Hardy Remarks

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Remarks on the Hardy Inequality

D.E. EDMUNDS

^a,* and

R. HURRI-SYRJ,,NEN

^b

Centre

for Mathematical Analysis anditsApplications,

University of

Sussex,

Falmer, Brighton,

East Sussex

BN1 9QH,

UK

b

Department

ofMathematics,University ofHelsinki, Hallituskatu15, SF-O0100Helsinki, Finland (Received3July1996)

LetDbeanopensubset ofIR (n>2)with finiteLebesguen-measure, letd(x)be thedistance from x IR"totheboundary0DofD,and let < p < o.We giveasimpledirectproof that ifIR"\Dsatisfies theplumpness^{conditionofMartio}and Viisili 10],then the inequality ofHardytype,

holdswhenever/^>max{0,^ot }.Wealsoshow that theplumpnessconditionmaybereplaced byones whichenable domains with lower-dimensional portionsoftheir boundariestobe handled.

AMSsubjectclassification^{number: 46E35.}

Keywords: Hardy inequality; distance function; plumpness; Poincar6 inequality; Sobolev spaces.

1 INTRODUCTION

Let D

be an

open

subset of

R n,

let 1 < p < oe, andgivenx

D

let

d(x)

be the distancefromx totheboundary0

D

of

D. It

iswell known

(cf.

[6], p.

223)

that if ubelongstothe Sobolevspace

Wp(D)

^and

u/d Lp(D),

0

then infactulies in

Wp(D),

the closure of

C(D)

ⁱⁿ

Wlp(D).

^Thisholds

withno restrictions on 0

D.

Theresult inthe oppositedirection,namelythat

0

if u

W]p(D)

^then

u/d Lp(D),

would follow immediatelyif one knew thattheHardy inequality

*_Authorfor correspondence.

125

126 D.E. EDMUNDSandR.HURRI-SYRJ,NEN

fo ¹

(lu (x)l/d (X))

^pdx <_ C

IVu ^(x)l

^{pdx, u}

. ^{Wp(D) (1.1)}

was truefortheparticular

D

and theparticularp. Thevalidityof

(1.1)

^has beenextensively investigated: for

example,

Davies

[4]

has shown that if p 2and

D

isboundedand satisfies a certaintype ofcone condition,then

(1.1)

holds;it is clearthat hisargumentcanbeadaptedtopermitother values of p. ^Other work ontheHardy inequality

(1.1),

and weighted analogues of it, maybe found in the

paper

by

Ancona [2]

and thebook by Opic and Kufner

[11 ]. Moreover,

Lewis

[9]

has shown that if 1 < p < n and

R

ⁿ

\ D

is(1, p) uniformly fat,then

(1.1)

holds;ifn < p < c he shows that

(1.1)

holdsfor all

D # ^R n.

The uniform fatness condition which heimposeswhen 1 < p < nisthat there is apositiveconstant

;

such that for all x

R

ⁿ

\ D

and allr > 0,

Rl,p (r

^-1

^(B

^(x,r)fq

^(Rn\ D)))

^>_),

where

B

(x,

r)

isthe

open

ball in

R

ⁿwith centrex and radius r, and

R

_{1, p}is a certain Rieszcapacity.Sufficient conditionsfor

(1.1)

^{to hold}have also been givenbyWannebo 14]; theseareexpressedin terms of acapacity^introduced by Maz’jaandenablehim toreproduceLewis’sresultsforp > n andtoshow that

(1.1)

holds if p > n 1 and

D ( Rn)

^issimplyconnected.

In

15]other sufficient conditionsfor

(1.1)

tohold aregiven.

In

the presentpaperwe show that if

D

has finite volume and

R

ⁿ

\ D

satisfiestheplumpnessconditionofMartioand Viiisiilii 10],then notonly does

(1.1)

holdbut also moregeneral inequalitiesofthe form

fo ^{(iu (x)l /d (x))}

^{pd C}

^{f. (IVu (x)l/d tx))}

^pd,U

^C

(1.2)

where 1 < p < cx and

/3

^>_

^max{0, ot-1}.

This is established in Section 2 by comparatively straightforward procedures. When

D

has a Lipschitz boundary, our result

agrees

with Theorem 10.4 of Gurka and Opic [7], obtainedby entirelydifferentmethods and under the additional assumption

that/3

>

p

(p

1).

Ifot 1and

fl

^0,

^(1.2)

coincides with

(1.1)

but does not thengive anythingnew, as it can be shown that if

R

ⁿ

\

^D

is

plump

andunbounded,it is

(1, p)

uniformly fat forevery p > 1 so that Lewis’s resultapplies. Theplumpness condition, which will beexplained

in detail in Section2, isarather natural geometriccondition on

D

which is

easy

tocheck and hasnothingtodo with p. Like Lewis, ourarguments

depend

on awell-known lemmadue toCarleson [3];but wehopethat our direct use of theplumpnessconditionmayhavesomeappealfor those who areless familiar with notions ofcapacity, and maystimulatefurtherwork.

In

Section3, the

plumpness

condition is

replaced

by^{ones which}enableus tohandledomains with lower-dimensionalportionsof their boundaries, and here the rangeofpossible

p’s

forwhich,say,

(1.1)

holdsisdependent upon thepropertiesof

D.

Whilethese results can infact beobtained fromcapacity results,wehopethat the direct method ofproofwill beofinterest.

2 A WEIGHTED HARDY

^INEQUALITY

First we fix the notation andprovidesome basic definitions.Throughoutthe paperweshall assume

(unless

^otherwisestated) that

D

isanopen subset of

R

ⁿ

(n

>

2)

with finiteLebesguen-measure. Givenanysets

A, B

C

R n,

the distance between

A

and

B

willdenotedby

d(A, B)

andthedistance from x

R

ⁿ to

A

by

d(x,

A), writing

d(x) d(x, OD)

for shortness;if

A

has finiteLebesguen-measure

A In,

the

average

of a function u over

A

isdefined tobe

UA

IAI

^-1

[

u

(x)

dx.

Ja

The

open

ball in

R

ⁿ with centre x and radius > 0 will be denotedby

B(x,

t);when rn

N

^t_J

{cx}, C

ⁿ

^(D)

^{willstand forthe}

^space

^{of allrntimes}

continuouslydifferentiablereal-valuedfunctions withcompact supportin

D;

we write

Ilullp,o (fo lu (x)l

^p

dx)

^{1/pforall p}

^(1,

^o); k-dimensional Hausdorffmeasure on

R

ⁿ will be denotedby

7-/k

when k < n

Wp ^(D)

willstandfor the Sobolev spaceof all functions which,togetherwith their first-orderdistributional derivatives, are in

Lp (D).

Giventwonon-negative expressions (thatis, functions orfunctionals) R1,

Re

^{we shallwrite}

R1

^-<

R2

as

ashorthandforthestatementthatR1

^<

CR2

for some constantC (0,

)

independent of the variables in the expressions R1, R2; if

R1

^-<

R2

^and

R2

^-<

R1

^wewrite

R R2.

DEFINITION2.1 Given anyb

(0,

1], we saythat

R

ⁿ

\ D

^is b-plump

if

thereexiststr > 0such that

for

^{all y}

^D

^{and all}

^(0,

^{tr] there isan}

x

(R

ⁿ

\ D)

^tqB(y,

t)

with

d(x)

^> bt.

128 D.E. EDMUNDSandR. HURRI-SYRJ./i.NEN

This definition isdue to Martioand Vaisali 10];JerisonandKenig

[8]

^call thehypothesisofthedefinition acorkscrew condition.

Moreover,

there is a connectionwiththe exteriorregulardomainsof Triebel and Winkelvoss

13]"

if

D

coincides with the interior of itsclosure,then

D

isexteriorregularif, and onlyif, it isb-plumpfor some b.

Our

first result isthefollowing:

TIOIEM2.2.

Suppose

that

R

ⁿ

\ D

is b-plump

for

^{some b}

^(0,

^{1], let}

1 < p < cx and let or,

R

be such that

max

{0,

ot-

1}. (2.1)

Thenthereis a constant

C

> 0such that

for

^allu

C ^(D)

[ [

dD dD

(2.2)

Proof

^{Letu 6}

C ^(D)

^{wemayandshallsupposethatu}is defined on all of

R

ⁿ and is zero on

R

ⁿ

\

^D.

^Let

^{I/Vbe a}Whitney decompositionof

D (see [12],

p. 16);that is, IA; is afamilyof closeddyadiccubes

Q,

withpairwise disjoint interiors, suchthat

D U _aw ^Q,

1 < d(Q,

OD) /

^{diam (Q) < 4 forall}

^Q

⁶

142

(diam (Q) beingthe diameter ofQ)and

1/4

^<_ diam (Q1)/diam (Qz) < 4forall Q1,Q2

142

withQ1NQ2

.

For

each

Q

^{6 I/V}wefix an_XQ OD such that d

(OD,

Q) d

(XQ, Q)

^and

choose acube D

a

^{withcentre x}

a

such that diam

()

^diam

^(a).

Then

fo ^{(In (x)l/d (x))}

^pdx

^{fQ (lu (x)l/diam}

^a

^(Q))P

^dx

< ^QeW

f ^{(lu (x)l/diam (Q))P}

^dx.

^(2.3)

Since

R

ⁿ

\ D

isb-plump,there existstr > 0 suchthat for all

z

^{6 0}

D

and allt 6

(0,

cr],thereisay ⁶

(R n\D)

fqB(z,t) withd(y,

0D)

> bt.

We

_may assumethatcr >_ diam

(9)

^{for all}

^{Q W}

^{andsomaychoose}

t= diam

( )

for if there isamaximal

Qo

6Wsuchthat diam

(o)

^{> r,}

we simply takek > 0 such thatcr > k diam

(o)

^and then work with

k diam

()

^{insteadof diam}

(). ^It

^{followsthat foreach}

^{Q W}

^thereis

ay

(R

ⁿ

\ D) n B (XQ,

^diam

())

^withd(y) > b diam

(9);

^{we write}

,-- Q (y,

^{b diam}

(9)In),

the

open

cube with centrey and sidesof lengthb diam

() /n

^parallelto theaxes.

As A

C

R

ⁿ

\ D,

themeanvalue

u’

^{0. Thusfrom}

^(2.3)

^weobtain

fD ^{(lu (x)l/d ’ (x))}

^{pdx -< Q}

^E

^’Vv

^{(lu (x) /}

^diama

^(Q))P

^dx.

(2.4) Use

ofH61der’s and MinkowskPsinequalitiesnowshows thatforallc

R,

f,

^u

^{(x).- u’2l}

^{pdx <_ 2p}

^{(I O.In/I.ln)} f, ^{lu (x) cl}

^pdx

21)

(n/b)

ⁿ

f, ^{lu (x) cl}

^pdx.

The choicec u

’

^{in this}inequality, togetherwith the Poincar6inequality inacube

(see [6],

p.

243),

gives

f’ ^{Ill (x) u"l}

^pdx

^I

ⁿ

^{n+l--q (f’}

^[Vll

^(x)l

^qdx

^{)P/q (2.5)}

where p andq are related by 1 < q < p nq/(n-q); theconstant implicitinthe inequalityisindependentof

Q.

Since

Vu (x)

0whenever x

R

ⁿ

\ D,

^{we seethat}

if/

^> 0,

lVu (x)l

^qdx ^{Q6"V Q r’IQ}

T

^O

fn" ^{IVu (x)[}

^qdx

{diam

(Q1)/d (x)}

^q

IVu ^(x)l

^qdx

QIW, QInQTb.O

_ ^{(diam (_.))flq} _f, ^{(IVll (X)[ /d}

^fl

^(X))

^qdx.

^(2.6)

130 D.E. EDMUNDSandR.HURRI-SYRJ,,NEN

Hence

from

(2.4)-(2.6)

^wefind

fo ^{(lu (x,, /d}

^dx

(IVu ^(x)l/d (X))

^qdx

+/-

(IVu ^(x)l/d (x))

^qdx

[0.1

in^-p/q

aw (2.7)

the finalinequality beinga

consequence

of ourassumptionthat

fl ^-or-t-

^{1 > 0.}

To

concludethe

proof

weusethe following well-knownlemma first

proved

byCarleson

[3]

whenp 2 andn 1

(see [12]

forthegeneral

case).

LEMMA

2.3

Let Qo

beacubein

R

ⁿ and supposethat Qi is asequence

of

cubessuch thateachQiiscontained in

Qo and,

^Qi

ln

^<const

^IQ01

^let

v

Lp ^(Qo)for

^somep (1,

cxz).

ThenthereisaconstantC, independent

of

^{v,such that}

Z ^{IQilln-P It) (x)l}

^{dx < C}

^{Iv (x)l}

^pdx.

^(2.8)

We

apply this to the

Q,

noting that the basichypothesis of the lemma is satisfied sincefor a fixed cube

B,

It)In

diam" (Q)-<_

IBI..

QcB,QW QCB,Qel/V

Sincep/q > 1, ^Carleson’slemma shows that thefight-handside of

(2.7)

canbeestimatedfromabovebyamultipleof

(IVu ^(x)[/d (x))

^pdx,

and the theoremfollows.

Remark 2.4 (i)When

D

has aLipschitzboundaryit isplain that

R

ⁿ

\D

satisfiesthe

plumpness

condition, and soinequality

(2.2)

holds. Thisresult, underthe additionalassumptionthat

fl

^{> p}

/

^{(p 1),}was obtainedby Gurka andOpic ([7], Theorem

10.4).

^Their

paper

alsocontains sufficient conditions for

(2.2)

tohold whenO

D

is in the H61der classC^,K for sometc

(0,

1];

and itgives results concerning the inequality

analogous

to

(2.2)

but with the left-handsidereplaced by

D

lU ^(X)Iq/d

^q

^(x)

^dx

forsuitableq.

(ii) Whenot 1

and/

^0,

(2.1) reduces

^tothe Hardy inequality

(lu (x)l/d (x))

^pdx <C

fo ^{IVu (x)l}

^{pdx, u}

^{C (D) (2.9)}

mentioned in the Introduction.

As

explained there, the special case of our results, that

(2.9)

holds when

R

ⁿ

\ D

is plump and unbounded

and

1 < p < cx,iscontained in thoseofLewis

[9]. Note,

however, that inspection ofour

proof

shows that the constant

C

in

(2.9)

may be takentobe

where

p(n-3)+

-l+a(6a)ab-nn7

con

( 1)

¹

a=p 1--

n 3

and^(.On

B (0, 1)In;

^if

D

isconvex, then wemaychooseb

=

^1.

^In

^this

connection we are informedthatwhen

D

isconvex and hasC

boundary,

then

P.

Sobolevski and

T.

Matskewich have very

recently

shown thatthe best constant C in

(2.9)

is 1 seealso

’On

the bestconstant for Hardy’s inequality’,

M. Marcus, V.J.

Mizel,

Y.

Pinchover

(to appear).

When p n 2 and

D

is asector of a circle the best constantCin

(2.9)

hasbeen shown byDavies

[5]

to be4ifthe

angle

ofthesector isless

than/30

^4.856.

Ifwe use theclassical variationalcapacity argument,

Lemma

2.5 below, theHardy inequality

(1.1)

follows easily.

As

normal, for acompactsubset

E

ofanonemptyopenset

D

in

R

ⁿ we write

v6C(D),

^{0<v< lonDand|}

cap (E, D)

^inf

IlVvlIoP,

D_ v 1 in an

open neighbourhood

P

of

E

in

D.

LEMrA

2.5

[6,

Corollary 2.4/Chapter VIII]

Let Q

be a cube in

R

ⁿ and

define

^{any u}

C ^(D)

^tobe

^zero

^{outsidethedomain}

^{D. Let}

^{1 < q < p <}

n__q_

P

^{< n}

If

^q-

^cap (

^t3

^{(Rn\ D)} 2Q)

^>

^o,

thenforanyu

C(D)

n--q’

132 D.E.EDMUNDSandR.HURRI-SYRJ)NEN

lU(x)l

^pdx <

c(n’q)

diam

(a)n

(fQ )P/q

(q

^cap

(-

^fq

^(Rn\ - ^I2Q))

^p/q

^IVu(x)l

^qdx

Using

Lemma

2.5andthe

proof

ofTheorem2.2 we obtain thefollowing theorem.

THEOREM 2.6

Suppose

that

D

is adomain withconstants⁾ > 0,

co

> 0 suchthat

q

cap (-

^fq

(Rn\ D), 20)

^diam

^(a)q-n

^{>_ )}

^(2.10)

for

^{all cubes}

^Q

^{Q(y) with centre y OD and}

^O

^{< diam (Q) <}

co ^D 1In. ^Let

^{1 _<}q < P

.<

_{n_q}^nq p < n Then thereexistsa constant c > 0 suchthat

for

^allu

C

^(D),

lu(x)lP

fo

d(x-3P

^{dx c}

^IVu(x)l

^pdx.

Proof ^We

^usethe same notation as in the

proof

of Theorem2.2.

We

need toverifyonlytheinequality

lu(x)l

^pdx diam

(Q)-P <

^diam

^{(_)n(1-qe) IVu(x)l}

^qdx

(2.11)

where 1 _< q < p

_< n--n-q-n_q,p<n,"

otherwise the

proof

issimilar totheproof of Theorem 2.2.

However, Lemma

2.5and theassumptionof Theorem2.6 immediately yield

(2.11):

lu(x)l

^pdx diam

(Q)-P

<I

^{q- capc(n’q)}

⁽

^fqdiam

^{(Rn\D), (O)n-q}

^int

⁽²⁾⁾

x diam

()) ^n(1-q) IVu(x)l

^qdx

Ic(rt’q)l

^p/q

(f. )P/q

< diam (_)

n(1-q) _]Tu(x)[

^q_dx

Z

Remark 2.7

To

obtainthe

general

case

(1.2)

thecondition

(2.10)

should be

replaced

by q

cap (-

^q

^(Rn\

^D),

2Q)

^diam

^(Q)q(-)-n

^>

.,

where

_>0.

3

OTHER CONDITIONS ON THE BOUNDARY OF D

Firstwe establish thefollowingresult:

THEOREM3.1

Let D

be a domain in

R

ⁿ

(n

>

1)

andsuppose there ^are constants s 6

(0,1)

and

T

> 0 such that

for

^{each y O}

^D

^{and all}

(0,

T), thereisa k-dimensionalcube

Qk,t (Y)

COD, withy

ak,t ^(Y)

and^7-[k

(Q,t (y))

^>

^stk;

^{supposealso thatp (1,}

ⁿ⁾

^{issuch that}

for

^all

these k,n p < k < n- 1. Then thereis a constant C > O such that Hardy’s inequality

(lU (x)l ^/d (x))

^pdx C

]O ^{IVu (x)l}

^{pdx, u}

^{C (D)}

holds.

(3.1)

Our

proofof thistheorem hinges

upon

thefollowingtwo lemmas.

In

these allcubes are assumed to haveedgesparallelto the coordinate axes in

R n,

and the intersection of a cube

Q

in

R

ⁿ witha k-dimensionalplaneisdenoted by

Q’ (1

^< k <

n),

with theunderstanding that

Qn Q.

LEMMA

3.2

Let Q

beacubein

R

ⁿand letp (n,

cx)

q [1,

oe)

Then thereis a constant c c(n,p,q)suchthat

for

^everyu

W

^(Q),

lu ^(x) uat

^{< c}

(liu ^ua[I

^q(p-n’/pq,Q

IlVull,a )p/{np+(p--n)q) ^(3.2)

for

almost all x in

Q.

Proof

The result issimplythespecialcasem 1of

Lemma

5.18of

[1

], appliedto u u

a.

LEMMA

3.3

Let Q

beacubein

R n,

let1 <k < nand let0 < n p < k <

n.Then thereis a constant c c(n,p,q) suchthat

for

^everyu

wlp

^(Q)

(fQ ^[U

^{(y) UQ qdy}

)lie

^{<_ c}

(fQ ^IVu

^(y)lpdy

)liP ^(3.3)

where q kp/ (n p)anddy denotesLebesguemeasure on

R .

134 D.E.EDMUNDSandR.HURRI-SYRJfiNEN

Proof

^{Exactlyas intheproofof}

^Lemma

^{5.19of 1] we findthat}

fQ ^lu

^{(y) UQ}

fQ )

iz(p-v)/(p.)

qdy ^<

lu ^(x)

^{UQ qdx}

(fo ^{IVu (x)l}

^pdx

^(3.4)

where v is the

largest

integer less than p,/x

(nk_v),)t, (nk21_l)

^and

qo np/(n p).

By

Poincar6’s inequalityinthecube

Q

we can estimate the term

fQ lu ^(x)

^UQ q^{dx in}

^(3.4)

^bymeansof

^{fQ IVu (x)l}

^{pdx,and the}

resultfollows.

Proof of

^Theorem3.1

^It

^{isenoughtoprove}

^(3.1)

^foru

C ^{(D). Let}

^I/V

be aWhitney decompositionof

D.

Givenany

Q

l/V,fixx_Q OD such thatd(Q, O

D)

d

(Q, XQ)

^{fix a cube withX}Qas centre and such that

Q c

and diam

()

^c

⁽ⁿ⁾

^{diam (Q).Then}

o

(lu (x)l/d (x))P

dx

/_ ^{(lu (x)l/d (x))P}

^dx

"<

f

_JO^(lu

^{(x)l/diam (Q))P}

^dx.

^(3.4)

For

eachcube thereisa set

",d(a) ^:=

^{C O}

^D

^{such that}

(’;,d(a)) "

s diam

(9),

^{wherek (n p, n}

^1].

^{Since u 0 on}

,

l

(x)l

^pdx

f’6 ^{lu (x) ugl}

^dx.

^(3.5)

Moreover,

Minkowski’sinequalityand the Poincar6inequality^{in a}cube yield

(3.6)

where q np/(p

+ ^{k). Use}

of H61der’sinequality gives

(3.7)

where

)k

^isthe intersectionof and the k-dimensionalplane containing the cube

Qk,t (XQ)

for suitablet.

From Lemma

3.3 we have

(f’,l

^u

^{(x) u’}

^pdxl

^)lip

^{<_ c(k,n,p)}

(f,. ^{IVu (x)l}

^qdx

^{)l/q (3.8)}

where q np/(p

+ ^k),

^{p < n and n p < k < n.} Combination of

(3.4)-(3.8)

^nowshowsthat

(lu (x)l /d (x))

^pdx ^-.<

_

^{S-1 diam}

^{(Q)n-k-p IVu(x)l}

^qdx

s^-1 diam

(Q)-k-P ---Ix-p/q,n

QeV;

(f ^IVu(x)]

^qdx

)P/q

)"q

"< QVV

_

^S-1

^]_.11-p/q

ⁿ

^IVu(x)]q

^dx

(3.9)

since^np_q k p 0.

As

the

)

forma

sequence

of cubes to which Carleson’s lemma,

Lemma

2.3, maybeapplied,itfollows from

(3.9)

that

(lU (X)l/d (X))

^pdx <C

fD ^IVU(x)IP

^dx

forsomeC C

(k,

n,p)s^-1 Theproofiscomplete.

A

variant ofTheorem 3.1

along

the lines of Theorem 2.2can easily be given.

136 D.E. EDMUNDSandR.

HURRI-SYRJNEN

THEOgEM3.4

Let

_p

(1, n)

and

or,/3 R;

let

D

beadomainin

R

ⁿ

(n

>

1)

and suppose that thereare constants s

(0, 1)

and

T

>0such that

for

^each

y O

D

and all

(0,

T), there isa k-dimensionalcube

Qk,t (Y)

C

D

withy

Qk,t (Y), 7@ (ak,t ^(y))

^{> st}

^k,

^and

1 >_ max{O,

ot-

1},

n-p<k<_n-1.

(3.10)

Then thereis a constantC > 0suchthat

for

^{all u}

C ^(D),

(I. ^(x)l/ (x))

^edx

^{_< c} (IV. ^(x)l/e (x))

^dx.

Proof

^This follows the patternof that of Theorem3.1; justasbeforeand withthe same notation, itfollowsthat

L ^{(lu (x)l/d (x))}

^{pdx -<}

_.

^diam

^{(Q)--k-p ll-p/q,n}

QeW

)P/q

x

IVu(x)l

^qdx

(3.12)

whereq np/(p

+ ^k);

^{see the} inequalities leading up to

(3.9).

^Under conditions

(3.11)

theright-handsideof

(3.12)

canbeestimated fromabove byaconstant times

E

^diam

^{(Q)-k-ap+p ll-p/q,n ([Vu (x)l/d}

^E

^(X))

^qdx

Qsw

)P/q

E ^{Ii 1-p/qn (]Vu (x)l/d}

^fl

^(X))

^qdx

QV

Theresult nowfollowsasbeforeonapplicationof Carleson’s lemma.

Acknowledgements

It

is a

pleasure

^torecord ourthankstoTheRoyal SocietyandtheAcademy ofSciencesofFinlandfor the support givento

R.

Hurri-Syrj/inen.

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