Collected Works of Arne Beurling. Volume II Harmonic Analysis

408 Pages·1989·19.825 MB·English

by  Edited by L. Carleson, P Malliavin, J. Neuberger, and J. Wermer

Collected Works of
ARNE BEURLING
Collected Works of
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ARNE BEURLING
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Harmonic Analysis
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The Collected Works of
Arne BeurlingEd ited by
L. Carleson, P. Malliavan,
J. N euberger, and J. Wermer
Volume 2
Harmonic Analysis
Edited by
L. Carleson, P. Malliavan,
J. Neuberger, and J. Wermer
Bírkhauser
Boston . Basel . Berlín
Birkhiiuser
Boston . Basel . Berlin
Lennart Carleson John Neuberger
Royal Institute of Technology Department of Mathematics
Department of Mathematics University of North Texas
S-lOO 44 Stockholm Denton, TX 76203
Sweden USA
Paul Malliavin John Wermer
Department de Mathematiques Department of Mathematics
Universite de París 6 Brown University
10 rue Saint Louis en !'Ile Providence,. RI 02912
75004 París USA
France
Lennart Carleson John Neuberger
Royal Institute of Technology Department of Mathematics
Department of Mathematics University of North Texas
S-lOO 44 Stockholm Denton, TX 76203
Sweden USA
Paul Malliavin John Wermer
Department de Mathematiques Department of Mathematics
Universite de París 6 Brown University
10 rue Saínt Louis en l'Ile Providence,. Rl 02912
75004 París USA
France
Library of Congress Cataloging-in-Publication Data
Beurling, Ame.
CoIlected works of Ame Beurling.
(Contemporary mathematicians)
English and French.
Contents: v. l. Complex analysis -
v. 2. Harmonic analysis.
l. Functions of complex variables.
2. Harmonic analysis. 1. Carleson, Lennart.
11. TitIe. III. Series.
QA331.7.B48 1989 515.9 88-34968
ISBN 0-8176-3412-6 (set : alk. paper)
Library of Congress Cataloging-in-Publication Data
Beurling, Ame.
Printed on acid-free papero
Collected works of Ame Beurling.
(Contemporary mathematicians)
English and French.
© Birkhauser Boston, 1989
Conlents: v. 1. Complex analysis -
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v. 2. Hannonic analysis.
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l. Functions of complex variables.
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2. Harmonic analysis. 1. Carleson, Lennart.
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11. Title. III. Series.
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Preface
In 1985 one of us (Neuberger) met with Ame Beurling and obtained bis approval for
the publication of these volumes. Besides the published papers, Ame Beurling left much
unpublished work. This work inc1udes his UppsaIa Seminars (1938-52) and the Mittag
Leffier lectures (P19r7e7f).a ce
In accordance with Beurling's wishes, the editors have divided the papers, both published
and unpublished, into two parts: complex anaIysis and harmonic analysis. This division is
the basis of the two volumes of tbis collection.
The editors wish to emphasize that while Beurling approved of the publication of the
material in these two volumes, he did not get the time to review the unpublished papers as
they appear here. Therefore, all mistakes appearlng in the previously unpublished material
In 1985 one of us (Neuberger) met with Ame Beurling and obtained rus approval for
are entirely the responsibility of the editors.
the publication of these volumes. Besides the published papers, Ame Beurling left much
The unpublished material has the following orgins: The UpPSaIa Seminars were given in
unpublished work. This work includes his Uppsala Seminars (1938-52) and the Mittag
seminar lectures between six and eight in the evening on every second Tuesday ofthe month.
Leffier lectures (1977).
In 1951-52, Beurling asked Bertil Nyman and Bo Kjellberg to collect the available hand
In accordance with Beurling's wishes, the editors have divided the papers, both published
written notes from participants and to organize it and have it typed. The result was about
and unpublished, into two parts: complex analysis and harmonic analysis. This division is
480 pages, in Swedish. These seminar notes are to be available in their entirety at the
the basis of the two volumes of trus collection.
Mittag-Leffier Institute. In the present printed version we have removed those seminar notes
The editors wish to emphasize that while Beurling approved of the publication of the
wbich either were introductory in nature or else have since been published. The remainder
material in these two volumes, he did not get the time to review the unpublished papers as
were translated by Lennart Carleson and sorne editorial changes have been made.
they appear here. Therefore, all mistakes appearing in the previously unpublished material
In 1976-77 a "Beurling Year" was organized at the Mittag-Leffier Institute. Beurling gave
are entirely the responsibility of the editors.
a long series oflectures in wbich he described the development of his ideas in various fields
The unpublished material has the following orgins: The Uppsala Seminars were given in
of analysis. Earlier unpublished notes of bis were used in these lectures, inc1uding 1961
seminar lectures between six and eight in the evening on every second Tuesday ofthe month.
lectures at Stanford University (notes were taken by Peter Duren), lectures at the Institute
In 1951-52, Beurling asked Bertil Nyman and Bo Kjellberg to collect the available hand
for Advanced Study (notes were taken by Paul Cohen), and lectures on hydrodynamics (notes
written notes from participants and to organize it and have it typed. The result was about
taken by Michael Benedicks).
480 pages, in Swedish. These seminar notes are to be available in their entirety at the
The Mittag-Leffier lectures were written up and edited by Lennart Carleson and John
Mittag-Leffier Institute. In the present printed version we have removed those seminar notes
Wermer during the year 1977.
which either were introductory in nature or else have since been published. The remainder
We have added to Chapter III the Sections 7, 8, 9. Sections 7 and 8 are translations of
were translated by Lennart Carleson and sorne editorial changes have been made.
parts of manuscripts (in Swedish, approximately 1935) which Beurling prepared for his
In 1976-77 a "Beurling Year" was organized at the Mittag-Leffier Institute. Beurling gave
application for the Chair of Mathematics at Uppsala University. The original manuscripts
a long series of lectures in which he described the development of his ideas in various fields
are on fIle at the Mittag-Leffier institute. The material in Section 9 was handed to Neuberger
of analysis. Earlier unpublished notes of his were used in these lectures, including 1961
in January 1968. It is related to paper 44 of Beurling's published work.
lectures at Stanford University (notes were taken by Peter Duren), lectures at the Institute
We have included a Bourbaki report by Jean-Pierre Kahane describing unpublishedjoint
for Advanced Study (notes were taken by Paul Cohen), and lectures on hydrodynamics (notes
work by Beurling and J. Deny.
taken by Michael Benedicks).
Among other material related to tbis volume wbich is to be on fIle at the Mittag-Leffier
The Mittag-Leffier lectures were written up and edited by Lennart Carleson and John
Institute are notes from Beurling's course at Harvard University (1948-49).
Wermer during the year 1977.
We have added to Chapter III the Sections 7, 8, 9. Sections 7 and 8 are translations of
parts of manuscripts (in Swedish, approximately 1935) which Beurling prepared for his
application for the Chair of Mathematics at Uppsala University. The original manuscripts
are on fIle at the Mittag-Leffier institute. The material in Section 9 was handed to Neuberger
in January 1968. It is related to paper 44 of Beurling's published workv.
We have included a Bourbaki report by Jean-Pierre Kahane describing unpublishedjoint
work by Beurling and 1. Deny.
Among other material related to this volume which is to be on fIle at the Mittag-Leffier
Institute are notes from Beurling's course at Harvard University (1948-49).
v

Acknowledgements
Among the many who have worked to make these vohimes possible we wish to mention the
following: Mrs. Kerstin Rystedt-Ivarsson, Institute ofTechnology, Stockholm, and Dr. Kurt
Johansson, University ofUppsala, who put into TEX most ofBeurling's unpublished work
which is included here. We also wish to acknowledge the work of Professor Mohammad
Acknowledgements
Asoodeh and Mr. James Overfelt ofthe University ofNorth Texas. Professor Asoodeh made
the computer generated drawings in the Uppsala Seminars and Mittag-Leffier Lectures
following hand drawings made by the original note takers. Mr. Overfelt assisted with various
TEX revisions.
Among the many who have worked to make these vohimes possible we wish to mention the
following: Mrs. Kerstin Rystedt-Ivarsson, Institute ofTechnology, Stockholm, and Dr. Kurt
Johansson, University ofUppsala, who put into TEX most ofBeurling's unpublished work
which is included here. We also wish to acknowledge the work of Professor Mohammad
Asoodeh and Mr. James Overfelt ofthe University ofNorth Texas. Professor Asoodeh made
the computer generated drawings in the Uppsala Seminars and Mittag-Leffier Lectures
following hand drawings made by the original note takers. Mr. Overfelt assisted with various
TEX revisions.
vii
vii

Arne Beurling in memoriam*
:::;::::
~
by
a~
LARS JAHLFORS LENNARTfcARLESON
Ame Karl-August Beurling was bom the third of February 1905 in Gothenburg,
Sweden, and died in Princeton, New Jersey, the twentieth of November 1986. He
*
Arne Beurling in memoriam
studied at Uppsala University and obtained his Ph.D. in 1933. He was Prafessor of
.;::::::::- ::;::::::
Mathematics at Uppsala fram 1937 to 1954, at which time he resigned to become
Permanent Membebry a nd Prafessor at the Institute for Advanced Study in Princeton.
While on leave fram Uppsala he was Visiting Professor at Harvard University 1948-49.
LARS jAHLFORS and LENNARTfcARLESON
He was a member of the Royal Swedish Academy of Sciences, the Royal Swedish
Society of Sciences, the Finnish Society of Sciences, the Royal Physiographical
Ame Karl-August BeuSrloincige tyw ains Lbuonmd, Sthwe edtheinr,d thoef DFaenbirsuha Sryo ci1e9t0y5 oifn S cGieonthceesn,b aunrdg , the American Academy
Sweden, and died in Porfin Acerttso na,n dN eSwci eJnecrseesy. ,H teh ew taws eanltsioe thh oonfo rNaoryv emmebmerb e1r9 8o6f. thHee Swedish Mathematical
studied at Uppsala UniSvoecrsieittyy . and obtained his Ph.D. in 1933. He was Professor of
Mathematics at Uppsala froHme w19a3s7 atwoa r1d9e5d4 , thaet wShwiecdhi sthim Ae chaed ermesyi gonfe dS ctioe nbceecso mPrei ze 1937 and 1946, the
Permanent Member and Professor at the Institute for Advanced Study in Princeton.
Celsius Gold Medal 1961, and the University of Yeshiva Science Award 1963. In his
While on leave from Uphposanloar hae " wBaesu rVliinsigt iYnge aPrr"o fweasss ohre aldt Hata trhvea rMd iUttangiv-eLresfiftyle r1 9In4s8t-i4tu9t. e in Stockholm 1976/77.
He was a member of the Royal Swedish Academy of Sciences, the Royal Swedish
Ame Beurling was a highly creative matematician whose legacy will influence
Society of Sciences, the Finnish Society of Sciences, the Royal Physiographical
future mathematicians for many years to come, maybe even for generations. Anybody
Society in Lund, Swedewn,h oth ew Dasa ncilsohs eS otoc iehtiym o fw Sacsi einncfleuse, nacnedd thbey Ahmis ersitcraann gA pcaedrseomnyal ity and by his intense
of Arts and Sciences. Hcoem wmaistm aelnsot thoo mnoartahreym amtiecms.b eHre o pf utbhlei shSewde dviesrhy Msealethcetimvealtyic aaln d only when all details
Society. were resolved, and a sizable part of his work has never appeared in print. There are
He was awarded tphlea nSsw toe dpiushb liAshc ahdise mcoyl leocft eSdc iwenocrkess iPn rtihzee n1e9a3r7 fuatnudr e1, 9a4n6d, tthheey will include much that
Celsius Gold Medal 196h1a,s annodt thbee eUn npivreervsiiotuys loyf Yaveasihlaivbale S tcoie nthcee Amwatahrdem 1a9t6i3c.a lI np uhbilsi co Beurling's personal
honor a "Beurling Yearf"r iwenadss h aenldd astt uthdee nMtsi twtaigll- Lneevftelre rfo Irngsetitt uhtise uinn qSutoecsktihoonlimng 1 l9o7y6a/l7t7y. and boundless genero s
Ame Beurling wasi tya . hHigish lrye acdrienaetsivs et om sahtaerme ahtiisc iiadne aws hwoases ulengsaeclfyi shw iilnl tihnefl ueexntrceem e.
future mathematicians for mTahney yweoarrks otof cAommee , Bmeauyrblien ge vfeanl lsf orin gtoen tehrraetieo nms.a iAn ncyabtoegdoyr ies: complex analysis,
who was c10se to him hwaarms oinnficlu aennacleyds ibs,y ahnids psotrtoenntgi apl etrhseoonrayl.i tyIn aan dc hbayra chties riisnttiecn wsea y he transformed all of
commitment to mathematics. He published very selectively and only when all details
these areas of mathematics and made them interact with each other. This unity and
were resolved, and a siczoanbflelu epnacrte ooff ohriisg iwnoalr kid heaass annedv emr eatphpoedasr emda kine phrimin tu. nTiqhueere a maroen g analysts of our time.
plans to publish his collected works in the near future, and they will include much that
For convenience we shall nevertheless consider these areas separately, while
has not been previously available to the mathematicaI publico Beurling's personal
endeavoring to highlight the ways in which he enriched them aH.
friends and students wiII never forget his unquestioning loyalty and boundless genero s
ity. His readiness to sha* rRee phriinst eidd [eroams Awcatas Muantshe., l1f6i1s h(1 9in88 )t, h1e- 9e. xtreme.
The work of Ame Beurling falls into three main categories: complex analysis,
ix
harmonic analysis, and potential theory. In a characteristic way he transformed all of
these areas of mathematics and made them interact with each other. This llnity and
confluence of original ideas and methods make him unique among analysts of our time.
For convenience we shall nevertheless consider these areas separately, while
endeavoring to highlight the ways in which he enriched them al!.
* Reprinted [ram Acta Math., 161 (1988), 1-9.
ix
l
1. Complex analysis
Beurling's thesis [1] was published in 1933, bu! parts of it had been written already in
1929. It was not a mere collection of interesting and important results, but also a whole
program for research in function theory in the broadest sense. As such it has been one
of the most influential mathematical publications in recent times.
During the 1920's and 1930's T. Carleman and R. Nevanlinna had shown the
importance of harmonic majorization of the logarithm of the modulus of a holomorphic
function in1 .a pCloamnep lreexg aionna lDys. isL et E be a subset of the boundary of D. They considered a
harmonic function in D, later denoted by w(z,E,D) and known as the harmonic
Beurling's thesis [1] wasm peuasbulirseh aetd z i no f1 E9 3w3,i tbh uret sppaertcst otof Dit .h Iatd i sb deeenfi nwerdi tbteyn h aalvrienagd yb oiunn dary values 1 on E and
1929. It was not a mere cOo ollne cthtieo nre osft ionft etrhees tbinogu nadnadr yim. Ipfo ritt ains tk rneosuwlnts ,t hbautt 1a0lgsolf( za) l=w::h;;owl eo n the whole boundary,
program for research in tfhuennc,t iboyn tthhee omrayx iinm tuhme bprroinacdiepsIte ,s e1n0sgelf.(z Ao)sl= :s:;u;wch(z oit, Eh,a Ds )b efeonr eovneer y interior point zo.
of the most intluential matheBmeuatrilcinagl 'ps ulbelaidciantgio nidse ian wreacse ntot tfiimnde s.n ew estimates for the harmonic measure by
During the 1920's and 1930's T. Carleman and R. Nevanlinna had shown the
introducing concepts, and problems, which are inherently invariant under conformal
importance of harmonic majorization of the logarithm of the modulus of a holomorphic
mapping. The novelty in his approach was to apply the majorization to entities, mostly
function in aplane region D. Let E be a subset of the boundary of D. They considered a
of a geometric character, which are not by themselves invariant, but whose extreme
harmonic function in D, Iater denoted by w(z,E,D) and known as the harmonic
values, in one sense or another, possess this property. The method may have been used
measure at Z of E with rbeesfpoercet, tbou Dt .n oItt iisn d tehfiisn esdy sbteym haatviicn gm baonunnedr.a ry vaIues 1 on E and
O on the rest of the boundarAy . sIimf iptl eis ekxnaomwpnl et hwaitl l1 i0lgllufs(trza)tle~ wth oen w thoerk winhgo olef bthoius niddaeray. ,L et D be a region with.
then, by the maximum fpinriintec iapriee,a InogRl2f•( zCo)oln~swi(dzoe,r Etw, Do ) pfooirn etvs ezr ya inndte rZioo rin p oDi,n ta znod· let Q(z, zo, D) be the inner
BeurIing's Ieading didisetaa nwcea sb etotw feinedn zn aewnd eZso tiinm tahtees s efonrs et hoef thhaer mgroenaitce smt leoawsuerr eb obuyn d of the lengths of arcs
introducing concepts, ainnd D p jrooinbilnegm zs ,a nwdh zioc.h F aorrem i nthhee rreanttiIoy I (izn, vzoa,r iDa)n=t Qu(nzd, ezor, cDo)n/Rfo. rTmhaelr e is no reason why this
mapping. The novelty ins hhoisu ladp pbreo caocnhf owrams atlol ya pinpvlya rtihaen tm, bajuotr iizfa wtieo nc oton seindteirti e,l.s,(,z ,m zoo, sDtl)y= sup I(z* , zt, D*) for aH
of a geometric charactetrr,i pwlehsi c(zh* ,a zret, Dno*t) boyb ttahienmedse bIvye sa pipnlvyairnigan tth, eb usat mweh ocsoen feoxrtmreaml em apping to z, Zo and D,
values, in one sense or atnhoetnh eitr ,i sp oqsusietes so tbhvisi opurso ptehratty .,l .,T(zh, zeo m, De)t hios da m caoyn hfoarvme able einnv uasrieadn t. Beurling calls it the
before, but not in this seyxsttreemmaatli cd imstaannnceer .b etween z and Zo.
A simple example will Iilnlu csatsraet De tihs es iwmoprlkyi ncgo nonfe tchtiesd i dBeeau. rlLinegt Dgo ebse oan r teog iporno vwei tthh.e identity e-2G+e-;·2 = 1,
finite area nR2• Considwerh etrweo G p=oGin(zts, zzo ,aDn)d i sZ ot hien GDr,e eann'ds fluetn cgt(izo,n z Oo, Df D) bwei tthh pe o¡loen zeor. The proof is relatively
distance between z and Zeoa siny tbheec aseunses et hoef tchoen fgorremataels tin lvowareiar nbcoeu nmda koef st hiet pleonsgstihbsle o tfo a rrecpsl ace D by the unit disk.
in D joining z and zo. ForAm mthoer era tdiiof fIi(czu, lZto , aDn)d= Qal(szo, zom, Dor)e/R i. mTphoerrtea inst nroe sreualts oins wthhey tehsitsi mate for the harmonic
should be conformally imnveaasriuarnet ,e xbpurte isfs wede tchornosuigdhe rt hAe(Z i, nZeOq, Dua)l=it ysu p I(z*, zt, D*) for all
triples (z*, z~, D*) obtained by applying the same conformal mapping to z, Zo and D,
then it is quite obvious that A(Z, zo, D) is a cooformal iovwar=ia::n;;te.x pB(e-,ul.r,2li+o1g) calls it the (1)
extremal distance betweeo z aod zo.
In case D is simply wcohneroee ,cl.t,=e,ld., (zBo,e Eur, lDin)g i sg oneosw 0 t0h teo e pxrtorevme tahl ed iidsteanntcitey bee-t2wG+eeen- PZo= a1n,d the boundary set E. lf
where G=G(z, zo,D) is Eth eis G croenefoo'rsm fauHncyt ieoqnu iovfa Dle nwt ittho paonl ea zroc. Tohne thpero oufn iits rceirlactliev etlhye re is also an opposite
inequality
easy because the conformaI invariance makes it possible to re place D by the unit disk.
A more difficuIt and aIso more important result is the estímate for the harmonic
w;:::: exp( -,l., 2). (2)
measure expressed through the inequality
w ,;:; exp( -A2+ 1) (1)
x
where A= A(ZO, E, D) is now the extrema! distance between Zo and the boundary set E. If
E is eonformalIy equivalent to an are on the unít circIe there is aIso an opposite
inequality
(2)
x
These inequalities were strong enough for an independent proof of the Denjoy conjec
ture concerning the number of asymptotic values of an entire function of finite order.
The idea of extremal distance was a forerunner of the notion of extremal length.
Originally, Beurling measured lengths and areas only in metrics of the form
Qldzl=I<1>'(z)lldzl, where <1> is a conformal mapping. The more general case of arbitrary Q
was worked out later in collaboration with L. Ahlfors. Beurling's contribution to their
joint papers was always substantial. This is particulary true of [26], in which the
decisive idea was entirely due to Beurling. It deals with the boundary values of
These inequalities were qsutraosnicgo nenfoorumgahl fmora panp iinngdse. pIefn hd(exn)t ipnrcoroefa soefs t hfoe rD -eonojo<yx <coonoj etch ere is a quasiconformal
ture concerning the nummbaeprp oinfg a soyfm thpet outpicp evra lhuaelsf -oplfa anne eton tiirtsee fluf nwcittiho nb ooufn fdinairtye voardlueers. h(x) on the real axis, if
The idea of extremal distance was a forerunner of the notion of extremal length.
and only if for sorne constant Q> 1
Originally, Beurling measured lengths and areas only in metrics of the form
Qldzl=I4>'(z)lldzl, where 4> is a conformal mapping. The m1o/Q re,-c :-:g: hen(xe+ratl) c-ahs(ex )o -,fc-: ::aQ r bitrary Q
h(x)-h(x-t)
was worked out later in collaboration with L. Ahlfors. Beurling's contribution to their
joint papers was aIways substantial. This is particular y true of [26], in which the
for all x and t.
decisive idea was entirely due to Beurling. It deals with the boundary vaIues of
The impact of extremallength on present day mathematics can hardly be overstat
quasiconformal mappingeds.. IIft his( xa) t inthcer ebasaesiss foorf q-CuXals<iXco<nOfOo rtmhearl em isa pap iqnugass, icaonndf otrhmerael by also of Teichmüller
mapping of the upper half-plane to itself with boundary values h(x) on the real axis, if
theory and the new theory of dynamical systems. Recently it has become more
and onIy if for sorne concustsatnomt Qar>y 1 t o replace extremal length by its reciprocal under the name of modulus of
curve families. It has the advantage of carrying over more easily to several dimensions,
1/ ,c:: h(x+t)-h(x),c::
for instancQe -in- hc(oxn)n-ehc(txio-nt) w"""i"tQh capacity and in the theory of quasiregular functions.
Except for the joint papers with Ahlfors, Beurling published less than expected on
for all x and t. extremallength. A possible explanation is that he may have been looking in vain for a
The impact of extresmataislfleacntgotrhy otnh eporreys eonft edxatyre mmaatlh memetartiiccss, cwanh ihchar dstIiyll bdeo oevs enrostta et xist today.
ed. It is at the basis of quaBseicuorlnifnogr'ms paIr omofa opfp iinngeqs,u aalnitdy t(h1)e rmebayd ea ulssoe ooff tThee ifcohlmloüwlilnegr fact. Iff(z) is holomor
theory and the new thpehoicry i no tfh ed yunniatm diicsakl wsiyths tDemirsic. hRleet cienntetIgyr ali t< hna sa nbdef(cOom)=eO ,m aonrde if If(eiO)I>M on E, then
customary to replace extremal Iength by its reciprocaI under the name of modulus of
curve fa m ilies. It has the advantage of carrying over more easiIy to several dimensions, (3)
for instance in connection with capacity and in the theory of quasiregular functions.
Except for thejoint papers with AhIfors, Beurling published les s than expected on
O. Frostman's thesis appeared in 1935 and created a solid basis for potential theory.
extremaI Jength. A possible explanation is that he may have been looking in vain for a
Beurling realized that the Dirichlet integral and the energy integral are dual norms, and
satisfactory theory of extremal metrics, which still does not exist today.
that mes E in (3) should be replaced by the capacity of E. This was carried out in the
Beurling' s proof ofi innfeluqeunatliiatyl p(1a)p mera [d5e] , uwseh iocfh t hbee cfoalmloew tihneg ofarcigt.i nI fof efz n) uims ehrooloums ostru dies of exceptional sets
phic in the unit disk witha nDdi rbiocuhnledta irnyt ebgerahla v<ionr a onfd fh(oOlo)=mOo,r pahnidc iffu Infecetiiofi)nIs>.M on E, then
Another central theme during the 1930's was quasi-analyticity. In a characteristic
(3)
way Beurling's treatment of this topic was combined with harmonic analysis and
potential theory. He published very liule, but soñjé of his results appeared in his
O. Frostman's thesis apSptaenafroedrd i nle c1t9u3r5e asnedr iecsr efartoemd a1 9s6o1l id[ 3b1a].s iHs ifso rc oplolteecntteida l ptahpeeorrsy .w ill contain a complete
Beurling realized that thtere Datimricehnlte. t integral and the energy integral are dual norms, and
that mes E in (3) should be replaced by the capacity of E. This was carried out in the
influential paper [5], which became the origin of numerous studies of exceptional sets
and boundary behavior of holomorphic functions.
xi
Another central theme during the 1930's was quasi-analyticity. In a characteristic
way Beurling's treatment of this topic was combined with harmonic analysis and
l
potential theory. He published very liule, but soñje of his results appeared in his
Stanford lecture series from 1961 [31]. His collected papers will contain a complete
treatment.
xi