) j ^\f\&m

vVm

< a-\- ka

M V m) m( y Vmn A) .

B ut, if t Vnn y V m ф 0, the definition of A* implies n > m, so th a t t Vn - y V m7 “ 1 Vm. Therefore

m( y Vmn A) < m{yVmVml y m) < o2m {Vm)

and we finally find th a t Mf(x) < (1 + kc2)a when x $ A*. Thus M is of weak type ( 1 ,1). As it is obviously of type (oo, oo), Marcinkiewicz inter­

polation theorem gives the desired conclusion. Q.E.D.

C

5. Suppose, in addition to the hypotesis of Theorem 8, that Vn <=. Un form a basis of neighbourhoods of e: {Un: n = 1 , 2 , ...}.

Then

lim — 7Б7 f f ( y ) d m(y) = № ) (a -e•) xeRe& m(E) J

m(R)-+ 0

for any locally integrable function f.

A sequence {7n} satisfying Lemma 2 and the additional condition of Corollary 5 can be obtained, roughly in this way: Assume e e P |H nVn,

П so th a t changing Vn by some tn Vn we have e 6 f ) F n = Z. Write

П

Vn = \JzV W (disjoint union) such th a t each sequence {zV$}n is as in

z e Z

Lemma 2 (but not being fundamental domains for QjHn). Then, the point

is to translate each z V ^ so th a t they become close enough to e for large

n, but they keep on satisfying the proceedings conditions. We omit the

details.

If ^ is defined 1 to be only consisting of the sets tVn (( e H n), n — 1 , 2 , t hen Theorem 8 is valid without assumptions (4.1) and (4.2).

The proof of Theorem 8 contains a decomposition of Calderon- Zygmimd type, which is valid under assumption (4.1) only.

R e f e r e n c e s

[1] T. J. B o k s, Sur le rapport entre les méthodes d'intégration de Biemann et de Lebesgue, Rendiconti del Circolo Matematico di Palermo 46 (1921), p. 211-264.

[2] B. J e s s e n , On the approximation of Lebesgue integrals by Biemann sums, Aun.

of Math. 35 (1934), p. 248-251.

[3] H. R e ite r , Classical harmonie analysis and locally compact groups, Oxford Math. Monographs (1968).

[4] J. L. R u b io , Sobre integracion en los grupos clasicos y abstractos y aplicacion al Analisis de Fourier, Publ. Dpto. Ta Funciones Zaragoza.

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