Representing Measures on the Royden Boundary for Solutions of Δu = Pu on a Riemannian Manifold Citation Chow, Kwang-nan (1970) Representing Measures on the Royden Boundary for Solutions of Δu = Pu on a Riemannian Manifold. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/D80C-CD98. https://resolver.caltech.edu/CaltechTHESIS:07302015-141209767 Abstract Consider the Royden compactification R* of a Riemannian n-manifold R, Γ = R*\R its Royden boundary, Δ its harmonic boundary and the elliptic differential equation Δu = Pu, P ≥ 0 on R. A regular Borel measure m P can be constructed on Γ with support equal to the closure of Δ P = {q ϵ Δ : q has a neighborhood U in R* with U ʃ ᴖR P ˂ ∞ }. Every enegy-finite solution to u (i.e. E(u) = D(u) + ʃ R u 2 P ˂ ∞, where D(u) is the Dirichlet integral of u) can be represented by u(z) = ʃ Γ u(q)K(z,q)dm P (q) where K(z,q) is a continuous function on Rx Γ . A P ~ E -function is a nonnegative solution which is the infimum of a downward directed family of energy-finite solutions. A nonzero P ~ E -function is called P ~ E -minimal if it is a constant multiple of every nonzero P ~ E -function dominated by it. THEOREM . There exists a P ~ E -minimal function if and only if there exists a point in q ϵ Γ such that m P (q) > 0. THEOREM . For q ϵ Δ P , m P (q) > 0 if and only if m 0 (q) > 0 . Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Glasner, Moses Thesis Committee: Unknown, Unknown Defense Date: 3 April 1970 Record Number: CaltechTHESIS:07302015-141209767 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:07302015-141209767 DOI: 10.7907/D80C-CD98 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9069 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 31 Jul 2015 16:16 Last Modified: 29 May 2024 18:16 Thesis Files Preview PDF - Final Version See Usage Policy. 13MB Repository Staff Only: item control page

REPR~&J TING SOLUTIONS OF HEASURDS OH Tl:-'. .i.<.: ROYDcl'• BOUlWA.ttY FO R ~u = Fu ON A RIEHANN l fu\l HAiHFOLD Thesis by Kwan g-nan Chow In Partial Fulfillment of t he Requirement s For t he Deg r ee o f Doctor of Philosophy California Institute of Technolo ~y PasB.dena , California. 1 970 ( Subm~.tted April 3, 1 <;7 0) ,i . :"? ii I a.!11 gr eatly ind'?.bt ed to my ad visor '.)r . M. ;_Jl,q~> nP r for bis constant guidance, inte rest and patience which made. this thesis possible . Hany thanks are du e to the California In s titute of Technology for the ['.;raduate teaching assi stantships and summer f ell owship s for the past four years which enabled me to complete my graduate studies . It gives me special pleasure to express my appr eciation to my wife ifart;;uerite whose encouragement, help and many other contributions d urin g the preparation of this thesis have meant a great d~al to me. I ,o;la.dly dedicate this t hesis to her. iii ABSTRACT Cons i de r the .Royden compactifica.tj_on it* of a Ki. e mannian r = n*'- R i ts Royden b oundary , fl its harmonic b oundar y n-mani fold it, and t he elliptic c iffer ential eq u ation /:;.u = i'u, r egular .3o rel me as u r e mf' can be constructed on t h e closur e of i < 00 }. u("\ rt ,• ") J u"f < 00 i~ u( 7..) by Rx ci. fj'r = {q E f1 l > 0 : q has a n e i ghborhood i..i in rt* wi t h Dv e ry ene r~y-finite s olution u \ i. 8 . .r:,(u) = f u( ri ) iC ( z , q )dm.l:' (q) = U(u) + c an be r epr es ented wher e i(z , q ) i s a conti nuous fun c tion on rv r. A !-~function i s a nonn e r,;;ative s o lution whi c h i s th e i nf irnUill of d o wnward directed family of ener gy-finite s olu t ions . function is call ed PE-minimal ,,...., A nonzero .i: ~ if it i s a cons tant multiple of ev e r y ,...,., ,,....,, nonzero P Z-function dominated by it. Ti!illRci:·i. There exists a P .:..- minimal function if and only if there e xists a point lllL' (q) A r wi th suppo rt e.:iual t o wher e D( u ) i s the Jiri chl e t i nte ,;;r.nl of u) ' on lt. > 0. m0 (q) > o. TEEOR2H . For q E !1r . mF (q ) > 0 q E r s uc h tha t if a nd only if iv TABL E OF' CONT£NT~> 1 I i\'r 1te.J uuc·1 IOt~ ••••••••••••••••••••••••••••••••••••••• • •••••••• • 1. 1. i:'H~Ii·il.l:J;\ i·U I.i:::> •••••••••••••••••••••••••••••••••••••••••••••••• 4 II . .:\ 33TRAC'f HAPJ·lO!~IC CLA:3S.&S •••••••••••••••••••••••••••••••••••• 14 III . :\01 Dt!:L~ COi..lf ACTIFIC.ATION •••••••••••••••••••••••••••••••••••••• 28 I '.f . rlAIU~i.Oi; rc BOUND1\RY •••••••••••••••••••••••••••••••••••••••••••• 37 ., y • VI . R.~F ass1!.ij TI i·J G i"i E..1.\.3uH.!~5 . • • • • • • • • • • • • • •••••••••••••• • ••••••••••• 4 3 • •• 56 'fll . i3If~J~OGRAf h"'Y.••••••••••••••••••••••••••••••••••••••••••••••••• SJ 1 INTHDDUCTION ,..,, The notion of HD-minimal functions was first introduced by C Consta.ntinescu and A. Cornea in 1 <}5e.• The objective of this dissertation is to give a complete characterization of the more general class of all f-f!:-minimal functions on Ri.emannian manifolds following the pattern that H. Nakai has ,...., established in the case of HD-minimal functions on Hiemann surface, rv and finally to show that PB-minimal functions are closely r elated to fili_minimal functions. In finite dimensional real Euclidean space, the boundary of a set has a strong influence over all harmonic functions on t he set. A Riemann surface generally can not be embedded in a l arger surface, and therefore it has no natural boundary. For this purpose, Royden introduced, and Nakai developed the concept of Royden boundary for R."\.emann surfaces which is suitable for the study of all harmonic functions with finite Dirichlet integral (called HD-functions). For instance, HD-functions satisfy the maximum principle on the Royden boundar-J. Nakai also constructed a bounded positive r egular Borel harmonic representing measure on the Royd.en boundary so that to gether with a harmonic kernel every HD-function has an integral r epresentation r.J on the boundary and can thus be cha.racterized. Since '.10- fu."'lctions , in ,,...., particular HD-minimal function s , are infima of downward directed f amilies of non."legative HD-functions, it is not surprising that t:-iey can also be represented, although in a different fashion, as integr als on the ,..., Royd en boundary. One striking r esult is that there exists an HD-minimal function if and only if the r o exists an atom on the Ro,yd <-m boundR.r y ( 2 with respect to this harmonic representing measure. Recently, M. Glasner and R. Katz have shown that the concept of Hoyden boundary can be carried over to R:i.emannian manifolds, and that the Royden boundary is suitable for the study of not only HDfunctions, but ~C:-functions, i.e. the solutions of the elliptic differential equations 6u = ru with finite energy integral. In this thesis, we will see that a representing measure together with a kernel for the solutions of liu = Pu can be constructed similarly on the .Royden boundary, and most resultsfor HD-, .fill_ and ®-minimal functions can be generalized to F?-, fT;;._ and PS-minimal functions. The first main result is that there exists a [."'E:-minimal function if and only if there exists an atom on the Royden boundary with respect to the solution representing measure. ·1·J hat is more significant is that under certain circumstances ,,....., there exists a P.2:-minimal function if and only if there exists an HD-minimal function. In Chapter I, some preliminaries about Riemannian manifolds are given. We will prove that the family of all solutions of fiu = ?u on a Riemannian manifold forms a harmonic class in Chapter II. Although the procedure has been referred to, this is the first systematic exposition which brings the local properties of s olutions of elliptic differential equations in ~uclidean space up to a Riemannian manifold . In Chapter III, the Royden compactification and the Royden boundary are described. Several maximum and minimum principles are I given in Chapter IV. The representing measures on the Hoyden boundary rJ are constructed in Chapter V. PE- and ~ £-function s are carefully 3 s tudied in Chapter VI. Finally, the two main r esults are c iven in Chapter VII. 4 I. P RHLIHIN Afil ES In this chapter the definition of a Riemannian manifold is given. ·rhe calculus of exterior differential fonns is also sketched. We shall see that the equation Au = Pu on a Riemannian manifold, where P is a c1 n-fonn, in tenns of local coordinates is a self-adjoint uniformly elliptic second order differential equation. For a c1 -function · the uirichlet integral as well as the energy integral with respect to a n-form Pare introduced.Finally, Green's formula and the Dirichlet principle are derived from Stokes' theorem. Ia. family DEFINITIONS. A = { \Ua,~a) : a€ I Let R be a connected Hausdorff space. A } , where I is some index set, is an~ for R if 1) for each a, Ua is an open relatively compact subset of R is a homeomorphism, where En denotes n-dimensional Euclidean space, 2) { Ua : a e I } covers R, 3) if -1 ~lo ~ 2 is 2 . a C -function n-manifold. The homeomo r- whenever it is defined. The space R, with A, is called a c 2 phism ~ associated to each open set U i s called a system of local coordinates ~. and the open set U is called a parametric refion. U is called a parametric ball if ~(U) is the open unit ball in ; n. ~ is also called a parametric cube if ~(U) is a n-dimensional cuba 5 A cf m.-submanifold (m < n) that for all x e M, ~ ( U r't N) there is a pair M of R is a connected subset such (u,;) e A with x e and is an open subset of an m-dimensional subspace of En. Ib. DEFINITION. 2 A c n-manifold R is orientable if there is -1 an atlas A on R such that det(~ 1• f, 2 ) > 0 whenever U for any ~1'; 2 eA ~lo ~2_1 is defined. Ic. THEOREM. Every c 2 n-manifold is arcwise connected. Id. TH&> REl1. A C n-manifold is separable if and only if it 2 has countable base of parametric balls. These theorems are analogues of Theorems 2-17, 2-18 of [18]. Ie. DEI''INITION. A Riemannian manifold R is a connected, orientable, separable, noncompact (for our purpose) c2 n-manifold with a fundamental tensor gij yielding a positive definite form 1 are C -H8lder continuous. From now on, R will always denote a Ri.emannian manifold . If. We denote inverse matrix of (gij) by (gij) and the determinant of ( gij) by g. If we denote the local coordinates of R by n x1 , ••• ,x, then the arc element ds in each parametric r egion i s given A (ili.emannia.n) metric can be defined on R in terms of the arc element ds in a natural REN.ARK. way. .R is a metric s pace where the metric topology of 1t is equivalent to the locally Euclidean topology of rt. Indeed, if we take a countable base balls for R (see Id). Note that parametric balls having 3 1 :::: { l3 :::: { U } of para'Tletric U'} , which is the family of same centers and l/2 radii as those in B, covers R. Theri on each u'e B' in terms of local coordinates constant c such that for every vector (y1 , ... ,y11)e t1 we have there is a because l:gijy iyj i s continuous on U x j , where :i is t he unit ball in Ig. f Let AP be the set of all differentiable p-forms = ''f ~ a d x a , where tensor of rank p and l -< a.1.<a ·+l <:n , J. a a dxa :::: clx 1 /\ •••A dx P . fa is ~ covariant 1\n i nner n roduct i s dete rmined for AP by the rule b (1) dxa. dx = det( g aib.J) . In the 1-dimensional space An this product gives the volume el ement W by W•W:: 1. Th e Hodge s tar operator * is de.fined as 7 follows. If f e AP, then *f € An-p is det e r min ed by i ts ext e rior p roduct s h A *f for all = (f·h)w h e AP . Th i s gives an isomorphism between AP and i\n- p . I h. Th e ext e rior derivative df of a p- form f = ~fadxa /::,.u = *d *du. is the (p+l) -form (2) wher e df = "f(Ofa/o-? )dxi. 1 The Laplacian of a function (0-form) u i s Ii. To deternine /1u in local coordina tes, we be.:;in by compu- tine; t he local exp r e s s ion of *f for an a r bitrC1.ry p- fo nn f. Let f ::: Lfadx~A 2: a f\d x P , • a *f = ( 3) bn-p • /\ dx (*f\dxbi" • b For a given b = (~,. •• , bn-p), take the compl ement of I b and left multiply both sides of (3) by ' b dx 1/\ 0 f • /\ dx P left we obtain I bl I bl dxbi.I\ .•• f'.dx P f\ *f = ((dxb:i_.I\ · • ·/\dx P)· f )w = L a while on the ri Bht we have I fadet( gaibj)w, On t '.1e 8 where Eb' b is the siEnature of the pennutation ,Je concludf~ t hat L{ b . Since we shall only deal with functions \0-forms) u on ii , we proceed as follo ws . 6.u = *d*du, ( !+) dx du= d*du i , = \' 0 ~~j J =I:~ . . c xJ 1,J w ..; . [; ' !. In t h i s thes is, we shall study t he s olution s of the ea uation b.u = *Pu, where P is a nonnc~gative n-form. Locally it is a self- adjoint elliptic second order differential equation as we have just l;'urthe nnore , we s hall see that, locally /1u = *.i:·u s hown . is a uniformly elliptic differential equation . Tak e n counta ble base (cf . Id) . Le t b'= 8 = { U } of par a[lletric ball s for R {u'} bo Uw family of pa ramntri c l)<t1l.~~ which havo 9 same centers an d l/? radii <JS those in t: . ifote t hat L' cove r s ;{. In e ach U e i.3 , b,y the p revious result , tlrn equation /1u = "'Pu h R. ~• the form where pdxl" . . . 11 dxn is a local representation for i· . If u'e 3 ' which has same center as U but 1/2 r adius , then th e re exists a constant c:::: 1 on u' where such that (;>1_ • ••• ,yn) € ff1 is an arbitrary point. This was es tablished in If. Thus 6u = * Fu is a unifonnly elliptic second order r· I. and diffe r ential eq uation. Hence, in particular, t he r e sults in ~5 [) 9] can be applied T' -J. in parametric balls U • DEFINITION. A subset ·J of H is said to have a piecewise smooth boundary bG if bG consists of a countable number of c 2 ( n-1)- e.nd c 2 (n-2)-submanifolds which do not accumulat-:i in R. DEFINITION. An exhaustion of R i s a family {-\ of r elatively compact open sets such that 11 ~ CRn+l : n = 1, 2 , ••. } for all n and Ut.H.n : n = 1 , 2 , ••• } = R. ThiOR!!:i.'l. There exists an exhaus tion { H11 J each f{n is a r e p;ion with piecewi se s moo th bounuary. of 0. s uch t hat 10 Proof. By Id, we can choose a countable base {Ui} of parametric R:i_ = u1 , and d efine inductively Rn to balls for R which covers R. Let be a connected open set which is a finite union of U's such t.~at i\i-l u Un. Clearly {1\i Ik. lb ~ l meets the requirement. Integration on manifolds can be d efin ed by means of partition of uni t y . For t he d etails see [171. Consider any relatively compact subset S of R. For functions u, v e c1 ( s ), t he Dirichlet integral of u and v over S i s defined a s D5 (u,v) = ~duA*dv. s In addition, i f F is a n-form on R, the energ;y integral with res pect ~ of u and v over S i s defined as Es (u,v) = D3 (u,v) + ~uvP . :::> If S i s any subset of R and u, v e C1 ( S ) , the Dirichlet and en er gy- integr al s of u,v over S are def i ned as D5 (u,v) = lim D~ (u ,v) .:>n and E5(u,v) = lim Es (u,v ) , n if the s e limits exis t, whe re Sn = SA 1\i fl . If S = R, we simply write and { 1\i J i s an exhaustion of 11 D(u,v) instead of DR( u, v) and E(u,v) and ER( u, v). We also denote and and drop the letter S when S = R; these are the Dirichlet and energy integrals of u over s. Since D n(u) and 5 E5n(u) are monotonic, we can conclude by the Schwarz inequality that the definitions of Dg(u,v), E5 (u,v) are independent of the choice of [Rn}• Suppose S is contained in a parametric region. tie shall calculate the expression of D8 (u,v) in tenns of the local coordinates. By (4) and (5) with v in place of u we see that Thus Il. STOKES' THEOREM. Let G be a relatively compact open set in R whose boundary bG is piecewise smooth. Then for all have Jr bG r· \..1 f€ An-l, we 12 COROU.ARY (Green Is .fonnula). If u e c1 (G) and v e c 2 (G), then DG(u,v) + fud*dv = fu•dv. G bG Proof. f bG In fact, by Stokes' theorem u*dv = fd(u*dv) G = Jdu A *dv + Jud*dv G G = DG(u,v) + Jud*dv. G Im. DIRICHLET PRINCIPLE. Let G be a relatively compact open set in R whose boundary bG is piecewise smooth. If v € c1 (G) u 6 c2 (G) and such that u is equal to v outside G and u satisfie s the equation d*du = u.P Proof. in G where P is a cl n-fonn on R, then By Green's f'onnula EG(u - v, u.) = DG(u - v, u) + f cu - v)u? G = DG(u - v, u) + f G = f<u - v)*du bG = 0. <u - v)d*du ' 13 Hence Now 0 ~ EG(u - v) = EG(u) - 2EG(u,v) + EG(v) = EG(v) - EG(u), i.e. 14 II. ABSTRACT HARMONIC CLASSES Let P be a c1 n-form on a Rianannian manifold R. By identifying n-forms and 0-forms which correspond under the * isomorphism. we can write the equation d*du = uF as (1) ~u = Fu. Beginning with a few simple axioms, M. Brelot [l J has developed the abstract theory of harmonic functions (also cf. [9]) which contains the classical results of the theory of certain elliptic differential equations as special cases. In particular, the complete presheaf of solutions of the equation (1) on open subsets of a Riemannian manifold R satisfies the axioms as we shall see in this chapter. Indeed in this thesis, the concept of abstract harmonic classes will be used as a unifying tool and also as a mean for making the transition from a few well-known local properties of (1) to the global properties that will be needed. Throughout the exposition we use the following notations: a) for any function f and G a subset of its domain, flG is the restriction of f on G, b) for any functions f and g (fv g )(x) = max( f(x), g (x)) (fl'\ g )(x) = min(f(x), g (x)), 15 and c) instead of lim supx~A.x ......x f(x), 0 1~6 A,x-x f(x), lim infxGA,x .... x f(x) and 0 we simply write 0 respectively whenever no confusion arises. Ila. M. Brelot has introduced the following definition of a hannonic class (cf. p. 169, [9]). Let W be a connected, non-compact, locally compact, Hausdorff space. If G is an open subset of W, then by bG and Gwe mean the boundary and the closure of G respectively. The functions that we shal.l consider on W are extended realvalued functions with the usual lattice ordering ~ • By an upward directed family of functions we mean a nonempty family such that for any two functions :f and g in the f am.ily there is a third function h in the fa.>nily with h(x):: (fv g)(x) f'or all x e H. A downward directed family is similarly defined. DEFINITION. Let ~ be a class of real-valued continuous functions with open domains in W such that for each open set family GC W the 3f ( G), consisting of all functions in W< with domains equal to G, is a real vector space. An open set G of W is said to be regular for 8f or regular if for every continuous real-valued function f defined on bG ther~ is a unique continuous function ori G such that H(f,G)lbG:: f, H(f,G), or simply H(f), defined H(f,G)!Ge I{, and H(f,G)==::O if' :f~O. 16 Moreover the class 8( is called a hannonic class .2!! W if it satisfies followin g three axioms. domain U, then C h!V e M for all open subset V CU, and (2) if fha: aeI} X , where ha has domain Ua and I is some index set, such that hal ua n~ = hblUa('\~ domain regular such that I h ua = ha for all a E I. There i s a base for the topoloe:y of \.i consisting of regions. Axiom III. x hE 9{ with for all a,b EI, then there is an Uf ua : a e I} Axiom II. and h Eal with I{ is a. complete presheaf, i.e. (1) if Axiom I. If G is a reg ion in W, A a compact subset of G a point of A, then there is a constant h E ~(G) nonnegative function M> 1 such that every satisfies the inequality h(z) ~ .Mh(x) at eve ry point z of A. IIb. e~ uival ent TE.'.XJR'.:il·f. Given Axioms I and II, the following axioms are to Axiom III. '1 i s an upward directed subfa:nily of ~ ( G) where G is a region in W, then the upper envelope of 'J i s either -+«> Axiom III • 1 or a function in If ~ (G). Axiom III • 2 If f~} is an increasing sequence in G is a region in \.J, then either lim 1\-i Axiom III function in ~(G) 3 = +oo or lim ~ is in ):{(G) where ~(::;). • If G is a r egion in \V, then every nonne gative is either identically 0 or has no zeros in G. Furthermore, for any point x in G, tho set 1x = {h e ~ ( G) : h > O and i s equicontinuous at x. h(x ) =1 } 17 For the proof, see p. 373 and p. 378 of [ 2] as well as p. 598 of [10]. Ile. THEOIDM. on U and U is an open subset of R} Proof. T< = { h : h is a solution of (1) The family Obviously the set is a hannonic class. J((G) of all solutions of (1) on an open set G is a real vector space. Axiom I is clearly satisfied. The collection B'of all para.metric ball s with same centers and i radii as those in B,which is a countable base for the topology of Ras in Id, is a countable base for the topology of R. To see these parametric balls in B' are regular with respect to corresponding balls in .Efl J\ it is sufficient to see the the are regular with respect to the corresponding elliptic differential equations. At this point we appeal to the results in [5]. Finally, 1< Ild. satisfies Axiom III because of the following lemma. LH1A . Let G be any region in R and x, y E G. Then t here exist constants a and b depending on x and y such that a u(y) S u(x) S b u(y) for any nonnegative solution u 6 T< (G). Proof. Indeed, if x and y are in a parametric ball, then the lemma is an easy consequence of the Harnack' s theorem of [5] . Otherwise, there exists a finite number of points z1 , ••• ,zn e G such that and U. C G where and . z 0 are in some parametric ball = x, zn+l = y. 1 Hence by previous observat~on, z1 i = 0,1,2, ••• ,n there are 18 constants 8i•••••an = 0,1,2, ••• ,n. i 11_ .... ,bn and such that Consequently Together with Axiom III , we draw the following useful Ile. 3 consequence from IId. THEORE21. k(x,y) u Let G be any region in R. For any = inf f c :(l/c)u(y) ~ u(x) ~ cu(y) e J.< ( G) J , function x,y € G, let for all nonnegative functions then i) k(x,y) ::_ 1, ii) u(y) /k( x ,y) < u(x) < k(x,y)u(y) - u 6 1< ( G) , - iii) k(x,y) = k(y,x) for all iv) lim k(x,y) x,y-z =1 v) for fixed y, Proof. for any nonnegative x,y e G, for all z e G, k(x,y) is a continuous function of x. i) and ii) are obvious. From ii) we see that u(x)/k(x,y) ~ u(y) ~ k(x,y)u(x). Thus k(x,y) :S k(y,x). Hence iii) follows . o be such that By symmetry we also To prove iv), if f> 0 have is any positive number, let N = max(l +o,1/(1 - o)) < l + E. • asserts that the family k(y,x) :S k(x,y). H.ecall Axiom Il1J tz = f u/u(z) : u e J( (G), u > 0 J is equicontinuous at z. Thus there is a neighborhood U of z such that for 19 all x e u j u(x)/u(z) - 1 J < ~ , or equivalently (1 -o )u(z) < u(x) < (1 + ~)u(z). Hence u(z)/N < u(x) < Nu(z). Consequently, 1 ~ k(x, z) ~ N < 1 + E , i. e . lim x-z k(x,z) = 1. Now, from the inequalities u(z)/k(x,z) ~ u(x) ~ k(x,z)u(z) u(z)/k(y,z) ~ u(y) :S k(y,z)u(z), we obtain u(y)/(k(x, z )k(y,z)) :S u(x) ~ k(x,z)k(y,z)u(y). Thus by definition and i) 1 ~ k(x,y) :S k(x,z)k(y,z). vi) Hence k(x,y) tends to 1 as x and y tend to z because k (x,z) and k(y,z ) do by the previous argument. It remains to prove v). Note that vi) is true for all x,y and z e G. Thus l/k(z,x) = k(x,y)/(k(z,x)k(y,x)):S k(x,y)/k(z,y) ~ k(x,z) k(y ,z)/k (z, y ) = k(x, z ). Hence we have i.e. k(x,y) or limx is a continuous function of x. -z k(x,y) = k(z,y), 20 IIf. From now, a function u e }< with open domain Oc R solution .2.!L!l· will be called a The particular hannonic class of harmonic '£unctions with open domains in R, i.e. of R} {h : h satisfies b.h = 0 on U, U is an open subset will be denoted by J.(. IIg. A regular exhaustion of R with respect to a DEFINITION. M is an exhaustion of R such that each of its members is hannonic class regular with respect to ~. THIDRDI. both There is a regular exhaustion on R with respect to 0 T< and J( • Proof. Consider the exhaustion {}\.} constructed in Theorem Ij. Note t hat for any parametric ball U with local coordinates ,, ~(b~ ri U) satisfies the cone condition (see [8J , p.)29). Hence there exists a barrier function at every point of ,(bl\i n U) on ~(l\i ('\ U). This means that ~(bl\i (') U) i s regular in Iffl for the Laplace equation ·~}u/ ox1 + • • • • + ·clu/ ~ x11 = o. 2 2 Hence by Theorem 10.2 of [19] elliptic equation bl\i n U of ~(b£\ir'I U) is regular in Ff1 for t he (cf. Ii) is thus regular with respect to both [9], and hence bRn J(° and is regular with respect to both by Brelot's comparison theorem (cf. Theorem J.2, [9]). f.<. by Theorem J. 3 J< and T< 0 21 IIh. Consider again an abstract hannonic class at. on W as in IIa. If G is a regular open set. let C(bG) be the set of all contim,1ous functions on bG. For each f 6 C(bG), we consider the function H(f,G) given in the definition of regularity (Ila). For each x e G, it is easy to see that H(f,G)(x), as a function off, is a positive linear functional. There is a positive regular Borel measure r(x,G) defined on bG such that H(f,G)(x) for each f e C(bG). Since = Lfdr(x,G) r(x,G)(bG) = H(l,G)(x) < oo, this measure, hence also the functional, are bounded. DEFINITION. respect to 8::( IIi. r(x,G) is called the hannonic measure with for G at x. Let f be lower semi-continuous on bG where G is as in IIh. DEFINITION. f is integrable with respect to »:{ is integrable with respect to = sup [ f gdr(x, G) r(x, G) x e G, i.e. for all g e C(bG), g ~ f } < 00 for all case, we set H(:f,G)(x) = H(:f)(x) = on bG if it f x e G. J fdr(x, G) In this :rdr(x,G). bG IIj. Brelot ([l], p. 65) has proved the following lemma which in fact is an easy consequence of Axiom III 1 • 22 LEMHA. Let f be lower semi-continuous on bG where G is a ~ regular open set. If f is integrable with respect to H(f) e on bG, then ~(G). IIk. Let G be a regular open subset of W for THEOREM. ~ , f a lower semi-continuous function on bG and x e bG. If f is bounded from above, then If f is bounded from below, then lim infbGf(x) ~ lim infGH(f)(x). For the proof, see [ 91 on p. 173. Let W be as in Ila. A family Ill. ~ of extended real-valued lower semi-continuous function with open domains in w is called a superharmonic class with respect to 1) for all v ):( if e ~ with dome.in G, v(x) < o::i for some x in each component of G, and 2) for all x e G such that v(x) < co and for any neighborhood u of x, there is a regular region v with such that v is integrable on bV and v(y) ?_ H(v, V)(y) x e v eve u for all y in V. A useful observation is LEMMA. au are in ~ if If u,v a> o. with same domain, then Uf\V, U+ V and 23 Functions in 8:{ are called superhamonic function;; with ~• respect to We say an upper semi-continuous function u belongs to the family »l_ , the subhannonic class with respect to 3'.{, if u is called a subhannonic function with respect to Wt. -u E 8:{. and If G is an open subset of W, we let ){(G) denote the class of functions in with domain G and ){ ~(G) the cla.ss of functions in J!. with domain G. IIm. THEOREM. Obviously »:{ C ~ n ~. Proof. u E ~ n 9:t Conversely, for any with domain G, u being both lower and upper semi-continuous must be continuous on G. Furthermore, for any x E G, there is a regular open neighborhood U of x with UC G. By the super- and sub-harmonicity of u, we have ul u :::, H(u, U) ::::., ulu, e )t ( G) • ~ n ~ C 8={ • u Hence IIn. THEOREM. a,b are constants with Assume 1 e i.e. ulu = H(u,U). By Axiom I, Bf. (W). Let G be a region in W and a S 0 Sb. 1) If v E W{(G) and v ~a, then either v > a or v:: a. 2) If u e ){(G) and u Sb, then either u < b or u::: b. 3) A nonconstant function in ~(G) takes neither a nonnegative maximum nor a nonpositive minimum in G. For the proo~of this and the next theorem, see [9], p. 178. 24 IIo. THFDREM. Let G be a relatively compact open subset of W and v a. nonconstant function in c ~ lim infGv(x) such that 8t(G). Let c be a nonpositive number for each x E bG. If l E af(G), then on G. v > c In view of this, we shall henceforth make the ass'l,Jmption that l E 3\(W). IIp. Some relations between hannonic functions and solutions on R are given below. + Let f( 0 = THIDREM. u ~ o}. - Then T< +c T< and f(c 0 Proof. that f h e J.<•: h:: 0} T( T{ = f u E Ji: and In particular, l e K(R). To prove the first assWlttion it is sufficient to show 1<.• = f u 6 T< : u :: 0 0 } C - T< • So let u' e J( 0 + and any z ' in the domain of u. For any regular neighborhood U of z which is contained in the domain of u. let v be the element of 1<(U) such that vlbU = u{bU. We want to show that u(z) :::_ v(z). If this is false, then let = {x E U : u(x) < v(x)} • Note that v :::_ 0 on U by the regularity of u. and /j.(v - u) = fJ.v = Pv:: 0 in V. Hence v - u is constant on V (see V [4], p. '.326), which means that Vis empty. Thus u> v onU, i.e. u€T<. The second assertion can be proved similarly. IIq. Fron now on, functions in :rt and 0 T< will be called super- and sub-hannonic functions respectively, while !'unctions in and Ji will be called super- and sub-solutions respectively. l< 25 Some classical theorems about solutions of 6.u = Pu Ilr. are particularly easy when viewed in terms of harmonic classes. THEOREM. Let { un } be a sequence in »<( ~J) . If {un) uniformly on each compact subset of W to a function u, then Proof. For all x e W, un u(x) = This is true for all u S 8{ (W) IIs. u+, u- € 3{( U) udr(x, U). bU x e U, hence u is integrable and u € a:{ ( U). L et U be any relatively compact regular open LEMMA. u e 3{ (w), there exist nonnegativ e functions such that Horeover, u+ olbU, f by Axiom I. subset 01' W. For any uv r(x,U ) converges to u unifonnly on compact sets, in particular on bU, we have Hence 81 (i~). there is a relatively compact regul.ar open neighborhood U of x. In terms of harmonic measure Since u e converges u = u+ - u - and u- u-jbU = -(Uf\ O)jbU. Proof. Le t COHOLLARY. on U. are continuous on U, and u+ = H(uv O,U) u+ < sup) uj , and and u- u+l bU = = H( -(uri 0) ,U). 26 Proof. IIt. in f< ( R) ( This is an application of ThHorem Ilo. Let THWRJ!M. { un} be a bounded sequence of fW1ctions Jr (R), resp. ) • Then there is a subsequence which converges uniformly on compact subsets to a function u € R (R) ( J{• (H.), resp.). Proof. to prove that Consider Ras a metric space (cf. If). We are going { un} is an equicontinuous sequence of qontinuous functions on any compact set. Because of the boundedness of { un}• Ascoli 1 s 1 theorem asserts that { un is a normal family, -i.e. { un} has a subsequence converging uniformly on compact subsets to a function u. rly Theorem IIr, u e 'f< (R). Indeed, for a:ny point q 6 R and any relatively compact regular open neighborhood U about q, we have by Theorem Ile u~(q)/k(p,q) for all p 6 U, where u: < k(p,q)u~(q) (and u~ later on ) is defined in the Lemma. Hence (l/k(p,q) - l)u~(q) :S u:(p) - u~(q) < (k( p , q ) - l)u~(q). Thus < + un(q) max( k(p,q) - 1 , l - l/k( p , q ) ) < M max( k(p,q) - 1 , 1 - l/k(p , g) ) , where M is a bound for { un). \u~(p) - u~(q)I Similarly < H max( k(p,q) - 1 , 1-1/k(p , q ) ). 27 . ;' As a consequence, < 2.M max( k(p,q) - 1 • 1 - l/k(p,q) ) , where the right hand side tends to 0 independently of n as p tends to q since k(p,q) is a continuous function. Hence q. Since R is a metric space, subset of R. {uni is equicontinuous at {un} is equicontinuous on any compact 28 III. ROYDEN COMPACTIFICATION In a region of Euclidean space, the boundary of the region plays an essential role in the study of solutions of elliptic differential equations. The Hoyden compactification R• of a Riemannian manifold R is used to give the manifold a boundary r = R• ' R, the Boyden boundary, with which the class of all solutions with finite energy can be analyzed. The Boyden compactification is constructed by means of Hoyden algebra which is the collection of all bounded Tonelli functions with finite Dirichlet integrals. In this chapter, we shall study the 1 properties of the Royden algebra and describe the construction of the Royden compactification. An intrinsic part of the Royden boundary is the hannonic boundary. Its importance will be examined in next chapter. Illa. DEFINITION. A real-valued function f on R is a Tonelli function if for ea.ch parametric cube U with local coordinates ~. where 1) for ea.ch i the function is absolutely continuous for almost all (with respect to the Lebesgue measure) ,b . )ic. • .J<(a ,b ) • .L+1 . 1+1 n n (a.i. 2) ~(U) • or/ oxi are square integrable over any compact subset of 29 IIIb. Let M(R) be the set of all bounded Tonelli functions on R with finite Dirichlet integrals. M(R) will turn out to be an algebra as we shall see in the next theorem. We call M(R) the Royd.en algebra. THEDR.&1. M(R) is a commutative algebra with identity under the usual algebraic operations. Forany feM(R), Proof. Clearly H(R) is a real vector space. It is also clear l/feM(R) ifandonlyif infRlfl>O. th.at the constant function 1 is also in M(R) serving as the identi~y. Now we are going to show M(R) is an algebra. For any f,g 6 M(R), K be a bound for r2 and g2 • fg is a bounded Tonelli function. Let Then on any relatively compact 5, DS(fg) = Jd(fg) A •d(fg) s = J (gdf + fdg) A (g*df + f"'dg ) s = J 2 g df A *df + 2fg df A *dg + :r2ds h *dg s < K (DS(f) + 2D.,::> (f,g) + D,,;:;> ( g )) Thus i.e. fg e M(R). Hence M(R) is indeed ·an algebra. JO Finally if f e M(R) with inf RI f I > o. Then supRll/fl < L for some L. Note on any relatively compact set S D5(1/f) = ~d(l/f) A *d(l/f) s Thus 4 D(l/f) ~ L D(f) < oo • i.e. l/f E M(R). Ille. THEOREM. operations v and Proof. M(R) is a lattice under the usual lattice (\ • Let G = {x e R : f(x) > g (x) . }. Then D(f v g) = DG(f v g) + DR ,GD(f U g) < D(f) + D(g ) Similarly, D(f f\ g) ~ D(f) + D( g ) < oo. IIId. We shall make use of the following notions of convergence in M( R). Let { fn J be a sequence in M( R). We write 1) f = C-lim f'n if { fn} converges to f unifonnly on any compact subset of R. 2) f = B-lim fn if { fn} is bounded and J) f = U-lim f'n if { fn} converges to f unifonnly. 4) f= D-lim fn if lim D(f - fn) = o. 5) f = £-lim fn if lim E(f - fn) = o. 6) f = XD-lim fn (or f = XE-lim fn) if and f= E-lim fn) IIIe. f = C-lim fn• f = .D-lim fn (or f = X-lim fn, where X can be C, B or u. THIDREl-1. M(R) is SD-complete , i.e. complete wi. th respect to the ED-convergence. Cf. [12], Lemma 1.5. f = B-lim fn and This theorem means, in part, if lim D(fn - fn+ p) = 0 for all p, then fn e H(R), f e H(R) and f = BD-lim f n. IIIf. THIDREM. c1(R) rt H(R) is dense in H(R) with respect to the DD-convergence. In fact, this is Theorem 1 of [12]. We are more interested in the follo'Wing corollaries. COROLLARY (Green 1 s fonnula). Let G be a relatively compact 32 open subset of R with piecewise smooth boundary bG. If u e M(R) and v e c2 (G'), then COROLLARY (Dirichlet Principle). Let G be a relatively compact open subset of R with piecewise smooth boundary bG. If u, v E M(R) such that IIIg. u = v on and u satisfies 6u = Pu R '-..G in G, The Royd.en compactification of R is a compact Hausdorff space R* such that to 1) R is open and dense in R*, 2) functions in M(R) can be extended as continuous functions R•. and 3) M(R) separates points of R*. The compa.cti.fication is unique up to homeomorphism fixing R elementwise. We shall give a description of the construction of the Royden compactification in section IIIi. For the details. see p . 184 of [14] and p. 159 of [13] or cf. any exposition on the Gelfand representation. IIIh. For all functions f 6 M(R), we define a nonn Then THIDfill'i. H(R) with tho nonn II· II is 11 IJmiach algebra . JJ Cf. Ilib and IIIe. IIIi. l'l e denote the dual space of l'l (R) by N(H)* and conside r 1 the weak* topology of M(R)*. Let us consider the set R* of all multiplicative bounded linear functionals L with L(l) = 1. It can be proved that R* is a subset of the closed unit ball and is closed in the weak* topology of M(R)*. Furthermore R* is compact and Hausdorff. '.-le define a mapping a from R to R* such that for every cr(p) (f) = f(p) for all f p e ii e N(R). It is easy to see that o we have is one-to- one and continuous. Moreover o(R) is open and dense in R*. \..Je now identify R with o(R). We can see that the topology of R is the same as the relative topology of R in R*. Now we are ready to extend every function in H(R) to R* continuously •.for every f € M(R), we set f(L) = L(f) p G R, then f(o(p)) for all L € R*. In particular if =o(p)(f) = f(p). It turns out tha.t the function f is indeed a continuous extension of f. That H(R) separates points of R* is immediate. R* constructed above is the Royden compactification of R. From now on. if G is any subset of R 2£. R*, -:J always means the closure of G in R*. IIIj. THEOREM. H(R) is dense in C(R*) with respect to t h e uniform nonn. Indeed, the proof i s theorem. B.n application of Stone-\veiers trass 34 IIIk. This and the Urysohn lemma l ead to the following consequence. THEOREM. (Urysohn property) Let Ki. and K2 be any two disjoint compact subsets of R*. For any two distinct real number r 1 < r 2 , there always exists a function r 1 ~ f ::;, r 2 I1 f .K. = r and Proof. 1 , such that = l, 2. g e C(R*) By Urysohn lemma, there is a function = r2 + 2. gl11_ = rl - 2 and hE M(R) hi 11.. < rl - 1 with i f e 11(R) g IK2 f = (h U r 1 ) n r 2 e M(R) Illl. r 1 , r 2, and with By IIIj, there exists a function hlK2 > r 2 + 1. Then the function meets the requirement. DEFINITION. r = R* '-.R is called the Royden boundary of R. The following proposition is due to Nakai (Proposition 6, [16]). PIDPOSITION. IIIm. G "bG is open in R* for any open subset G of R. If we let M0 (R) be the set of functions in M(R) with compact support in R, then we have the following theorem. THEOREM. Proof. r = {q. GR* : f(q) = 0 for all f If g E Mo (R). then € rl r = f IR*'-. R = 0 M0 (R)}. because the support of f is in R. On the other hand, for any q 6 R, there is a relatively 35 compact open neighborhood U of q in R. Note that R*. By virtue of Theorem IIIk, g(q) = l glR*'-.U = -l. and supp f CUC R and IIIn. Let MA(R) there is a function Let f(q) > o. R• '-.,U is compact in g e M(R) with f = g v 0 € M(R). Note that Thus f e l1 0 (R). be the ED-closure of M0 (R) fl= {q e R* : f(q) = o for all in H(R), then f e MA(R)} is called the harmonic boundary of R. The following is obvious. LEMMA. IIIo. ~ is a compact subset of f". Now we consider the RoyS.en compactification of subregions. Given any region G in R, G itself is a Riemannian manifold. Hence we can consider the Royden compactification G* of G. It is not surprising that there is a canonical relation between the Royden boundaries of G and R. The next two theorems are Propositions 7 and 8 of [l6l. IIIp. l'HEDRlli. There exists a unique continuous mapping j from G* onto G fixing G elementwise, where G* is the Royden compactification of G and G the closure of G in R*. IIIq. Clearly G '-G = bG V ( (G '\.bc)f\ l). If we let 36 B(G) = (G'\ bG) n r, THIDRF.M. then G*'\. G = j -1 - (bG) v j j is a homeomorphism of -1 Moreover ( B(u)). GU j -1 (B(G)) onto GUB(G). Later on (cf. VIIe) we shall talk about representing measures on the Royden boundaries. and shall show that a Borel subset E of B(G) has positive measure with respect to R* if and only if j positive measure with respect to G*. -1 (E) has IV. HARMONIC BOUNDARY We now change to the following more traditional notations. For any open subset G of R, we consider the solutions of /1u = Pu, and let PX.(G) = f u € T< (G) : u satisfies the property X on G} where X can be a. boundedness property like N (nonnegative), B (bounded), D (finite Dirichlet integral), E (finite energy integral) or combinations of these. Also for harmonic functions, we let HX(G) = { u 6 T((G) : u satisfies the property X on G} where X can be N, B, D or combinations of these. We say that u is a PX-function~ G if u e PX(G), or u is an RX-function ~ G if u e H.X(G). As we shall see, b. will play a significant role in determining the energy-finite solutions. Indeed, in this chapter, we shall state that energy-finite solutions, in particular Dirichlet-finite harmonic functions s atisfy the maximum principle on /1 • Noreover, an energy-finite Tonelli function f can be uniquely decomposed into a sum as f =u + g where u e PE(H.), ulfl = flfl, E(u) ~ E(f) and lul ~max~ Ir!. Glasner and Katz [6] have proved the followin g result for the harmonfo boundary b. : b. =¢ if and only if H i.s Green's function on R). pa.raoollc (i.<i. thor t:.l is no hannonic Since we want to study the Riemannian manifolds with nontrivial energyfinite solutions which are always nonparabolic [7], we shall always have nonempty harmonic boundaries. At the end of this chapter, we shall introduce a subset /lp of /J. which will eventually determine all PS-functions. lVa. Then M(R) and () • LEl1i"iA. Let M(R) = {f : f is Tonelli and D(f) < oo }. is a lattice with respect to the usual lattice operations v Moreover, functions in M(R) have continuous extended real- valued extension to R*. ,v The proof is simple, for if f is a nonnegative function in N(R), then h = f/(l + f) is a nonnegative function in N(R) which has a continuous extension ti to R* by IIIg. It is easy to see that f = h/(l ~ h) (f(q) f on R*. In general = oo if h(q) if f is not nonnegative, then f is a difference of two nonnegative ones in M(R), i.e. IVb. = 1) is the continuous extension of f = (f v 0) - ((-f) v O). Clearly, HD(R) and PD(R) are subsets of h(R). Furthermore we have the following theorems. THEOREM (Maximum and minimum principles). on R takes its maximum and minimum on 6'very HD-function ~ • Cf. [14], p. 192. lVc. THIDREM (Maximum principle). 1). Every nonnegative .39 6. PD-function on R takes its maximum on 2). A PD-function on R is nonnegative if it is nonnegative on ~ • Cf. [7], Theorem 2. IVd. Since Let E(R) = {r: f is Tonelli on Rand E(f) <oo }. E(R) C M(R), functions in E(R) have extended real-valued continuous extension to R*. THIDREM (Royden-Nakai decomposition theorem). f E E(R) has ,the unique decomposition e FE(R), e E(R) f = u + g with 1) u 2) lul ~ sup8 1rl, 3) if v is a supersolution on R and 4) B(u) ~ E(f). g &'very function and IA = o, f - u v > f, then v ~ u, For the proof, see Theorem 3 of [7] and cf. p. 190 of [14]. IVe. DEFINITION. A compact set K CR* compact set if b(K I'\ R) is piecewise smooth and is a distinguished K fl R = K. A useful generalization of the previous theorem is function THEX:lREl'l. Let K be a distinguished compact set in R* . Ever-.f f E E(R) has the unique decomposition 1) u e PE(R'\.K)fl E(R), g e E(rt) 2) jul ~ sup(ll"K)Vb(Kf\R)lrl, and I f = f - u + g with ul~vK = O, 40 if v is a supersolution on J) then v > u to every on R' K and v > f on R '-. K , R '\. K, 4) E(u) ~ E(f). Cf. [7], p. J50. IVf. Let us denote by nK the correspondence which associates f e E(R) theorem IVe, i.e. the unique u e PE(R'-K) in the decomposition we have We denote n = n¢ where ¢ is the empty set. nK is called the solution projection on R with respect to K (or simply the solution projection on R if When P: K = ¢). 0, the corresponding projection is denoted by and is called the hannonic projection on R with respect to K (or simply the harmonic projection on n if K = ¢ and denoted by n°). It is worthwhile to note that projections are linear mappings. IVg. LElwll'iA. If F is a closed subset of r "~ ' then there is a positive superharmonic function v which is in N(R) such that v = 0 on lJ. and v = oo on F. er. [J], Hilfsatz 9.1, p. 101. IVh. MINIMUM PRINCIFLE. Let G b e an a rbitrary region in H. and u a supers olution on G bounded from below. If for s ome nonpos iti ve 41 lim inf Gu(q) ? c number c, for all q E bG V ( 6 (') G), u ? c. then We extend u to u' as a lower semi-continuous function Proof. on G by defining u'(p) = lim infGu(p) for all p E G '-G. U is open in bG V G'\.G (6 f l G) CU. = u. b < c, let Note u'IG U = { p E G'\.G : u'(p) > b } • since u' For any is lower semi-continuous, and Note- F = (G'-G) '-.U is a compact subset in R*, in particular, F is a closed subset of r '-. ~. By the lemma, there is a nonnegative superharmonic function v E M(R) such that Since v is also a supersolution by IIp, the function = oo • vlF w = u' + v/n is a supersolution on G by Lemma Ill, where n is a positive integer. w> b Observe that on G '-G. Being a lower semi-continuous function w takes its minimum a on "O'.. By minimum principle IIn, w can only take its minimum on the boundary G '-G. w> b on G. u ~ c on G. Consequently w ::::_ c IVi. Hence w ? a ;:!, b, and indeed on G. As n tends to oo , we see that We have seen that the harmonic boundary /::. plays a significant role determining the energy-finite solutions, and hence Dirichlet-finite harmonic functions. However, in [7] for energy-finite solutions, subset 6p of /). it is shown that l!J. is too big. More precisely, an open to be given below has a strong influence on solutions. DEFINITION. The n-form P is said to have finite integral at lf-2 _ p e b. if there is a neighborhood set of all such points p e A I P < oo. The then there is a of p such that U C R* will be denoted by tJl\R b.p. An immediate observation is b.p is open in LEMM.A. f 6 E(R), then IVj. LE:-1MA. If Proof. If f(p) > 0 neighborhood U of p such that 00 > E(f) ~ [ _r2p > o2J P, for some f ILl"-.AP = o. PE A-....... /).f, .rl u > 0 > 0 for some 0. We have hence I p < oo, i.e. p € b.p. This un R Uf'\ R Uf'\ R fJ. • is, however, impossible. IVk. As a corollary of IVc and IVj, we have THIDml. (Maximum principle). function on R takes its maximum on l). Flvery nonnegative fE- [j,li. 2). A PE-function on R is nonneg ative if it is nonnegative on IVl. It is worthwhile to note that AP = A if ~p <co. In particular, AF =fl when 1-' : O. 4J V. REPRESENTDJG MEASURES A representing measure on the i\oyden boundary for the class of HD-functions was constructed by Nakai [14]. In this chapter we Eeneralize his results to the equation fiu = Pu. A relation between these two kinds of representing measures is given in Ve. Va. Let ~(R) be the set of all bounded, energy-finite Tonelli fwictions. E(R) is a subalgebra as well as a sublattice of N(R). Tl!EORliH. E(R) has identity if and only if JP < oo • R Proof. For any f,g E E(R), it is obvious that af E E(R) for any real number a. Also by Schwarz 1 s inequality E(f + g) = E(f) + 2E(f, g) + E(g) ::: E(f) + 2./E(f)E(g) + H:(g) < 00 . Moreover, < D(fN) + sup g·2 Jr2p o R <co • So E(R) is a subalgebra. On the other hand, we let f(x) > g(x)}, tlHm E(f u g) = t!.:G(f v g) + EH"-Li(f u E'J G = {x E R 44 ::=. E(f) + E(g) Similarly E(f f'\ g ) < Vb. oo • THIDRE11. Proof. Hence E(R) is also a sublattice. E(R) is BE-complete. Let{~} be a BE-Cauchy sequence in E(R). Clearly {"b} is also a BD-cauchy sequence in M(R) which is B.D-complete. Hence a BDlimit . u e M(R) >..(G) exists. Consider a measure on R defined by = JP for any Borel set G and the complete space L2 (R,A.) of all. ·G f'\iJ is square integrable functions with respect to>.. on R. Since E-Caucby, it means J(~+m - ~) 2P)= 0 lim E("b+m - '\i) = lim(D('\i+m - '\i) + In particular, for all m. for all m. Hence {"h \ is a Cauchy" sequence in lim "h = u, we have Final.l.y, 11.m J(u - '\i) 2 P = 0, E(u) !5 E("h) + E(u - "h) <co lim [("b+m L2 (R,>.. ) • i.e. '\i/p = 0 Since u = BE-lim '\i• by the triangle inequality. Thus u e E(R). Ve. Since E(R) is a subset of M(R) whose elements are continu.- ous on R*, the restrictions of functions in E(R) on 6.P on f:! . are continuous What we have more is THEX>REM. E(R) Itl = { Itl f f e E(R)} is dense in C0 ( 6,p ) with respect to t he unifonn norm. Proof. By Lemma IV j For any positive f, the set f K =0 on /.). "- d> for all = {p e ti : f( p ) ~ f} f € E (R). i s a compact 45 subset of 6. and also a compact subset of Hence E(R)l~Pc; C0 (f:!). which excludes q 1 f ltt' K < f. We are to show that E (R) separates points of tl. For any two distinct points in R* of. q 6.p • Obvious ly 2 /!, there is a neighborhood U q 1 ,q 2 e J P < oo. By the Urys ohn such that un R property (IIIk), there is a function f E M(R) f(q ) = 1 E(f) = D(f) + Jf2P ~ D(f) + 1 < oo. and Hence supp f C U. f E E( R) and We have z 0 JP Utl R 2 1 E(R) theorem assures the denseness of fixed 0 ~ f ~ l, f(q ) :-f. f(q ). As a by-product, we see that E(R) vanishes i d entically at no point of Vd. with i}'. The Ston e-Weiers tras s \tf in C /! ). 0 ( 'de consider the projection TT introduced in IVf. For a E R, we define a linear functional s on I E(R) tf by sf = TTf(z 0 ) for all fin E(R). By the decomposition theorem IVd and Theorem IVk, lsfl .::;; sup~P I fl. Hence s is a positive bounded linear funct ional on llsll = s up{jsrl : f E E(R), s up6p lfl= l}.::;; 1. Si n ce E(R)ltt with norm E( R) IK i s dense in C0 (t[) which is complete with r e spe ct to the s up norm, we can extend s to C0 (/! ) · by an obvious limit p roce s s . Thus s be comes a positive bounded l i n ear functional on C0 ( t! ) with II s 11 .::;; 1. By t he !ti.es z repres entation theorem, there i s a uniq u e bounded po s itive r et.ula r :3or el measurA m on tl such t hat sf = f tf We can regard m as a measure on r d m. r by definin g 46 m(G) = m(G f'\ ti' ) for all Ger. When P = O, we denote the corresponding measure by m0 • The measures m and m0 will be called the representing measure with center z 0 for solutions Ve. THEDREM. 2) v(z ) > 1) and harmonic functions respectively. m is a positive bounded regular Borel measure, 0 ffvdm for all supersolutions v e E(R), - t! (where support of m is considered relative to the topology of tl and by definition m( r "-.. .tl) = o ) , 3) supp m = 4) m, satisfying 1), 2) and 3), is unique, 5) m( /).p) :;;, 1; Cf. [14]~ Proof. m( tl) = 1 if and only if F =o. Theorem 2.1. l) is true by Riesz representation theorem. For any supersolution v in E(R), v 2:, rrv by IVd. Hence v(z0 ) : : rrv(z0 ) = sv = f vdm. Th:u,s 2) is proved. Now suppose clos ed in b.p, the set q e S =supp m is a proper subset of {).P . 3ince Sis t:! . . . . _ S is open in tl. Thus for an arbitrary fl '- S, ther e is an open neighborhood U in R* of q such that Un 6 p is disjoint from S and I P < oo • LJ('\ R a functi on fin M( R) such that By the Urysohn property, there is 0:;;, f :;;_ 1, supp f c U and f(q) = 1. 47 2 [£ 1-' < D(f) + J E(f) = D(f) + Note that f- < oo Uf"\R - we can apply n to f. Note also that rrf > 0 Hence • Thus f e E(R), so on R, in particular fpfdm = sf > o. pdm > Uf"\ /1 - f /1 Thus the contradiction establishes J). I f !!! is a measure with support in f1p satisfying 1) and 2), then for all f in E(R), f rc:4! Jfdm = 1Tf(z 0 ) = /).p tl by 2) and the decomposition theorem IVd. Since E(R)lf1P C0 ( /1P), such equalities are also true for functions in !!! is dense in C ( /).r). That 0 = m is then a consequence of the uniqueness part of the Riesz representation theorem. Hence 4) is true. If m( /).p) = Us II~ 1 then /).p = Ii P ;: 0, by Vd and the Riesz representation theorem. and m = m0 f m(/j') = m0 (!1) = • \·le have dm0 = n°l(z0 ) = 1 , ~ since in this case solutions are hannonic functions . m( flP) = 1, t hen llsll= 1. such that 11 - sfil < l/i. Since E(R) ld, is dense in C 0 { f~} C E(R) if For any integer i, however large, there is a nonnegative function ri in C0 ( f1P ) a sequence Conversely ( sup f 1 = 1 ~P), to each i of nonnegative functions such that I 1 im ( sup 'P r n - oo l:s. sup., f i /).• n 1 - =1 r1n I )= and 0. there is 48 ri ) otherwise) ri/(sup (replace by tl n n suptf Note that By sfi IIt, it and lri - r~I < l/i. = limn- co sf~. {lTfi} is a sequence of bounded solutions. contains a subsequence, again denoted by { lTfi} converging uniformly on compact subsets to a solution u. Observe that lim sfi i and < l/i +!Is II sup t! jr1 - ril < l/i + l/i == 2/i. Hence u(z0 ) = 1. p u z 0 e R. u is also bounded by 1. Thus u takes its maximum at (see IVd), Hence {nri\ is a sequence unifonnl.y bounded by 1 Since =1 by maximum principle IIn. This is possible only if =o. Vf. fixed point The r epresenting measure m is obtained with respect to a Zo 6 R. Similarly, we can construct a representin g measure w:i.. th respect to any other point P: O). and deno t e it by mz (m0 It is useful to know the following theorem. THIDREM. solution. z e R If f e C0 ( tl), then u(z) = frdmz. is a r z if 49 Proof. f(z) = nf(z) and hence = ffdm.z, by V'e,2). Now if such that f = nf If f is a PE-function, then f e C0 ( /:/), f = U-lim fn on then there is a sequence t:.P. Note that nfn frn} c E(R) are PE-functions and supR* lnrm - nfnl ~ sup plnfm - nfnl ts. which tends to 0 as m, n tend to ~ Thus {nfn { being a uniformly convergent sequence of continuous functions on R*, - limit u which is also continuous on R*. By IIr, has a .E is a solution and \ = lim = f Jfn~ t:P fdmz tl = u(z) , i.e. u is a solution. Vg. LEMMA. Proof. u(z) = f fdmz mz is absolutely continuous with respect to m. For all nonnegative function is a nonnegative solution by Vf. By Ile, 50 see that ~ is absolutely continuous with respect to m. Vh. LlliMA. For any z e R, there is a Radon-Nikodym derivative ,!(z,q) of ~ with respect to m with the following properties. tl. l) ]i(z0 ,q) = 1 2) K(z,q) = 0 on R x ( r ~ ~p), J) K(z' ,q)/k(z,z') S !(z,q) S k(z,z')!(z' ,q) for all except on a set E(z,z 1 ) C 6p Proof. q € on R ic tl of m-measure zero. 1) is satisfied because ~ 0 = m by definition. 2) tl . It remains to prove J). f e C0 ( tl) we have by Vf and Ile, is obtained since m is concentrated on For any nonnegative (l/k(z,z')) Jrdmz' -< or (l/k(z , z' )) }ramz -< k(z,z') Jrdm, z J !(z' ,q)f(q)dm(q) Sf!(z,q)f(q)dm(q) S k(z,z') f !(z',q)f(q)dm(q). Hence < K(z ,q) < k(z,z')K(z',q) -K(z',q)/k(z,z') -except on a set E(z,z' ) c fJ.P of m-measure zero. COROll.ARY. l/k(z, z0 ) S !(z,q) ~ k(z, z0 ) on H " tl except .5l for q in a set E(z,Zo) of m-measure O. Proof. Vi. z' = z0 Let THEOREM. in 3) and apply 1) • There is a real valued function (kernel) K(z,q) R"t such that it is continuous on R, and defini3d on tl. 1) K(z0 ,q) = 1 on 2) K(z,q) =0 on '.3) l/k(z, ~) ~ K(z , q) $ k(z, Zo) 4) for any fixed function on .5) for any supersolution K(z,q) When K0 (z,q) Proof. E = tl. is a nonnegative Borel v E E(R) , P.: 0, except for q in a set of m-measure o. the corresponding kernel will be and it was introduced in [14] , p. 196. Let T be a countable dense subset of R containing z • LJ { E(z, z') : z, z' e- T} where that m(E) = o. For any q e t:l '- E and Set R x ~K(z,q)v(q)dm(q), K(z,q) e PN (R) REMARK. denoted by z e R, on r . v(z) 2:: 6) R..c(r'-t:l) , 0 E(z, z') is as in Vh. z , z' e T, we have Note lS_(z' ,q)/k (z ,z') :S .!f(z,q) ~ k(z,z')_!(z',q), or (l/k(z,z') - l).!£(z' ,q) :S ,!(z,q) - ,!(z' ,q) < (k(z,z ' ) - l)K(z ' ,q): thus 52 j!(z,q) - !,(z' ,q)I S!(z',q) max(k(z,z') - 1, 1 - J/k(z,z')) $ k(z' ,z0 ) max(k(z,z') - 1, l - l/k(z,z')). q e ~P, E We obtain, for :• (7) lim. for all z 6 R by Ile. by l'K (z,q) - lf(z' ,q)j = 0 z,z' e T;z,z'-z" - Vh, 2). Note that (7') is also true :for all r' t::,.P We then define K(z,q) lim ·; T- i ; q e K(z ,q) = 1 q e For :fixed see this let r '-E if q E if q E E. r , K(z,q) is a continuous function on R. To q ¢ E and , z e R be any ~ints. Pick any sequence converging to z. For each n, let distance between ~· and ~ E T fzn} such that the Riema.nnian zn is less than l/n, and I!(~,q) - K(zn q) I < l/n. 1 By the triangle inequa1ity, \~\ is a sequence converging to z. Thus K(z,q) = lim lf(sn ,q) by definition. Then jK(z,q) - K(zn,q)\ $ IK(z,q) - K(~,q)I + IK(~,q) - K(zn,qA. where both tenns on the right tend to 0 . We have K(z,q) = lim K(zn,q). Thus K(z,q) is continuous on R when continuous when q ~ E. But obviously, K(z,q) i s q e E. 1) and 2) are satisfied since K(z,q) agrees with!(z,q) when z e T or when K(z,q) q ¢; t!. satisfies J) because of Corollary Vh and the continuity 53 of k(z,z0 ). 4) is true by the construction of K(z,q). ,._ 'l'o prove 5), let v be any supersolution in E(H), ~ E T such that z a lim zn. z e. R and Note that K(z,q) = lim K(z ,q) - m a. e. n and by Ve,2) By the continuity of v and Lebesgue'sdominated convergence theorem v(z) ?!:, ~K(z,q)v(q)drn(q). Thus 5) is proved. Now it remains to prove 6). w(z) = ff(q)drn (q) z K(z,q) f ~ C0 ( t}') is a solution by Vf, and hence is continuous. Thus w(z) = lim w(z) = limT J T Recall For any nonnegative f(q).!f(z, q) drn(q ) = limT !(z,q) and apply the Lebesgue dominated convergence theorem again, we have w(z) = Now cover R by a family any z e R, z € Ui [ Ui f r(q)K(z,q)drn(q). i=l, 2. , ••• J for some i. of pe.rametric balls. For Recall the hannonic measure with res pect to the harmonic class f( for Ui r( z ,Ui) at z as defined in IIh, and hence obtain Jr(q)K(z,q)dm(q) r =f f r bUi = f f(q)K(x,q)dm(q)dr(z, Ui )(x) f f(q) r K(x,q)dr(z, u )(x)dm(q) 1 bU1 by Fubini 1 s theorem. K(z,q) Thus for any z E Ui J = K(x,q)dr(z, Ui) (x) bUi except on a set with m(~(z)) Ai(z) C: IJ.p ~ = LJ {Ai (z) : z e T " u1 J A = EU ( = o. Let and Note m(A) = o. For all ,!(z' ,q) LJ {Ai : i = l, 2, ••• } ) • q e r '- A and z'e T f"\ U., 1 = K(z',q) = f K(x,q)dr(z',Ui)(x). bUi which is the evaluation at z' of the unique solution on continuous boundary value K(x,q) for an arbitrary z'' e ui, K(z" ,q) = we have J (see IIh and IIk). with the As z' tends to K(z" ,q) = lim , ., K(z' ,q) z-z K(x,q)dr(z", Ui) (x) bU1 u1 and z" 5.5 for all q E r' A. Thus K(z,q) e PN(Ui_) if q € r' A. Since U. is arbitrary, 1 K(z,q) E PN(R) m a. e. VI. THE CHARACTERIZATIONS OF PE- AND P&-FUNCTIONS A PE-function, as we defined before, is a solution of 6u = I'u with finite energy integral. A nonnegative solution is called ,.., a PE-function if it can be obtained as the infimum of the family of all ,,..., PE-functions above it. Both PE- and PE-functions can be represented by the representing measure m and the associated kernel K(z.q) as u(z) if u e 'PE(R). = ~u(q)K(z,q)dm(q) r and u(z) = if u e PE(R). continuous f on ~(liJo supR u(q))K(z,q)dm(q) Furthermore, for each u E PE(R). there is an upper semi- r which is the infimum of the family of all PE- functions above it on fJ. lim sup such that R u(q) = f(q) m a. e. Most of the results mentioned above are generalizations of the work of Nakai [14] for HD- and HD-functions where an H'D-runction is a nonnegative harmonic function obtained as the infimurn of the family of all HD-functions above it. Vla. For any functions ' \ and u 2 on R we define the following ·' 57 = the greatest solution minorant of ll:l. and Uz• and =the least solution majorant of u._i and THEOREM. PE(R) 1 v1 fl. v 2 I~ = v1 n v2 ~ Uz· is a v ector lattice under A , v • v1 v v 2 1Ll = v1 u vdil for any and In fact, v ,v E PE(R). 1 2 Cf. [13] , Proof. ,., Obviously, PE(R) is a vector space. Note that I I which is a lattice under PE(R) C E(R) ~ n u in PE(R), Theorem 3.11. e PE(R), <tt:i. n u2 )1~ and u , 2 and For any tt:I. and ·Uz then for all n(~" Uz) :;; "11. n Uz and rr(~ n u2 )1~ = 0 dominated by both ~ q E ~ lim inf by By the Hoyden-Nakai decomposition theorem IVd If v is any solution which is > u • 2 is a supersolution by Lemma IIl and is in E(R) the previous observation. n(~ n u 2 ) n R (n(u n u ) - v) (q) 1 2 - by the continuities of fu..'l'lctions in E(R) on R• . Theorem IVh implies 58 that on R. Hence '1:J.. /\ '1z = TT( 11_ () '1z) and is in PE(R). It can be proved similarly that is in PE(R) -('-1.. V Uz) by observing that COROLLARY. '-1.. v Uz = rr('1.. v Uz) is a supersolution. Every PE-function u is a difference of two non- negative ones. In fact, u = (u V 0) - ( (-u) v 0). VIb. THEDREl{. If u(z) = u E FE(R), then ~u(q)K(z,q)dm(q). r Note that both u and -u are supersolutions in E(R), Proof. and apply Vi,5). Vlc. THEOREM. Proof. u is nonnegative. If u Ir e L1 ( r 'm). By Corollary VIa, it suffices to prove the case where Indeed u(z) = Thus u E FE( R.)' then f ~u(q)K(z,q)d:m(q). r u(q)dm(q) = u(z)<oo 0 59 VId. THEDREM. f € L1 ( If r ,m), then u(z) = ~f(q)K(z,q)dm(q) r is a solution. Cf. [14] , Theorem 2.2. Proof. E(R) Since which is in turn dense in 1 L (f ,m) in L 1 -norm. ~ = nfn Ll( 0 r .m) in ( Ll-norm, B(R) Ib.p is dense in Let us pick a sequence ( fnl C E(R) lim Let Itl is dense in C /:!) in sup nonn such that Jr jrn - rjmn = o. which is in PBE(R) and ~IA.P = rnll~.P by IVd, thus ~(z) = ~fn(q)K(z,q)dm(q) r by VIc. Now iu(z) - u,,(zlJ :S ~lf(q) - fn( q),K(z,q)dm(q) ~ k(z,z0 ) ~ !r(q) - fn(q)j dm(q) r by Vi, 3). Hence [ unJ tends to u unifonnly on each compact subset of R by the continuity of k(z.z0 ). VIe. Thus u is a solution by IIr. By Axiom III1 , the lower envelope of a downward directed family of nonnegative PE-functions is again a nonn egative s olution. We 60 denote by the collection of all nonnegative solutions obtained PE(R) in such fashion. Solutions in PE(R) are called PE-functions. HD-functions are defined similarly. VIf. THOOREM. A nonnegative solution is in the family PE(R) if and only if it is the C:..limit of a decreasing sequence in PE(R). Proof. - The sufficiency is clear. To prove the necessity, let u e PE(R) such that u(z) = inf { v(z) : v e <iJ }, where q. is some downward directed family in PE(R). For any fixed point x E R, there is a sequence fun } c CJ such that u(x) = lim '1n (x). Let and vn = '1n A vn - 1 By VIa, fvn} is a decreasing sequence in PE(R). Axiom III 2 insures = lim vn is a solution and Dini s theorem provides that actually v = C-lim vn• Observe that v - u ~ 0 and v(x) = u(x), hence that v 5 u v 1 by Axiom III. Thus u is the C-limit of a decreasing sequence in PE(R). VIg. only if THEOREM. ,...,,, A nonnegative function u is in PE(R) if and u(z) = inf { v(z) : v E CJ.Cu)} where CJ Cu) = fv E r E(R) v ~ u } • Proof. The sufficiency is obvious because fJ<u) is a 61 downward directed family of nonnegative solutions in PE(R) by Theorem VIa. On the other hand, being in PE(R), is a decreasing sequence in PE(R). u ( z) u = lim vn { v0 } c Clearly where { vn } q(u), hence = inf { vn ( z) : n = l , 2 , • • • } ~ inf' { v(z) : v E CJ.Cu) } However, we always have u(z) :S inf { v(z) v 6 CJ (u) } • Thus u(z) = inf { v(z) : v e (}Cu) } I REMARK. VIh. fl, CJCu)jA is closed under n. For any nonnegative function f on the harmonic boundary we define 'J (f) = { v e PE(R) v ~ f on fJ. } • and U(f:i) = { f : f is a nonnegative function on 6 f(q) = inf' { v(q) : v E t (f) } } VIi. LEl1MA. Proof. And v 1 /\ v (v /\ v 2 1 2 E )1 fl r(f). 1 (f) For any is a downward directed family. v1 ,v2 e r(f), = (v (\ v 2 )lfl 1 and - ~ f v1 A v 2 :S vi' by Theorem Vla. i = 1,2. Thus 62 REMARK. Certainly u(z) = inf { v(z) : v E 't(f)} is a ,,._ ~ PE-function. However, there is no guarantee that PE-functions are continuous on R* or have finite energy. to ..prove that any function in on U(.6) In particular, we can not hope is .ite restriction of a function fJ. which is the limit of a decreasing sequence of PE-functions. VIj. subset of tl LEMMA. is in The characteristic function of a compact U(/J.). Let K c t! Proof. be any compact subset and characteristic function. For any number of open subsets ~ •••• ,Un q f 0 cK its IJ.' K, we cover K by finite ~ . f"\ of R* such that ·rl < ~: . ·. and J. ~ ~ u1 = 1,2, •• ,n.The Urysohn peoperty guarantees the for any i f e M(R) existence of a nonnegative function r\K= l and supp f < f and E(f) = D(f) + J r2P < c f K- e E(R). cK(q 0 ) Thus f ~ 1, c LJ{u1 : i = 1,2, ••• ,n}. Observe that D(f) + tJ . - u = Tif E PE(R) and = 0 = f(q = u(q 0 such that ) 0 ). Uf"\R i I u b. = f P < .co • Hence I!J. ~ cK. Now Also for any point q 1 e K, cK(q ) = 1 = f(q ) c u(~). Since q and~ are arbitrary and 1 0 1 cK(q) ~inf v(q) : v e t(cK)} ~ u(q) for all q e fJ. , we have f cK(q) = inf { v(q) : v Vlk. e j:(cK) } • In order to prove the next theorem, we have to prove a lemma for technical purposes. 63 LEMMA. X is a locally compact Hausdorff space with a bounded positive regular Borel measure k. If er. is a family of nonnegative extended real-valued continuous functions on X which is closed under J x f inf { f e t }dk = inf { J fdk ; f x Let Proof. .f(q) = inf [ f(q) : f t } e for all q e x. F'irst we shall prove that 1 is upper semi-continuous. For a fixed continuous. f :S. Clearly neighborhood U of q e U. for all 0 E' X, there is a. decreasing sequence = lim f n (qo ). f(q ) 0 - such that q q 0 Thus Let { fn} C t which is upper semi- r: For any positive f , there is an open E = -f ( q 0 ) + E f' ( q) < f ' ( q ) + such that 0 f(q) < .£(q 0 ) + E q e U. for all Hence ! is upper semi-continuous. In particular f is integrable. Now denote t There is a decreasing sequence { ~} C f e t}. r = lim I gndk. If theorem. Clearly g = lim ~, f :S. g, hence then I gdk = r r = inf ( jfdk such that by monotone convergence f fdk < J gdk = r. We claim that actually the equality holds. Suppose on the contrary such that the set A(o) f fdk < r, then there is a positive = { q E X : .f(q) < g (q) - o } has positive measure. By the regularity of the measure k there is a compact subset A C A(6) with positive measure k(A) = a. The i goroff theorem s tates that there is a compact set K such that {~} converges uniformly to g on K and k(X " 1\) < a/2. ~ k(A) - k(X '.K) Hence > a - a/2 = a/2: k(K I\ A) = k(A) - k(A '- K) in particular K f'I A is not empty. 64 :fq(q) < g(q) - 0/2. Since g is continuous on continuous on X, there is h = fq f\ • · • • f\ fqn, 1 Note that ~ n h e If we set t 1 < - x '(Kn A) e j: K f"'\ A K f"'\ A is compact, so V(q ), ••• ,V(~) 1 which cover g - h > 6/ 4 h and g n h = lim ~ f\ h. . However, l X'-(KflA) and Kf"'\ A. then 1 (g n h)dk = r = lim J (~ n h)dk = K f"'\ A and :f is q q' E V(q). :for all there are finite number of such K f"'\ A. with an open neighborhood V(q) of q in 6/4 s g(q') - fq (q ') with fq e ~ q e K f"'\ A, there is a :function For any (g n h)dk + on 1 (g n h)dk Kf\A J hdk < 1 gdk + 1 <g - 6 I 4) dk Kn A - x ' (Kf\ A) K() A gdk + = Jgdk - (6/4) J dk :S r - (6/4)(6/2). X Kn A Hence the contradiction r proves the lemma. vn. THIDREM. u rv e PE(R) f E U( 6) u(z) = inf { v(z) : v e "'f (f) } such that if and only if there is a :function u(z) = fK(z,q)f(q)dm(q). In this case, C:f. [14] , Lennna J.2. Proof. by VIg. for all v E - I:f u is any .PE-:function, then u(z) = inf ( v(z) : v E CJ. (u) i f(q) = inf { v(q) : v e CJ.(u)} Let q E 6 j.(f) } • Cle arly 9Cu) C t(f), But we always have that hence f(q) ~ inf f(q) ~ inf { v(q ) f v(q) : v E 'J (f)}. 65 Thus v = inf f v(q) : v e 1 (f) } u(z) = inf and U( ti ) • f € By VIg £v(z) : v E <;}. (u) } = inf { Jv(q)K(z,q)dm(q) v E CJ.Cu) } • Now observe that, by Theorem VIa, the family of all functions in Cj.(u) restricted on /J. is closed under 11 ('1 11 • u(z) = = Thus the lemma VIk implies that J J inf { v(q) : v e <;J.<u)} K(z,q)dm(q) f(q)K(z,q)drn(q). , To prove the sufficiency, let f be any function in U< 6. ) , i.e. f (q) = inf { v (q) : v e 1 (f) } • Note that by Theorem VIa r e stricted to the family of all functions in /1 is closed under 11 f\ 11 • u(z) = f = J '}(f) Again by Lemma Vlk, r(q)K(z,q)drn(q) inf { v(q) : v e j=(f)} K(z,q)drn(q) = inf { Jv(q)K(z,q)drn(q) : v e j:(f)} =inf { v(z) : v e "J(f)} which is in PE(R) since 'J(f) n egative solutions in PE(R). is a downward directed family of non- 66 ,-..J VIm. TH:OOREM. u(z) = ff(q)K(z,q)dm(q) lim supR u(q) S f(q) lim sup R u(q) = f(q) Proof. u e PE(R) If for some f for all q € !:. m a. e. on For any such that e U<l:l), , then and moreover /1 • v e ~ (f), we have that v € PE( R) and = Jf(q)K(z,q)dm(q) u(z) f ~ v(q)K(z,q)dm(q) = v(z). Thus for all q € /1 , lim sup R u(q) < lim sup v(q) = v(q) R by the continuity of PE-functions on R*. Hence lim supR u(q) S inf { v(q) : v e 1 (f) } = f(q) for all e !:. • q Observe that lim sup R positive z u(q) = f(q) = 0 f(q) = 0 for all q e l:l "- t:.P , hence for any q €' !:. "'- tl c and some compact subset for all q € K. K of' Now Consider the characteristic function suppose for some lim supR u(q) < f(q) - fj,P , We are going to show that w(z) = . m(K) = O. cK of .K and EfcK(q) K(z,q)dm(q). 67 w(z0 ) = E m(K). Note that D:t, ••• ,Un ·in R* can cover K by open sets for every i g e M(R) i = 1,2, ••• ,n. such that We claim that w is bounded. glK = E J such that P< oo U1 ("\ R By the Urysohn property O~ g ~ E, In fact, we there is a function supp gc U{u : 1 and = 1,2, ••• ,n} • Observe that E(g) = D(g) + I gzF s n(g) + s2I: J · < oo • g E E(R). Hence rm e PBE(R) Thus As a consequence ngl~ and P u.nR 1 = gl~ ~ f cK. J g (q)K(z,q)dm(q) ~ eJ cK(q)K(z,q)dm(q) = w(z). rrg (z) = rrg S su% g S E , hence w is bounded by f • By Lemma VIj U( ~) cK is in ;v and therefore w is a .P~ . function. By the part of the theorem just proved, f'or all q e lim supRw(q) S f cK(q) fl , in particular =0 for all q E l1 '- K w(q) < E for all q € K. lim supR w(q) and lim sup R Thus no matter lvhether - q E K or q E fl '- K, lim supR (u(q) + w(q)) S lim sup[{ u(q) + lim supR w(q) < f(q). Consequently for any v E t(f) lim infR(v(q) - u(q) - w(q)) ~ lim infR v(q) - lim supR(u (q ) + w(q)) ~ f(q) - f(q) = o. Since v - u - w is bounded from below, we have v > u + w by the 68 minimum principle IVh. Thus v(z ) > u(z ) + w(z ) = u(z ) + Em(K). 0 0 - 0 0 Now together with the previous theorem, u(z0 ) = inf { v(z0 ) ~ u(z0 ) Hence 0> m(K). Thus m( K) = 0. : + fm(K). v e t(f)} 69 VII. r.,; CtlAR4CTER!ZATlON OF PE-MJ:NJ.MAL .FUNCTIONS This chapter consists of two parts. The first is a generalization of Nakai's work [14] on HD.minimal functions to P'E-minimal. - functions. The second is a new result which shows that HD-minimality is ,,..., closely related to PE-minimality. VIIa. DEFINITION. - A nonzero PE-function u is called a ,..., ,....., PE-minimal function if for any PE-function v such that u ::=, v we have cu = v for some constant c. ,....., H;~minima.1 functions ar,e defined similarly. VIIb. THEDREM. ' .,...., I There exists a PE-minimal function u on R if and only if there exists a point q 0 e f.! with positive mo.measure. More precisely, if u is PE-minimal, then there is a point q 0 ~ t! that m(q0 ) > 0 arid u(z) Conversely, if m(q0 ) > 0 = aK(z,q0 ) for some positive constant a. for some q 0 E /:!, then K(z,q .P'E-minimal function. Cf. [14], Theorem J.J. Proof. Let u be a P"E-mini.mal function and K1 = f q e /! : lim supRu(q) ;::: l/i } • S= : lim supRu(q) > 0 Since = and such i=l,2, ••• } } 0 ) is a 70 m(S) > 0, there is some n such that m(K ) > o. n K. Note that K is compact. By Vlj in U<b.) For simplicity, we denote K by n the characteristic function cK is and by Vil w(z) = (l/n) ~K(z,q)cK(q)dm(q) ,...... is a PE-function. By VIm lim supRw(q) = (l/n)cK(q) a. e. on 6. • Now we are going to show that the measure on K is concentrated at an atom in K. To see this, let A be any compact set in K such that Again we conclude from VIj, Vll and VIm that m(K '-A) > O. v(z) = (l/n) ~K(z,q)cA(q)dm(q) rv is a PE-function and lim supRv(q) = (l/n)cA(q) a.e. - on /).. Clearly u ::=, v. If m(A) > 0, then v is a nontrivial P E-function dominated by u. Thus there is a nonzero positive constant c such that cu =v by the minimality of u. However, note that c/n :S c lim supRu(q) = lim supRv(q) = 0 for almost all q e K'- A. This is absurd. Hence no atomic point q0 such that m(K) = m(q0 ), B and C of K such that m(C) > o. K = B v C with m(A) = O. If K has then there exist subsets Bn C=¢ and m(B) > 0 By the regularity of m, there are compact subsets L and M of 71 B and C respectively both with positive measure. However, the foregoing argument implies that both m(M) and m(L) must be zero. such that m(K) = m(q ). contradiction establishes the existence of q0 Since Ki c Ki+ j indeed m(S) = m(q ). 0 f(q) = lim supRu(q) By Vll and VIm a.e. 0 lim m(K. ) = m(S), and for all j This 1 f E U<~) for some we have with we conclude that u(z) = JK(z,q)f(q)dm(q) = f(q )m(q )K(z,q ). 0 0 0 To prove the sufficiency, let m(q ) 0 = l/k > o. q0 e t:l such that By VIj and VIl K(z,q0 ) = k J K(z,q)c{q }(q)dm(q) 0 is a P'E-function. We are going to prove that ,,...., If v is a PE-function such that K(z,q0 ) ,....; is PE-minimal. v(z) < K(z,q ) , - 0 then lim supRv(q) < lim sup K(z,q ) z e R,z-q o a. e. i.e. lim sup v(q) = 0 We set lim sup v(q0 ) = a. v(z) = f Then K( z,q)(lim supRv(q))dm(q) on ~ 72 = (a/k)K(z,q ). 0 - is PE.minimal. VIIc. THIDREM. Let q e /! . 0 Then m(q ) > 0 0 if and m0 (q 0 ) >. 0. only if Indeed, if neighborhoods of q 0 m(q ) > o. o f Un } Let be a sequence of open in R* such that - Un:::> Un+ l lim m(Un) = m(q0 ) lim m0 (Un ) , = m (qo ) 0 and • By the Urysohn property, there is an f E E(R) n such that 0 < f < 1, - n rnlun+ 1 = 1 and supp f fn ~ fn+l Clearly f'n E E(R) we have lim fn 0 and m • Note hhat 11°fn E HBD(R), 0 rr rn\ /:::. and I 6 = c{ qo} = 11fn \ Ii = fn I b. • n c U n for all n. By the choice of (Un}' almost everywhere with respect to both m By IIp Tif E PBE(R) n rr0 f n and is a supersolution, hence so 73 is rr0 f n - rrfn • q e fl Observe that for all = o. We have rr0 f > TTf n - n by the minimum principle IVh. Note that {rr0 fn} and frrfn} are both decreasing sequences. Since ) (K°(z,q)f (q)dm0 (q) = TT0 f "(z) 1 n n > TTf (z) - n = ~K(z,q)fn(q)dm(q), , as n tends to Setting oo , z = z0 we have by the monotone convergence theorem gives m0 (q ) > m(q ) > o. 0 - 0 Thus the necessity is proved. To prove the sufficiency, we need the lemmas of VIId - VIIg. VIId. The following lemma is a modification of a result on Riemann surf'ace due to Nakai ([16], :Proposition 9). generalize completely to Riemannian manifolds . sufficient for our purpos es. His result does not But the followin g i s 74 be a point such that Let LEl~!A. and U be any neighborhood of q 0 in R*. neighborhood V of q 0 in R* such that m0 (q ) > 0 0 Then there exists a vcu and is a v " l{ region in R with piecewise smooth boundary. To prove this lemma, we simply replace the triangulations by parametric balls in Nakai's proof; then everything follows. VIIe. Consider a region G in R, its closure G in R* and its Royden compactification G*. We recall (see IIIo - IIIq) that B(G) = (G '- G) () r and j is the unique continuous mapping from G* onto G fixing G elanentwise. We have Moreover, by theorem IIIq, j is a homeomorphism of GU j-1 (a(G)) to G U B(G). Let z 0 ~ G which is the center of the representing measure m on the Royden boundary r. If we denote the Hoyden boundary rG' harmonic boundary by ~G and the representing measure for solutions on rG by then following is true. of G by ~. ·J LEMMA. Let G be a region in R with piecewise smooth boundary JP <oo. such that Let Ebe any Borel subset of E(G) , then G m(E) > 0 Dt(j-1 (E)) > o. if and only if Nakai ( [16 J Proposition 8) has proved the case where Proof. compact and E c ~·Jithout 1:::,.P. loss of generality Note that p = o. we may assume that i; is since EC B( G) . 75 There exists a sequence { Vn} E C Vn + l C Vn C B(G) v G • m(E) = lim m(Vn () r )• such that Thus for all n. - n- Take f fn} is a decreasi~g sequence and as functions on G, then E(R) C E(G). fn e fn E E(R) E(G) i.J"e denote the extension f* n IG*'- then n 0 < f* < 1. = 0 ~ fn ~ l, because we always have that and F < oo for all n and / Vn()R mG(j-l(E)) If we view fn to G* by :f*, of open sets in R* such that r 1 CVn ) = 0 Now we set ~(z) = JK(z,q)fn(q)dm(q) r and ~(z) = JKG(z,q)fri(q)dmG(q), rG where KG(z,q) is the associated kernel of mG. Let us consider the exhaustion f l\i} of R constructed in Theorem Ilg. where TT is the projection defined in IVf. sequence bounded by sup f Rn and the Dirichlet principle of Illf, Note that Then 1lfo set {vn,m 11 m is a E(vn,m + 1 ) < - E(vn,m ) S E(fn) and thus by 76 by Green's formula of Illf. Consequently, Note that vn. ml R'- G = O. Also, being a bounded sequence of solutions on has a convergent subsequence, again denoted by {vn mlm• ' which is BE-Cauchy on G. Let vn = limm-oovn ,m • Note that vnlG € E(G),which is BE-complete, and vnlH'\.. G = O. going to show that vn is continuous on Rand hence 1,-/e are v0 E ~ (R). In' fact, if x is any point on bG and N is a parametric ball about x, we let w be the strong barrier function for x on respect to the differential operator Note that L(w) ~ -1 w(x) = 0 (see p. wl NnG'-fx} and L =!::. - P (see p. 341, [ 4]). is strictly positive while 341. [4]). Let c be a positive constant such that Then L(cw - vn , m) = L(cw) - L(vn,m ) < -c ~ Since Nr'I G with cw - vn,m >- 0 o. b(Nn G), we have on n,m >0 - CW-V 77 on N() G by the maximum principle (p. 326, [ 4]). Let tine; m- oo gives cw-vn>0 on Thus N" G. vn is continuous at x. Since x is arbitrary, we see that vn is i.e. continuous at every point of bG and is continuous on R. Now we let (1) Similarly, {'1n. mlm h.a.s a BE-Cauchy subsequence, again denoted by ' on R. We set u } { n,m m' Bl!:-limm-+OO '\i m• ' ~ E PBE(R). Note that ~16 = vnj6 = Then fnl /::::. Thus ~ = 1-\i by maximum principle IVc. Note also that f'b} and fvn} are decreasing sequences since {fn} is decreasing on on R. Thus un,m- > - vn on f\n for all m. 6. Clearly vn i s a subsolution Hence U_ on R as well > V n - n as on R* . We denote the extension of vn,m to ~· by Vri m• Since fvn,mlm is a BE-Cauchy sequence on G and vn,m we have Vri m -~= 0 ' I ' v* v* n = -n· Also = fri 6G , Hence where vri on f n == 0 on bG U B(G), rG, vn E PBE(G) and vri_ \ 6 c = Yri I~G is the ex.tension of vn to G*. Therefo re we have v*>v* n - n+ 1 Note that 78 (2) and Now suppose is a subsolution on 1\. Also note that u.. Hence u ~.m - u and n.m E PN(Rm) u 1 ,m - un,:n >v 1 - vn - on R. l,m - un , m >v: - 1 - vn on bRm ( 3) In particular. implies that l\,m+l - "1n,m+l ~ ~.m - u n,m \... ' ) on ;...i'"m· By the minimum principle IIo, (3) is true on i\n· Therefore (3) holds on It. a s well as on rt*. Conseriuently , (4) lim m-oo (u.....L,m - u n , m) = u....J.. u n > l\,m - u n,m for all m. Since Axiom III 2 • (1), we have Hence lim v n = 0 limn-oo u·n,m = 0 on b Rm. As on Hm by the maY.imum principle. ( 4) implies that limn-oo ( u..J.. - u. ·n ) > - limn-oo ( u l,m - u n,m ) = '1:1., m on f\n for all m. Since on R. As a re s ult lim '1n ~ 0, and therefore u.. - lim u.. J. m-oo J.., m' we h ave lim ~ = O. Conseci u ently 79 m(E) = lim un(z 0 ) = O. Together with (2), we see th.at m(~) > 0 if and only if m ~ (j-1cl!.:)) > O. J [)5]. V:Lif. The following lemma is due to Nakai LEZ'iN.A. If P is any c1 n-form on lt such that Jl ' < =, then K there exists an isomorphism T of HB(R) onto FB(R) with the following properties: 1) u € HB(R) {un} C HB(R) if and only if is a decreasing sequence with limit f Tun } is a decreasing sequence in l-B(R) with limit Tu 'E ?B(R). 2) Let {.i:\i} be a regular exhaustion of i{• .for any let Tiu be a continuous function on R such that Tiu\K'-~\ = u!R'-Hi. Then for all COHDLLARY. h E HB(n). Proof. T.u!K € PB (H.) ). l l and Tu= B-lim Tiu. for all VIIg. u E HB(H.) h E HBO( R) u e nB(Il). if a nd only if Th € i .6ii:( R) In this case Note that the sequence {Ti h} C h.J(Ri) converges to Th unifonnly on compact subsets. If h e HBD(R), then by hypothesis and the Dirichlet principle (IIIf) E(Ti+ j h) :S ;_.; (Tih) :S l:<:\ h) <: "" • i'ior0ove r Gr oen':> formula (Illf) impl;l..e~; that 80 0 = i!:H. (T . ju, T. .u - 'l'. u) •j_ + j J. + J. + J 1 = f£ (T l. +. J. u) - C:(Tl. .+J. u, TJ..u). Thus = !i:(T.]. + J.u) "C,,:!,... ('I' • + . U , 1'. u) + !!:(T.u) J. J. ]. J = i (T.+ .u) - E(T.u). 1 J ]. i s a BE-Cauchy sequ ence. Since ~ (R) i s 8~complete, we i:ience [Tiu} have Tu E FBE(R). Furthennore , i t is clear that T.u - u 3ince l'Ill(R) is the i:D-closure of 1': (R), we have 0 ( Tu - u) It::. = 0 , i . e• u J. E Vi (H) 0 by IIIm. Tu - '.: E' Lf::.( i.(). Hence Iti = Tu j fi . Th e proof for the sufficiency is exactly the sa:ne . VIIh. Let q 0 € ~p Now we are going to c ompl e t e the proo f of The orem VIIc. such t hat neighborhood Q of o-'O in R* m0 (q 0 ) > O. l3y VIld, the r e exists a f . }' < oo G("I R piecewise smooth boundary. and Q ~ R is a r e5ion i n lt with Cons i der the r e e;ion G in H* and i ts Royden compactification G*. :.J = 2 f'"\ l~.. its closure :.-;e al s o consider the continuous mapping j from G* onto G g iven in 1111, which fixe s 3 vr1 CBC; )) to elementwise. Since j i s a homeomorphism from 0 .:J v B(G) i s a po int in by Iliq, we have i s the Royden boundary of G. ry Vlle , m~ (q ) 1 > O. I,.,l1 which 81 By VIIf, HB(G-) is isomorphic to i·B(;J) under the isomorphism T since JP < cc Note that (i h(z) = K~(z,q1 ) = J(K~(z ,o)cf }(q)dm~(q) u • ql 'J rG is a nontrivial function in l J rv HD(G) ("\ Hi3( G) 'J h(z ) = 1. since f Un} be a sequence of open sets in G* such that 0 Let Dn ::> :Jn+ 1 7 q 1 , and f' E "'(;,.: ) n that by the i.Jrysohn property: of H(G). to 0 mG a.e. ·;1e see that as well as un where such ffn} is a decreasing sequence converging mG a. e. Let on = n~fn 'J n~ is the harmonic projection on G (see IVf). Thus 11n I6G = fn j 6 G a nd h enc e p rincipl e IVb. ~ E HBD ( G), { ,-\i} is a decreasin!S sequence by maximum de have = lim J~ u~(z, q)un (q)dm~(q) u rG = JrGKG (z,q) c{ ql} (q)dm~.u (q) = h(z) by the monotone ~onve rg ence the orem. and 'i'un I{),,, = n b. r· by Vllg . Also u U J •J 82 Th = lim T~ by VIIf. Th ( z) Thus = lim '1'1\i (z) = lim fr:, K,..(z,q)'\i(q)dinr (q) u v: Lt = J'rG r; (z,q )°{ q l} (q )ilinu ( q ) by the monotone convergence t."1-ieorem a gain. Sinc e supG Th we have = sup~ h > 0, \.1 mG(ql) > o. Ive apply VI Ie once more and see that .-1 ( \ J q J 0 = q 1° m(q 0 ) > 0 since SJ BIBLIOGRAPHY [l]. M. Brelot, Lectures on Potential Theory, Tata Institute of Fundamental Research, Bombay, [2]. C. Constantinescu and A. Cornea. On the axiomatic of harmonic functions [3]. 1960. I, Ann. Inst. Fou rier, Grenoble, 13 (1963), 373-388. c. Constantines cu and A. Cornea , Ideale Rander Riemannscher FUichen, Springer- Verlag, Berlin, 1963. r,l q. - • ! L. R. Courant and D. Hilbert, Methods of ifathematical Physics, Vol. II , Interscience, [5]. New York, 1962. W. Feller, Uber die LOsungen der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptis chen Typus, Math. Ann ., 102 (1930) , 63)-649. [61. M. Glasner and R. 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