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A NNALES DE L ’ INSTITUT F OURIER

P ETER L OEB

B ERTRAM W ALSH

The equivalence of Harnack’s principle and Harnack’s inequality in the axiomatic system of Brelot

Annales de l’institut Fourier, tome 15, no2 (1965), p. 597-600

<http://www.numdam.org/item?id=AIF_1965__15_2_597_0>

© Annales de l’institut Fourier, 1965, tous droits réservés.

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Ann. Inst. Fourier, Grenoble 15, 2 (1965), 597-600.

THE EQUIVALENCE OF HARNACK'S PRINCIPLE AND HARNACK'S INEQUALITY

IN THE AXIOMATIC SYSTEM OF BRELOT

by PETER A. LOEB Q AND BERTRAM WALSH (2)

During the last ten years, Marcel Brelot [2] and others have investigated elliptic differential equations in an abstract setting, a setting in which the Harnack principle is assumed to be valid. When necessary, the Harnack principle has been replaced by another axiom which establishes a form of the Harnack inequality. In 1964, Gabriel Mokobodzki showed that the two axioms are equivalent when the underlying space has a countable base for its topology (see [I], pp. 16-18) We shall show that this restriction is unnecessary. First we recall some basic definitions.

Let W be a locally compact Hausdorff space which is connected and locally connected but not compact. Let & be a class of real-valued continuous functions with open domains in W such that for each open set Q s W the set ^Q, (consis- ting of all functions in ^») with domains equal to D, is a real vector space. An open subset Q of W is said to be regular for ^ or regular iff its closure in W is compact and for every continuous real-valued function f defined on OQ there is a unique continuous function h defined on 0 such that

AJOQ = f, h\Qe^, and A > 0 if /•>0.

Moreover, the class ^ is called a harmonic class on W if it satisfies the following three axioms which are due to Brelot [2] : AXIOM I. — A function g with an open domain 0, c W is an element of ^ if for every point x e Q there is a function h e ^ and an open set w with xe w cf) such that g\w = h\w.

(^G^-Sies^ectS^81 science Foundati011 research ^ts (i)GP-1988 and

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598 PETER A. LOEB AND B E R T R A M WALSH

AXIOM II. — There is a base for the topology of W such that each set in the base is a regular region (non empty connected open set).

AXIOM III. — If % is a subset of ^)Q, where Q. is a region in W, and ^ is directed by increasing order on 0., then the upper envelope of ^ is either identically -}- oo or is a function

in ^Q*

It follows immediately from Axiom I that if h is in JpQ, then the restriction of h to any nonempty open subset of its domain is again in ^). Given Axioms I and II, Constanti- nescu and Cornea ([3], p. 344 and p. 378) have shown that the following axioms are equivalent to Axiom I I I :

AXIOM IIIi. — If 0. is a region in W and \h^ is an increa- sing sequence of functions in ^)Q, then either lim h^ is identi- cally + oo or lim h^ is in ^RQ.

n

AXIOM IIIg. — If 0. is a region in W, K a compact subset of Q, and XQ a point in K, then there is a constant M ^1 such that every nonnegative function h e ^)Q satisfies the inequality

A(^)<M./i(0 at every point x e K.

Given Axioms I and II, we shall show that the following axiom is equivalent to Axiom III.

AXIOM Ills. — If 0. is a region in W then every nonnegative function in <§)Q is either identically 0 or has no zeros in tl. Fur- thermore, for any point XQ <= 0. the set

^ = [h e ^)Q : h > 0 and h[x^ = 1}

is equicontinuous at XQ.

Axiom IIIi is, of course, just the Harnack principle, and Axiom IIIg gives a « weak » Harnack inequality for ^)Q. On the other hand, a consequence of Axiom 1113 is the fact that for any region Q, and any compact subset K c Q there is a constant M ^ 1 such that for every nonnegative h e ^?Q and every pair of points x^ and x^ in K the relation

(1) — • A(^i) < A(^) < M.A(a;i)

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holds. Moreover, for any point x in Q. and any constant M > 1 there is a compact neighborhood K of x in which (1) holds fo^ft M01? I \3 establishes a ^"g Harnack inequality for ^Q Mokobodzki has established the equivalence of IIL and III for the case in which the topology of W has a coun- table base; it is this restriction which we shall now remove

That Axioms III and III, are equivalent follows from the

THEOREM- — Let ^ be a harmonic class and Sl be a resion

ln^ ? \, a point in D? and set q) = ^^a: h>0 and h{x,} = lj. Then $ is equicontinuous at x,.

k pkroo/,' ~t Let <0 be a regular ^e§ion and K a compact neigh- borhood of ^ such that ^ K c <o c d c 0. Each continuous function f on ^ has a unique extension H(/-) e ^ , and for each x^ co the mapping f-^ H(f)^) from C(^) into the reals is a nonnegative Radon measure on ^, which we denote by p,. Axiom III, (which follows from Axiom III) gives for each pair of points x, and x, in <o a constant M (depending on those points) for which H(/-)(^) < M.H(/-)(^), i.e.

p^<M.p^

in the usual ordering of measures on Oco. Hence all the measures iPxjx^ are absolutely continuous with respect to one another and the Radon-Nikodym density of any one with respect to any other is essentially bounded (((essentially)) beinff unambiguous because all the measures have the same null sets). Following an idea of Mokobodzki's, we now consider for each x.-co the Radon-Nikodym density of p, with respect to p which we denote by ^; each g, is essentially bounded and d^ = g,-d^.

Let A = ^: A^|. Axiom III, states that the func- tions in A are uniformly bounded on Oco, and of course they are continuous there. Thus, if S is any countably infinite subset of A, there is a function /•<. L^pJ which is an accumu- lation point of S with respect to the weak* topology of L-fp ) (i.e. the topology determined by L^pJ; see [4], p. 424';.

since L (pj c L (pj, f is also an accumulation point of S with respect to the weak topology of L^pJ (i.e. the topology determined by L°°(pJ.) Thus by the Eberlein-Smulian theorem

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600 PETER A. LOEB AND B E R T R A M WALSH

([4], p. 430), any sequence in A has a subsequence which converges weakly to an element of L^paJ. Since each

g.eL^pJ^L^pJ*,

it follows that any sequence \h^\ in 0 has a subsequence (which we may also denote by {h^) for which

W =f^

h

n{y)8^y)

d

^{y)

converges for each xe co; the pointwise limit function h on co belongs to ^ since it is the extension in ^ of the weak limit (in L^paJ) of the sequence ^|0coj. By a result of R.-M. Herve ([5], p. 432) __

h = sup (inf hn)

n k>n

where f(x) = sup (inf f(y)) as S ranges over the neighborhood

8 ye^

system of x. Thus h is the limit of the increasing sequence of lower-semicontinuous functions inf /^, and that limit is attai-

k>n

ned uniformly on the compact set K. It follows that hn -> h uniformly on K, and thus ^ [ K is relatively sequentially compact, hence relatively compact, in the uniform norm topology of C(K). So ^ [ K is equicontinuous (Arzela; see [4], p. 266), whence ^ is equicontinuous at the interior points of K, and in particular at XQ.

BIBLIOGRAPHY

[1] M. BRELOT, Axiomatique des Fonctions Harmoniques, Seminaire de Mathematiques Superieures (fite 1965), University of Montreal.

[2] M. BRELOT, Lectures on Potential Theory, Tata Institute of Funda- mental Research, Bombay (1960).

[3] C. CONSTANTINESCU and A. CORNEA, On the Axiomatic of Harmonic Functions I, Ann. Inst. Fourier, 13/2 (1963), 373-378.

[4] N. DUNFORD and J. SCHWARTZ, Linear Operators, Part I, General Theory, Interscience Publishers, Inc., New York, (1958).

[5] R.-M. HERVE, Recherches Axiomatiques sur la Theorie des Fonctions Surharmoniques et du Potentiel, Ann. Inst. Fourier, Grenoble, 12 (1962) 415-571.

Manuscrit regu Ie 23 novembre 1965.

Peter A. LOEB and B. WALSH, Department of Mathematics,

University of California, Los Angeles, Calif. (U.S.A.).

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