Appendix

For the reader's convenience we summarize below some of the less elementary tools we have used in the text, together with standard references to where proofs may be found. In what follows X denotes a locally compact Hausdorff space which has a countable base for its topology (though some of the results are true in a more general context) and 13 denotes the collection of Borel subsets of X.

Theorem A.I. Let f-l be a measure on X, let f : X x [a, b] --t IR, and suppose that f(', t) is Borel measurable for each t E [a, b]. Further, suppose that for some to the function f(', to) is f-l-integrable, that of lot exists on X x [a, b], and that there is a f-l-integrable function g on X such that I(af lat)(x, t)1 :S g(x) for all (x, t). Then the function F defined by F(t) = J f(', t) df-l is differentiable on [a,b] and F'(t) = J(aflat)(-,t) df-l.

Reference. Bartle [1, 1966], Corollary 5.9.

Definition A.2. Let f-l and f-ll be signed measures on X, and let v be a finite measure on X. (i) We say that f-l is absolutely continuous with respect to v if f-l(E) = 0 whenever E E 13 and v(E) = O. (An equivalent formulation of this condition is that, for every E > 0, there exists 8 > 0 such that 1f-l(E) I < E whenever E E 13 and v(E) < 8.) (ii) We say that f-l is concentrated on A E 13 if f-l(E) = 0 whenever E E 13 and EnA = 0. (iii) We say that f-l and f-ll are mutually singular if there are disjoint sets A, C E 13 such that f-l is concentrated on A and f-ll is concentrated on C.

Theorem A.3. (Lebesgue decomposition theorem) Let X, f-l and v be as above. Then there exist unique signed measures f-la and f-ls on X such that: (i) f-la is absolutely continuous with respect to v, (ii) f-ls and v are mutually singular, and (iii) J1 = f-la + f-ls·

Reference. Rudin [1, 1974]' Sections 6.9, 6.10.

305

306 Appendix

Theorem A.4. (Radon-Nikodym theorem) Let X, /-l and v be as above. If /-l is absolutely continuous with respect to v, then there is a unique fELl (v) such that /-leE) = JE f dv whenever E E 8.

Reference. Rudin [1, 1974]' Sections 6.9, 6.10.

Definition A.5. (i) The signed measures /-la and /-ls in Theorem A.3 are referred to respectively as the absolutely continuous component and the sin­gular component of /-l with respect to Y.

(ii) The function f of Theorem A.4 (which is unique as an element of Ll(v)) is referred to as the Radon-Nikodym derivative of /-l with respect to Y.

Definition A.6. In what follows we use Co(X) to denote the vector space of all continuous functions f : X -+ 1R with compact support. By a positive linear functional on Co(X) we mean a linear function L : Co(X) -+ 1R such that L(f) ~ 0 whenever f ~ O.

Theorem A.7. (Riesz representation theorem) Let L be a positive lin­ear functional on Co(X). Then there is a (Borel) measure /-l on X such that L(f) = Jx f d/-l for every f E Co(X).

Reference. Rudin [1, 1974], Theorem 2.14.

Theorem A.B. (Lusin's theorem) Let K be a compact Hausdorff space and let /-l be a measure on (the Borel subsets of) K. Further, let f : K -+ 1R be Borel measurable and let E > O. Then there exists g E C(K) such that /-leE) < E where E = {x E K : f(x) f. g(x)}. In particular, there is an open set U containing E such that /-leU) < E and flK\U is continuous.

Reference. Rudin [1, 1974]' Theorem 2.23.

Definition A.9. A sequence (/-In) of signed measures on a compact metric space K is said to be w* -convergent to a signed measure /-l if J f d/-ln -+ J f d/-l for every f E C(K).

Theorem A.lO. Let (/-In) be a sequence of signed measures on a compact metric space K and suppose that the corresponding sequence of total varia­tions (1IMnll) is bounded. Then there exist a subsequence (/-lnk) of (/-In) and a signed measure /-l on K such that (/-lnk) is w* -convergent to /-l.

Reference. Rudin [1, 1974], pp. 263, 269.

Definition A.11. A collection F of real-valued functions on a set K is said to separate the points of K if, given any two distinct points x and y in K, there exists f E F such that f(x) f. fey)·

Appendix 307

We now state two versions of the Stone- Weierstrass theorem.

Theorem A.12. Let K be a compact Hausdorff space and let :F be a vector subspace of C(K) such that; (i) :F separates the points of K, (ii) F contains the constant function 1, and (iii) max{j, g} E F whenever f, 9 E :F. Then:F is dense in C(K) with respect to the supremum norm.

Reference. Hewitt and Stromberg [1, 1965), Theorem 7.29.

Theorem A.13. Let K be a compact Hausdorff space and let :F be a vector subspace of C(K) such that; (i) F separates the points of K, (ii) F contains the constant function 1, and (iii) f 9 E :F whenever f, 9 E :F (that is, F is a subalgebra of C(K)). Then F is dense in C(K) with respect to the supremum norm.

Reference. Hewitt and Stromberg [1, 1965], Theorem 7.30.

Theorem A.14. (Tietze's extension theorem) Let K be a compact sub­set of a locally compact Hausdorff space X and let f E C(K) . Then there exists 9 E Co(X) such that 9 = f on K.

Reference. Rudin [1, 1974), Theorem 20.4.

For the terminology in our final result we refer to p.373 of Edwards [1, 1973].

Theorem A.15. (Green's formula) Let K C ]RN be a compact N­manifold-with-boundary, and let f and 9 be C2 functions on an open set which contains K. Then

r (f11g - g11f) d)" = r (f ~g - 9 ~f) dcr, 1 K 18K une une

where a/one denotes the exterior normal derivative at points of oK.

Reference. Edwards [1, 1973), Theorem 7.3 and pp. 392, 393.

Historical Notes

Notes on Chapter 1

1.1. Laplace's equation arose in mathematical physics in the last quarter of the eighteenth century when it was discovered that the gravitational force of attraction due to a body K (a compact set in ffi.3) can be expressed in ~3 \ K as the gradient of a function v, called a potential. Laplace showed that .1v = 0 on ~3 \ K.

1.2. The first indication that harmonic functions possess the mean value property is due to Gauss [1, 1840J; a special case of a result of his asserts, with the above notation, that if B(x, r) C ~3 \ K, then v(x) = M(v; x, r). The fact that the mean value property, together with continuity, character­izes harmonic functions (Theorem 1.2.2) was proved by Koebe [1, 1906J. In Theorem 1.2.2 continuity can be relaxed to local boundedness and measur­ability (Levi [1, 1909]). Some authors (e.g. Brelot [12, 1965J and Doob [6, 1984]) take the spherical mean value property and continuity as the defining properties of harmonic functions. A comprehensive survey and bibliography of the mean value property is given by Netuka and Vesely [1, 1994J

Theorem 1.2.6 was proved by Bacher [1, 1903]. The proof we give is es­sentially that of Nelson [1, 1961].

1.3. The Poisson integral with N = 3 appears in Poisson [1, 1823] . The first rigorous proof of Theorem 1.3.3(ii) for a continuous function f appears to be due to H. A. Schwarz [1, 1872].

The Riesz-Herglotz theorem was proved by F. Riesz [1, 1911J. Herglotz [1, 1911] refers to Riesz's paper for the result. Theorem 1.3.9 is a special case of a result of Parreau [1, 1951] who dealt with general domains. A very general result of this type is provided in Theorem 9.4.8. Jensen's inequality, which was used in the proof of Theorem 1.3.9, states more generally that, if J-L is a unit measure on a set X and f : X -+ I is J-L-integrable, where I ~ ~ is some interval, then ¢(J fdJ-L) :=:; J ¢ 0 fdJ-L for any convex function ¢ on I (see (13.34) in Hewitt and Stromberg [1, 1965]).

Theorem 1.3.11 in full generality is due to Kuran [2, 1972]. The history of the result is given in Netuka and Vesely [1, 1994]. The theorem was sub­sequently proved under weaker hypotheses (the class of integrable harmonic

309

310 Historical Notes

functions being replaced by a smaller class) by Armitage and Goldstein [I, 1990) and further generalized by Hansen and Netuka [I, 1993). Theorems of this type have turned out to be important in investigations of best harmonic Ll-approximation (see Goldstein, Haussmann and Rogge [I, 1988)).

1.4, 1.5. Harnack gave his inequalities in [I, 1887). They are the key to the convergence properties in Section 1.5; the simplest of these, Corollary 1.5.4, is due to Harnack himself.

1.6. Lord Kelvin (W. Thomson) introduced his transform in [I, 1847).

1.7. The half-space Poisson integral representation theorem for positive har­monic functions (Theorem 1.7.3) was proved by Dinghas [I, 1940) indepen­dently of the Kelvin transform. The proof involving the Kelvin transform was given by Lelong-Ferrand [2, 1949J. A further, novel proof is due to Huber [I, 1956) (see also Kuran [I , 1970]) . The result for N = 2 was earlier proved independently by several authors (see Huber [I, 1956) for references) .

1.8. The proof of Theorem 1.8.5 shows that if h is harmonic in B(xo, r), then the Taylor series of h centred at Xo converges on B(xo, (J2 - l)r). Kisel­man [I, 1969) and Hayman [I, 1970) independently showed that the constant J2 - 1 can be replaced by 1/ J2 and that this constant is sharp.

1.9. The Harnack metric in Exercise 1.9 appears to originate with Bear [I, 1965) and Kohn [I, 1966). For Exercise 1.11, see Armitage, Bagby and Gau­thier [I, 1985).

Notes on Chapter 2

2.1-2.3. The approach taken in these sections is largely influenced by Brelot and Choquet [I, 1954). Other relevant works are Muller [I, 1966), Stein and Weiss [I, 1971J and Axler, Bourdon and Ramey [I, 1992) .

2.4. The two-dimensional case of Theorem 2.4.5 is due to R. Nevanlinna [I, 1925), who provided modified Poisson kernels for the construction of solu­tions. This was extended to all dimensions by Gardiner [I, 1981) (see also Finkelstein and Scheinberg [I, 1975]). Gardiner [5, 1993) characterized open sets fl with the property that for each f E C(afl) there exists h E 1i(fl) such that h(x) ~ f(y) as x ~ y for each yEan.

2.6. For an account of Runge's theorem for holomorphic functions, which constitutes the background to the results of this section, we refer to Conway [I, 1978) . Corollary 2.6.5 is due to Walsh [1, 1929), and Theorem 2.6.6 is

Notes on Chapter 3 311

given by Gauthier, Goldstein and Ow [1, 1983] . The existence of a harmonic function with the properties of that in Example 2.6.8 is due to Zalcman [1, 1982] in the case N = 2 and to Armitage and Goldstein [2, 1993] in the gen­eral case. The construction given for Examples 2.6.7 and 2.6.8 is an analogue of that given for entire holomorphic functions exhibiting similar behaviour by Armitage [1, 1994) (see also Burckel [1, 1995]). A comprehensive account of harmonic approximation and its applications is given by Gardiner [7, 1995].

2.8. Exercises 2.14-2.16 are based on work of Burchard [1, 1976) (see also Hayman, Kershaw and Lyons [1, 1984]). The idea of universality, of which Ex­ercise 2.18 provides an example, is surveyed in an article by Grosse-Erdmann [1, 1999].

Notes on Chapter 3

3.1. F. Riesz [2, 1926] appears to have been the first author to give a defini­tion of subharmonicity equivalent to Definition 3.1.2. Phragmen and Lindelof [1, 1908] proved, for holomorphic fUIlctions, theorems of the kind which now bears their names. Theorem 3.1.10 can be obtained from a result of Fuglede [2, 1975] but the proof in the text is based on work of Chen and Gauthier [1, 1992).

3.3. Results on the approximation of subharmonic functions by smooth ones were proved by F . Riesz [3, 1930). Theorem 3.3.6 is due to Avanissian [1, 1961). The hypothesis of local lower boundedness in this theorem has been successively relaxed by several authors (see Armitage and Gardiner [2, 1993]) but cannot be dispensed with altogether (see Exercise 3.6). On the other hand, the local boundedness hypothesis in Corollary 3.3.7 can be dropped; see Lelong [1,1961) .

3.4. Theorem 3.4.3 is due to Gardiner [2, 1985]. The proof in the text is based on Gardiner and Klimek [1, 1986], where the result is proved in a more abstract setting.

3.5. Convexity theorems for mean values of subharmonic functions were ob­tained by F. Riesz [2, 1926], Dinghas [2, 1965) and Solomentsev [1, 1967]. Hardy [1, 1915] had earlier proved Corollary 3.5.8 for 0 < p < +00. The part of Corollary 3.5.8 concerned with m(ifJ;O,r) is referred to as Hadamard's three circles theorem; it was discovered independently by several authors (see Hardy [1, 1915]). The convexity of the functions N(s ;· ) and £(s;·) in The­orems 3.5.9 and 3.5 .10 has been proved under various hypotheses by several authors. Rippon [2, 1983] obtained the conclusion of Theorem 3.5.10 under

312 Historical Notes

the hypothesis that log+ £(8;·) is locally integrable on (0, I). For a study of £(8; .) in the case where 8 is subharmonic but not necessarily non-negative, see Armitage and Gardiner [1, 1987].

3.6. Least harmonic majorants were studied by F. Riesz [2, 1926].

3.7. Lemma 3.7.4 is known as Choquet's lemma.

3.8. Some literature related to the Exercises is as follows: Armitage and Gardiner [2, 1993] (Exercise 3.6); Beardon [1, 1971] (Exercise 3.1O); Carroll [1, 1988] and Drasin [1, 1981] (Exercise 3.21); Armitage and Kuran [1,1985) (Exercise 3.24). For Motzkin's theorem, which is used in Exercise 3.24, we refer to Theorem 7.8 of Valentine [1, 1964).

Notes on Chapter 4

4.1. The Green function and also certain integral identities (see Green's for­mula in the Appendix) appear in Green [1, 1828). The second formula for GBo in Theorem 4.1.5 appears in Boukricha, Hansen and Hueber [1, 1987].

4.3. The distributional Laplacian is part of a wider theory of generalized func­tions, or distributions, developed by L. Schwartz [1, 1950]. Theorem 4.3.5 is known as Weyl's lemma.

4.4. Theorem 4.4.1 can be found in F. Riesz [3, 1930) and Corollary 4.4.5 in Jensen [1, 1899].

4.5. In the case where f? = IR3 Theorem 4.5.1 is due to Evans [1, 1935) and Vasilesco [1, 1935), and Corollary 4.5.2 is due to Choquet [2, 1957).

4.6. Lemma 4.6.3 is adapted from Zeinstra [1, 1989). In the case N = 2 Theorems 4.6.4 and 4.6.7 are due to Littlewood [1, 1928) and Fatou [1, 1906) respectively. Privalov [1, 1938) generalized Littlewood's theorem to higher dimensions. The corresponding generalization of Fatou's theorem is due to Bray and Evans [1, 1927]. Far-reaching generalizations of these theorems are given in Sections 9.3 and 9.4.

4.7. Hyperplane mean values of superharmonic functions (see Exercise 4.2) are studied in Armitage and Watson [1, 1986].

Notes on Chapter 6 313

Notes on Chapter 5

5.1. Theorem 5.1.11 is due to Maria [1, 1934] and Frostman [1, 1935].

5.2. Theorem 5.2.1 is taken from Section 1.V.5 in Doob [6, 1984]. Corollary 5.2.2 is due to Brelot [5, 1941]. Bouligand [2, 1926] proved Corollary 5.2.3 for a compact polar set E.

5.3. The systematic study of reduced functions appears to originate with Brelot [8, 1945]. The equivalence of (a), (b) and (d) in Theorem 5.3.8 is due to Myrberg [1, 1933].

5.4. A definition of capacity for compact sets in IR.3 was given by Wiener [2, 1924]

5.5. De la Vallee Poussin [1, 1932] extended the definition of capacity to an arbitrary bounded set. His capacity was subsequently renamed as inner capacity by Brelot [2, 1939], and Monna [1, 1940] introduced the notion of outer capacity. Corollary 5.5.7 is due to Cartan [2, 1945].

5.6. Theorem 5.6.4 is due to Choquet [1, 1954].

5.7. Theorem 5.7.1 was announced by Cartan [1, 1942] and proved by him [2, 1945]. Special cases and weaker versions were proved by Szpilrajn [1, 1933], Rad6 [1, 1937] and Brelot [1, 1938].

5.9. Hausdorff [1, 1918] introduced the measures associated with his name. Comparisons between capacity and Hausdorff measure were made by Frost­man [1, 1935] and others. Carleson [1, 1967] showed that capacity cannot be completely described in terms of Hausdorff measure.

5.10. The idea in Exercise 5.4 is due to Huber [1, 1956]. It was rediscovered and further exploited by Kuran [1, 1970]. The result in Exercise 5.5(ii) is sometimes referred to as the Rad6-Stout theorem. In full generality the result in Exercise 5.13 appears to be due to Brelot and Choquet [2, 1957].

Notes on Chapter 6

6.2, 6.3. The use of upper and lower classes P f, IJ! f and corresponding gener­alized solutions H f, H f originates with Perron [1, 1923]. Wiener [1, 1924], [2, 1924], [3, 1925J showed that a generalized solution of the Dirichlet problem can be associated with any continuous boundary function (compare Theorem

314 Historical Notes

6.3.8). A definitive paper on the PWB approach is Brelot [3, 1939].

6.4. Theorem 6.4.6 is due to Brelot [3, 1939].

6.6. The notion of regularity is much older than the PWB approach to the Dirichlet problem. Example 6.1.1 of an isolated irregular boundary point is due to Zaremba [1, 1911]; the more striking Lebesgue spine (see Theorem 6.6.16 and Remark 6.6.17) appears in Lebesgue [1,1912]. Generalizations of the Lebesgue spine are described in Gardiner [4, 1992]; see also Section 7.2. The idea of a barrier appears in work of Poincare [1, 1890]. Theorem 6.6.4 is an improvement, due to Bouligand [2, 1926], of a result of Lebesgue [2, 1924).

6.7. The role of 00 in relation to the Dirichlet problem is discussed in Brelot [7, 1944].

6.8. A characterization of regularity in terms of the Green function (see The­orem 6.8.3) is due to Bouligand [1, 1924].

6.10. Theorem 6.10.1 is similar to a result in Doob [1, 1954]. More recent work on superharmonic extension is given in Gardiner [6, 1994] and Gauthier [1, 1994]. For a discussion and further references, see Gardiner [7, 1995].

Notes on Chapter 7

7.1, 7.2. Historically, the notion of thinness preceded the introduction of the fine topology. Thinness was defined by Brelot [4, 1940] in terms of condition (b) of Theorem 7.2.3, and its equivalence with condition (c) of that result was noted in Brelot [6, 1944]. The fine topology was proposed by Cartan in correspondence with Brelot (see p.14 of Brelot [6, 1944]). The fact that it gives rise to a Baire space (Theorem 7.1.3) is due to Cornea (see p.118 of Constantinescu and Cornea [1, 1972]). In connection with Theorem 7.2.5, see the comments above on Theorem 6.6.16.

7.3, 7.4. Much of this material is due to Brelot (see [6, 1944] for Theorems 7.3.2, 7.3.4, 7.3.7 and 7.4.4, and [8, 1945] for Theorem 7.3.5). Parts (i) and (iii) of Theorem 7.3.11 are due respectively to Doob [5, 1966] and Choquet [3, 1959]. Theorem 7.4.2 appears on p.l71 of Deny [4, 1950], where it is at­tributed to Cartan.

7.5. For Theorem 7.5 .1, Corollary 7.5.3 and Theorem 7.5.5 we refer to Brelot [4, 1940], Section IX,6 of [12, 1965) and [10, 1946] respectively. Theorem 7.5.6 is due to Keldys [1, 1941].

:\'otes on Chapter 8 315

7.6. Brelot [7, 1944) defined the notion of thinness at 00 and identified the unbounded open sets in ]RN (N 2': 3) for which {oo} is negligible. In connec­tion with Theorem 7.6.6 we mention the book of Fuglede [1, 1972), which expounds the theory of "finely harmonic" and "finely super harmonic" func­tions on finely open sets. Theorem 7.6.8 is part of a result of Ahern and Cohn [1, 1983), who used thinness to characterize domains n in C such that every holomorphic function from B into n belongs to the Smirnov class.

7.7. Wiener [1, 1924) provided his criterion for closed sets in connection with the study of regularity of boundary points for the Dirichlet problem (cf. The­orem 7.5.1) , and Brelot [4, 1940) generalized it to characterize thinness for arbitrary sets. Corollary 7.7.4 is also due to Brelot [8, 1945).

7.8. Theorem 7.8.1 is due to Deny [2, 1948) and Theorem 7.8.2 is related to an observation of Brelot [4, 1940) . An early result in the spirit of Theorem 7.8.4 is due to Evans [1, 1935], who showed that the set E' has 2-dimensional measure 0 when N = 3. A generalization of Theorem 7.8.4, comparing fine cluster values with cluster values along lines may be found in Essen and Gar­diner [1, 1999). Theorems 7.8.6 and 7.8.7 are special cases of results of Hansen [1 , 1971) and Aikawa and Gardiner [1, 2000).

7.9. Theorem 7.9.2 is due to Keldys [1 , 1941)' Brelot [9, 1945) and Deny [1, 1945), while Theorem 7.9.5 is due to Keldys [1,1941] and Deny [3, 1949). (The latter result is the harmonic analogue of Mergelyan 's theorem for holomorphic functions; see, for example, Theorem 20.5 in Rudin [1, 1974).) Generalizations of these results which deal with approximation on non-compact sets, together with a variety of applications, may be found in the recent book by Gardiner [7, 1995), where appropriate references are given.

Notes on Chapter 8

8.1-8.4. Martin [1, 1941)' using a metric equivalent to that in Definition 8.1.3, constructed the compactification n, distinguished between minimal and non-minimal points of .d, and established the representation in Theorem 8.4.1.

8.5. The Martin kernel of a strip is given explicitly by Brawn [1, 1972] . The approach taken in the text follows Gardiner [3, 1990) .

8.7, 8.8. The boundary Harnack principle for Lipschitz domains was obtained independently by Ancona [1 , 1978), Dahlberg [2, 1977] and Wu [1, 1978), after Widman [1, 1967] had established it for less general domains. Hunt and Wheeden [1, 1970) showed that the Martin boundary of a Lipschitz domain

316 Historical Notes

can be identified with the Euclidean boundary. Other authors, in particular Jerison and Kenig [1, 1982] and Aikawa [1], have extended the boundary Harnack principle to increasingly general classes of domains and used it to show that in these domains, too, the Martin boundary can be identified with the Euclidean boundary. The approach adopted in the text is mainly based on ideas from these last two cited papers, and we are grateful to Professor Aikawa for sending us a preprint of his paper [1].

Notes on Chapter 9

9.1. Theorems 9.1.3 and 9.1.7 are due to Brelot (see [8, 1945] and [11, 1956] respectively) .

9.2, 9.3. Lelong-Ferrand [1, 1949], working in the context of a half-space, was the first to develop this notion of thinness at boundary points. The general theory is due to NaIm [1, 1957].

9.4. Theorem 9.4.6 is due to Doob (see [2, 1957] and [3, 1959] for probabilistic and non-probabilistic proofs respectively). For Theorem 9.4.8 see Parreau [1, 1951] and Brelot [11, 1956].

9.5, 9.6. Most of this material is taken from NaIm [1, 1957], but Theorem 9.6.3 is due to Doob [3, 1959].

9.7. For sets E as described in Theorem 9.7.1, condition (9.7.1) actually characterizes minimal thinness at 0 with respect to D: see Burdzy [1, 1987], and closely related earlier work of Beurling [1, 1965], Maz'ja [1, 1972] and Dahlberg [1, 1976]. The book of Aikawa and Essen [1, 1996] presents a helpful approach to such results based on a "quasi-additivity" property of capacity. Theorems 9.7.4 and 9.7.6 are due to Brelot and Doob [1,1963]' while Theorem 9.7.8 is due to Doob [4,1965]. The case N ~ 3 of part (i) and also part (ii) of Theorem 9.7.10 are due to Lelong-Ferrand [1, 1949]. The case N = 2 of part (i) was originally thought to be false, but was then established by Jackson [1, 1970]. The background to Theorem 9.7.11 is a classical result of Lusin and Privalov [1, 1925] for holomorphic functions. Part (i) of the theorem is due to Arsove [1, 1964], and part (ii) is due to Gardiner [8, 1996]. For a converse, see Essen and Gardiner [1, 1999]. Complementary results may be found in Rippon [1, 1978].

References

P. AHERN and W. COHN [1] "A geometric characterization of N+ domains", Pmc. Amer. Math. Soc. 88 (1983), 454-458.

H. AIKAWA [1] "Boundary Harnack principle and Martin boundary for a uniform domain" , J. Math. Soc. Japan, to appear.

H. AIKAWA and M. ESSEN [1] Potential Theory - Selected Topics. Lecture Notes in Math. 1633, Springer, Berlin, 1996.

H. AIKAWA and S. J. GARDINER [1] "Evaluation of super harmonic functions using limits along lines", Bull. London Math. Soc. 32 (2000), 209-213.

A.ANCONA [1] "Principe de Harnack ala frontiere et theoreme de Fatou pour un operateur elliptique dans un domaine lipschitzien", Ann. [nst. Fourier (Grenoble) 28, 4 (1978), 169-213.

D. H. ARMITAGE [1] "A non-constant continuous function on the plane whose integral on every line is zero", Amer. Math. Monthly 101 (1994), 892-894.

D. H. ARMITAGE, T. BAGBY and P. M. GAUTHIER [1] "Note on the decay of solutions of elliptic equations", Bull. London Math. Soc. 17 (1985), 554-556.

D. H. ARMITAGE and S. J. GARDINER [1] "The growth of the hyperplane mean of a sub harmonic function", J. London Math. Soc. (2), 36 (1987), 501-512. [2] "Conditions for separately subharmonic functions to be subharmonic", Po­tential Anal. 2 (1993), 255-261.

D. H. ARMITAGE and M. GOLDSTEIN [1] "The volume mean-value property of harmonic functions", Complex Variables Theory Appl. 13 (1990), 185-193. [2] "Non-uniqueness for the Radon transform", Proc. Amer. Math. Soc. 117 (1993),175-178.

D. H. ARMITAGE and 0. KURAN [1] "The convexity of a domain and the superharmonicity of the signed distance function", Proc. Amer. Math. Soc. 93 (1985), 598-600.

317

318 References

D. H. ARMITAGE and N. A. WATSON [IJ "Mean values of positive superharmonic functions on half-spaces", J. Math. Anal. Appl. 120 (1986), 561-566.

M.G.ARSOVE [IJ "The Lusin-Privalov theorem for subharmonic functions", Proc. London Math. Soc. (3) 14 (1964), 260-270.

V. AVANISSIAN [1) "Fonctions plurisousharmoniques et fonctions doublement sousharmoniques" , Ann. Sci. Ecole Norm. Sup. (3) 78 (1961), 101-161.

S. AXLER, P. BOURDON and W. RAMEY [1) Harmonic Function Theory. Springer, New York, 1992.

R. G. BARTLE [IJ The Elements of Integration. Wiley, New York, 1966.

H. S. BEAR [1) "A geometric characterization of Gleason parts", Pmc. Amer. Math. Soc. 16 (1965), 407-412.

A. F. BEARDON [IJ "Integral means of subharmonic functions", Pmc. Cambridge Philos. Soc. 69 (1971), 151-152.

A. BEURLING [1) "A minimum principle for positive harmonic functions", Ann. Acad. Sci. Fenn. Ser. AI, no. 372 (1965), 7pp.

M.BOCHER [1 J "Singular points of functions which satisfy partial differential equations of the elliptic type", Bull. Amer. Math. Soc. 9 (1903), 455-465.

A. BOUKRICHA, W. HANSEN and H. HUEBER [1 J "Continuous solutions of the generalized Schrodinger equation and perturba­tion of harmonic spaces", Exposition. Math. 5 (1987), 97-135.

G. BOULIGAND [IJ "Domaines infinis et cas d'exception du probleme de Dirichlet", C. R. Acad. Sci. Paris 178 (1924), 1054-1057. [2) "Sur Ie probleme de Dirichlet", Ann. Soc. Polan. Math. 4 (1926), 59-112.

F. T. BRAWN [1) "The Martin boundary of ]Rn xJO, 1[", J. London Math. Soc. (2) 5 (1972), 59-66.

H. E. BRAY and G. C. EVANS [IJ "A class of functions harmonic within the sphere", Amer. J. Math. 49 (1927), 153-180.

M. BRELOT [1 J "Sur Ie potentiel et les suites de fonctions sous-harmoniques", C. R. Acad. Sci. Paris 207 (1938), 836-838. [2J "Sur la theorie moderne du potentiel", C. R. Acad. Sci. Paris 209 (1939), 828-830. [3) "Families de Perron et probleme de Dirichlet", Acta Litt. Sci. Szeged 9 (1939), 133-153. [4) "Points irreguliers et transformations continues en theorie du potentiel", J.

References 319

Math. Pures Appl. (9) 19 (1940) , 319-337. [5] "Sur la theorie autonome des fonctions sousharmoniques", Bull. Sci. Math. (2) 65 (1941), 72- 98. [6] "Sur les ensembles effiles", Bull. Sci. Math . (2) 68 (1944), 12-36. [7] "Sur Ie role du point a l'infini dans la theorie des {onctions harmoniques", Ann. Sci. Ecole Norm. Sup. 61 (1944), 301-332. [8] "Minorantes sous-harmoniques, extremales et capacites", J. Math . Pures Appl. (9) 24 (1945),1- 32. [9] "Sur l'approximation et la convergence dans la theorie des {onctions har­moniques ou holomorphes", Bull. Soc. Math. France 73 (1945), 55-70. [10] "Etude generale des {onctions harmoniques ou surharmoniques positives au voisinage d 'un point-frontiere irregulier", Ann. Univ. Grenoble . Sect . Sci . Math. Phys. (N.S.) 22 (1946) , 205-219. [11] "Le probleme de Dirichlet. Axiomatique et frontiere de Martin", J. Math. Pures Appl. (9) 35 (1956) , 297-335 . [12] Elements de la Theorie Classique du Potentiel. Third edition, Centre de Doc­umentation Universitaire , Paris, 1965. [13] On Topologies and Boundaries in Potential Theory. Lecture Notes in Math. 175, Springer, Berlin, 1971.

M. BRELOT and G. CHOQUET [1] "Polynomes harmoniques et polyharmoniques", in Seconde Col/oque sur les equations aux derivees partiel/es (Bruxelles , 1954), Georges Thone, Liege; Mas­son, Paris, 1955, pp.45-66. [2] "Le theoreme de convergence en theorie du potentiel", J. Madras Univ. B 27 (1957) , 277-286.

M. BRELOT and J. L. DOOB [1] "Limites angulaires et limites fines", Ann. Inst . Fourier (Grenoble) 13 (1963), 395-415.

H. G. BURCHARD [1] "Best uniform harmonic approximation" in Approximation Theory II, G. G. Lorentz et al. (eds.), Academic Press, New York, 1976, pp.309- 314.

R. B. BURCKEL [1] "Entire functions which vanish at infinity", Amer. Math . Monthly 102 (1995), 916- 918.

K. BURDZY [1] "Brownian excursions and minimal thinness I", Ann. Probab. 15 (1987), 676-689.

L. CARLESON [1] Selected Problems on Exceptional Sets. Van Nostrand, Princeton, 1967.

T . F. CARROLL [1] "A classical proof of Burdzy's theorem on the angular derivative", J. London Math . Soc . (2) 38 (1988), 423-44l.

H. CARTAN [1] "Capacite exterieure et suites convergentes de potentiels" , C. R. Acad. Sci . Paris 214 (1942), 944- 946. [2] "Theorie du potentiel newtonien: energie, capacite, suites de potentiels", Bull. Soc . Math. France 73 (1945), 74- 106.

320 References

CHEN HUAIHUI and P. M. GAUTHIER [1) "A maximum principle for subharmonic and plurisubharmonic functions", Canad. Math. Bull. 35 (1992), 34- 39.

G. CHOQUET [1) "Theory of capacities", Ann. Inst. Fourier (Grenoble) 5 (1954), 131-295. [2) "Potentiels sur un ensemble de capacite nulle. Suites de potentiels", C. R. Acad. Sci. Paris 244 (1957) , 1707- 1710. [3) "Sur les points d'effilement d'un ensemble. Application a l'etude de la ca­pacite", Ann. Inst. Fourier (Grenoble) 9 (1959), 91-101.

C. CONSTANTINESCU and A. CORNEA (1) Potential Theory on Harmonic Spaces. Springer, Berlin, 1972.

J. B. CONWAY [1) Functions of One Complex Variable. Second edition, Springer, New York, 1978.

B.DAHLBERG [1) "A minimum principle for positive harmonic functions", Proc. London Math. Soc. (3) 33 (1976), 238- 250. [2) "Estimates of harmonic measure", Arch. Rational Mech. Anal. 65 (1977) , 275-288.

C. de la VALLEE POUSSIN [1) "Extension de la methode du balayage de Poincare et probleme de Dirichlet" , Ann. Inst . H. Poincare 2 (1932), 169-232.

J. DENY [1) "Sur l'approximation des fonctions harmoniques", Bull. Soc. Math. France 73 (1945),71-73. [2) "Un theoreme sur les ensembles effiles", Ann. Univ. Grenoble Sect. Sci. Math. Phys. (N.S.) 23 (1948), 139- 142. [3) "Systemes totaux de fonctions harmoniques", Ann. Inst. Fourier (Grenoble) 1 (1949) , 103-113. [4) "Les potentiels d'energie finie" , Acta Math. 82 (1950), 107-183.

A. DINGHAS [1) "Uber positive harmonische Funktionell in einem Halbraum", Math. Z. 46 (1940), 559-570. [2) "Uber einige Konvexitatssatze fiir die Mittelwerte von subharmonischen Funk­tionen", J. Math. Pures Appl. (9) 44 (1965), 223-247.

J. L. DOOB [1) "Semimartingales and subharmonic functions", Trans. Amer. Math. Soc . 77 (1954),86- 121. [2) "Conditional Brownian motion and the boundary limits of harmonic func­tions", Bull. Soc. Math. France 85 (1957), 431-458. [3) "A non-probabilistic proof of the relative Fatou theorem", Ann. Inst. Fourier (Grenoble) 9 (1959), 293- 300. [4) "Some classical function theory theorems and their modern versions", Ann. Inst. Fourier (Grenoble) 15, 1 (1965), 113- 136. [5) "Applications to analysis of a topological definition of smallness of a set" , Bull. Amer. Math. Soc . 72 (1966), 579- 600. [6) Classical Potential Theory and its Probabilistic Counterpart. Springer, New York , 1984.

References 321

D. DRASIN [1) "Some problems related to MacLane's class", J. London Math. Soc . (2) 23 (1981), 280-286.

C. H. EDWARDS [1) Advanced Calculus of Several Variables. Academic Press, New York, 1973.

M. R. ESSEN and S. J. GARDINER [1) "Limits along parallel lines and the classical fine topology", J. London Math. Soc. (2) 59 (1999), 881-894.

G. C. EVANS [1) "On potentials of positive mass. Part I", Trans. Amer. Math . Soc. 37 (1935), 226-253.

P. FATOU [1) "Series trigonometriques et series de Taylor", Acta Math. 30 (1906), 335-400.

M. FINKELSTEIN and S. SCHEINBERG [1) "Kernels for solving problems of Dirichlet type in a half-plane", Advances in Math . 18 (1975) , 108-113.

w. H. FLEMING [1] Functions of Several Variables. Addison-Wesley, Reading, MA, 1965.

O. FROSTMAN [1) "Potentiel d'equilibre et capacite des ensembles avec quelques applications a la theorie des fonctions" , Meddel. Lunds Univ. Mat. Sem. 3 (1935), 1-118.

B. FUGLEDE [1) Finely Harmonic Functions. Lecture Notes in Math. 289, Springer, Berlin, 1972. [2) "Asymptotic paths for subharmonic functions", Math. Ann. 213 (1975), 261-274.

S. J. GARDINER [1) "The Dirichlet and Neumann problems for harmonic functions in half-spaces", J. London M9-th . Soc. (2) 24 (1981) , 502-512. [2) "A maximum principle and results on potentia' " , J. Math. Anal. Appl. 108 (1985), 507- 514. [3) "The Martin boundary of NTA strips" , Bull. London Math . Soc . 22 (1990) , 163- 166. [4] "A fine limit property of fUllctions superharmonic outside a manifold", Com­positio Math . 83 (1992) , 239-249. [5] "The Dirichlet problem with non-compact boundary" , Math. Z. 213 (1993), 163-170. [6) "Superharmonic extension and harmonic approximation", Ann. Inst. Fourier (Grenoble) 44 (1994), 65- 91. [7] Harmonic Approximation. London Math. Soc. Lecture Note Series 221, Cam­bridge Univ. Press, Cambridge, 1995. [8) "The Lusin-Privalov theorem for subharmonic functions", Proc. Amer. Math . Soc. 124 (1996), 3721-3727.

S. J. GARDINER and M. KLIMEK [1) "Convexity and subsolutions of partial differential equations", Bull. London Math. Soc . 18 (1986), 41- 43.

322 References

C. F . GAUSS [1] "Algemeine Lehrsiitze in Beziehung auf die im verkehrtem Verhiiltnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungs-Kriifte", (1840) Werke 5. Band, Gottingen, 1877.

P. M. GAUTHIER [1] "Subharmonic extensions and approximations", Ganad. Math. Bull. 37 (1994), 46-53.

P. M. GAUTHIER, M. GOLDSTEIN and W. H. OW [1] "Uniform approximation on closed sets by harmonic functions with Newtonian singularities", J. London Math. Soc . (2) 28 (1983), 71-82 .

M. GOLDSTEIN, W. HAUSSMANN and L. ROGGE [1] "On the mean value property of harmonic functions and best harmonic L1_ approximation", 1tans. Amer. Math. Soc. 305 (1988), 505-515.

G. GREEN [1] An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Nottingham, 1828. Math. Papers, Macmillan, Lon­don, 1871, pp.9-41.

K-G. GROSSE-ERDMANN [1] "Universal families and hypercyclic operators", Bull. Amer. Math. Soc. (N.S.) 36 (1999), 345-381.

W. HANSEN [1] "Abbildungen harmonischer Riiume mit Anwendung auf die Laplace und Wiirmeleitungsgleichung", Ann. [nst . Fourier (Grenoble) 21, 3 (1971), 203- 216.

W. HANSEN and I. NETUKA [1] "Inverse mean value property of harmonic functions", Math. Ann. 297 (1993), 147-156. Corrigendum, ibid. 303 (1995), 373-375.

G.H.HARDY [1] "The mean value of the modulus of an analytic function", Proc. London Math. Soc. 14 (1915), 269-277.

A.HARNACK [1] Die Grundlagen der Theorie der logarithmischen Potentie/s und der eindeuti­gen Potentialfunktion. Teubner, Leipzig, 1887.

F. HAUSDORFF [1] "Dimension und iiusseres Mass", Math. Ann. 79 (1918), 157-179.

W. K HAYMAN [1] "Power series expansions for harmonic functions", Bull. London Math. Soc. 2 (1970), 152-158. [2] Subharmonic Functions, Volume 2. Academic Press, London, 1989.

W. K HAYMAN and P. B. KENNEDY [1] Subharmonic Functions, Volume 1. Academic Press, London, 1976.

W. K HAYMAN, D. KERSHAW and T . J. LYONS [1] "The best harmonic approximant to a continuous function", in Anniversary Volume on Approximation Theory and Functional Analysis, P. L. Butzer et al. (eds.), Birkhiiuser, Basel, 1984, pp.317-327.

References 323

L. L. HELMS [1) Introduction to Potential Theory. Wiley, New York, 1969.

G.HERGLOTZ [1) "Uber Potenzreihen mit positivem, reel en Teil im Einheitskreis", S. -B. Sachs. Akad. Wiss. Leipzig Math.-Natur. KI. 63 (1911), 501-511.

E. HEWITT and K. STROMBERG [1) Real and Abstract Analysis. Springer, Berlin, 1965.

A.HUBER [1) "On functions sub harmonic in a half-space", 'lrans. Amer. Math. Soc. 82 (1956),147-159.

R. A. HUNT and R. L. WHEEDEN [1) "Positive harmonic functions on Lipschitz domains", 'lrans. Amer. Math. Soc. 147 (1970), 507-527.

H. L. JACKSON [1) "Some results for thin sets in a halfplane", Ann. Inst. Fourier (Grenoble) 20, 2 (1970), 201-218.

J. L. W. V. JENSEN [1] "Sur un nouvel et important theoreme de la theorie des fonctions" , Acta Math. 22 (1899), 359-364.

D. S. JERISON and C. E. KENIG [1] "Boundary behavior of harmonic functions in non-tangentially accessible do­mains", Adv. in Math. 46 (1982), 80-147.

M.V.KELDYS [1] "On the solvability and stability of the Dirichlet problem", Uspehi Mat. Nauk 8 (1941), 171-231 (Russian). [English translation in Amer. Math. Soc. 'lransl. 51 (1966), 1-73.)

O. D. KELLOGG [1) Foundations of Potential Theory. Springer, Berlin, 1929.

C. O. KISELMAN [1] "Prolongement des solutions d'une equation aux derivees partielles a. coeffi­cients constants", Bull. Soc. Math. France 97 (1969), 329-356.

P. KOEBE [1] "Herleitung der partiellen Differentialgleichungen der Potentialfunktion aus deren Integraleigenschaft", Sitzungsber. Berlin. Math. Gessellschaft 5 (1906), 39-42.

J. KOHN [1] "Die Harnacksche Metrik in der Theorie der harmonischen Funktionen", Math. Z. 91 (1966), 50-64.

U.KURAN [1] "Study of superharmonic functions in IRn x (0, +(0) by a passage to IRn +3 " ,

Proc. London Math. Soc. (3) 20 (1970), 276-302. [2] "On the mean-value property of harmonic functions", Bull. London Math. Soc. 4 (1972), 311-312.

N. S. LANDKOF [1] Foundations of Modem Potential Theory. Springer, Berlin, 1972.

324 References

H. LEBESGUE [1J "Sur des cas d'impossibilite du probleme de Dirichlet ordinaire", C. R. Seances Soc. Math. France 17 (1912). [2J "Conditions de regularite, conditions d'irregularite, conditions d'impossibilite dans Ie probleme de Dirichlet", C. R. Acad. Sci. Paris 178 (1924), 349-354.

P. LELONG [IJ "Fonctions plurisousharmoniques et fonctions analytiques de variables reelles", Ann. Inst. Fourier (Grenoble) 11 (1961), 515-562.

J . LELONG-FERRAND [1J "Etude au voisinage de la frontiere des fonctions surharmoniques positives dans une demi-espace", Ann. Sci . Ecole Norm. Sup. (3) 66 (1949), 125-159. [2J "Sur Ie principe de Julia-Caratheodory et son extension a. I'espace a. p dimen­sions", Bull. Sci. Math. (2) 73 (1949), 5-16.

E. LEVI [1 J "Sopra una proprieta. caratteristica delle funzione armoniche", Rend. Accad. d. Lincei Roma 18 (1909), 10-15.

J. E. LITTLEWOOD [IJ "Mathematical Notes (8): On functions sub harmonic in a circle (II)", Proc. London Math. Soc. (2), 28 (1928), 383-394.

N. N. LUSIN and I. I. PRIVALOV [IJ "Sur I'unicite et la multiplicite des fonctions analytiques", Ann. Sci. Ecole Norm. Sup. (3) 42 (1925), 143-191.

A. J. MARIA [IJ "The potential of a positive mass and the weight function of Wiener", Proc. Nat. Acad. Sci. USA 20 (1934), 485-489.

R. S. MARTIN [IJ "Minimal positive harmonic functions", Trans. Amer. Math. Soc. 49 (1941), 137-172.

V. G. MAZ'JA [IJ "Beurling's theorem on a minimum principle for positive harmonic functions", Zap. Nauch. Sem. LOMI30 (1972), 76-90 (Russian). [English translation in J. Soviet Math. 4 (1976), 367-379.]

A. F . MONNA [1] "Sur la capacite des ensembles", Proc. Kon. Ned. Akad. Wetensch. 43 (1940), 81-86.

C. MULLER [1] Spherical Harmonics. Lecture Notes in Math. 17, Springer, Berlin, 1966.

P. MYRBERG [lJ "Uber die Existenz der Greenschen Funktionen auf einer gegeben Rie­mannschen FHiche", Acta Math. 61 (1933), 39-79.

L. NAIM [lJ "Sur Ie role de la frontiere de R. S. Martin dans la theorie du potentiel", Ann. Inst. Fourier (Grenoble) 7 (1957), 183-281.

E. NELSON [1] "A proof of Liouville's theorem", Proc. Amer. Math. Soc. 12 (1961), 995.

References 325

I. NETUKA and J . VESELY [1] "Mean value property and harmonic functions" in Classical and Modern Po­tential Theory and Applications, K. GowriSankaran et al. (eds.), Kluwer, Dor­drecht, 1994, pp.359-398.

R. NEVANLINN A [1] "Uber eine Erweiterung des Poissonsches Integrals", Ann. Acad. Sci. Fenn. Ser. A. 24 No. 4 (1925), 1-15.

M. PARREAU [1] "Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann", Ann. [nst. Fourier (Grenoble) 3 (1951), 103-197.

O. PERRON [1] "Eine neue Behandlung der erst en Randwertaufgabe fur ..1u = 0", Math . Zeit. 18 (1923), 42-54.

E. PHRAGMEN and E. LINDELOF [1] "Sur une extension d'un principe classique de l'analyse", Acta Math. 31 (1908), 381-406.

H. POINCARE [1] "Sur les equations aux derivees partielles de la physique mathematique", Amer. J. Math. 12 (1890), 211-294.

S. D. POISSON [1] "Addition au memoire precedent et au memoire sur la maniere d'exprimer les fonctions par les series de quantites periodiques", J. Ecole Roy. Poly technique 19 cahier, 12 (1823), 145- 162.

N. PRIVALOV [1] "Boundary problems of the theory of harmonic and subharmonic functions in space", Mat . Sbomik 45 (1938) , 3-25 (Russian) .

T . RADO [1] Subharmonic Functions. Springer, Berlin, 1937 (reprinted, Chelsea, New York, 1949).

T. RANSFORD [1] Potential Theory in the Complex Plane. Cambridge Univ. Press, Cambridge, 1995.

F. RIESZ [1] "Sur certains systemes singuliers d'equations integrales", Ann. Sci. Ecole Norm. Sup. (3) 28 (1911) , 33-62. [2] "Sur les fonctions subharmoniques et leur rapport a la theorie du potentiel I", Acta. Math. 48 (1926), 329-343. [3] "Sur les fonctions subharmoniques et leur rapport a la theorie du potentiel II", Acta. Math . 54 (1930), 321-360.

P. J. RIPPON [1] "The boundary cluster sets of subharmonic functions", J. London Math. Soc . (2) 17 (1978), 469- 479. [2] "The hyperplane mean of a positive subharmonic function", J. London Math. Soc. (2) 27 (1983), 76- 84.

W . RUDIN [1] Real and Complex Analysis. Second edition, McGraw-Hill, New York, 1974.

326 References

L. SCHWARTZ [IJ Theorie des Distributions (2 vols.). Hermann, Paris, 1950/l.

H. A. SCHWARZ [IJ "Zur Integration der partiellen Differentialgleichung 82u/8x2 +82u/8y 2 = 0", J. Reine Angew. Math. 74 (1872), 218-253.

E. D. SOLOMENTSEV [IJ "On mean values of subharmonic functions", Izv. Akad. Nauk. Armjan. SSR Ser. Mat. 2 (1967), 139-154 (Russian).

E. M. STEIN and G. WEISS [IJ Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, Princeton, 1971.

G. SZEGO [IJ Orthogonal Polynomials. Third edition, Amer. Math. Soc., Providence, RI, 1967.

E. SZPILRAJN [IJ "Remarques sur les fonctions sousharmoniques", Ann. Math. 34 (1933), 588-594.

W. THOMSON (LORD KELVIN) [IJ "Extraits de deux lettres adressees a M. Liouville", J. Math. Pures Appl. 12 (1847), 256-264.

M. TSUJI [1] Potential Theory in Modern Function Theory. Maruzen, Tokyo, 1959.

F. A. VALENTINE [1] Convex Sets. McGraw-Hill, New York, 1964.

F. VASILESCO [1 J "Sur la continuite du potentiel a travers les masses, et la demonstration d'un lemme de Kellogg", C. R. Acad. Sci. Paris 200 (1935), 1173-1174.

J. L. WALSH [1] "The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions", Bull. Amer. Math. Soc. (2) 35 (1929), 499-544.

K.-O. WIDMAN [1] "Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations", Math. Scand. 21 (1967), 17-37.

N. WIENER, [1] "The Dirichlet problem", J. Math. Phys. MIT 3 (1924), 127-146. [2] "Certain notions in potential theory", J. Math. Phys. MIT 3 (1924), 24-5l. [3] "Note on a paper by O. Perron", J. Math. Phys. MIT 4 (1925), 31-32.

J.-M. G. WU [1] "Comparisons of kernel functions, boundary Harnack principle and relative Fa­tou theorem on Lipschitz domains", Ann. Inst. Fourier (Grenoble) 28, 4 (1978), 147-167.

L. ZALCMAN [IJ "Uniqueness and non-uniqueness for the Radon transform", Bull. London Math. Soc. 14 (1982), 241-245.

References 327

S. ZAREMBA [1] "Sur Ie principe de Dirichlet", Acta Math. 34 (1911), 293-316.

R. L. ZEINSTRA [1] "Some properties of positive superharmonic functions", Compositio Math. 72 (1989),115-120.

Symbol Index

Basic notation is summarized in Notation and Terminology (pp. xiii-xvi). Important notation subsequently introduced in the course of the text is listed below.

[.J 34 ::::: 265 1I·llz 35 (-, -)2 35 L ·Jm 33 QE 157 aN 101 A(y; Ti, r2) 1 {3(y,r) 252 BJ-l 172 C(E) 137 c(E) 154 C(K) 134 c(K) 151 C.(E) 137 c. (E) 153 RCO 151 C'(E) 137 c' (E) 153 cO"(n) 101 Co(n) 100 D 22 a/ane 3 Ll (Laplace's operator) 1 Ll (Martin boundary) 237 Oy 274 dm,N 35 Dp 33 D(y, r) 261 Emf 285 E' 19 j 83 J 83 r 19 G 90 r_ (z, c) 260 r+(z, c) 260

,(y,r) 252 r.,a 116 GJ-l 96 Go 90 GOJ-l 96 1l(A) 226 Hf 165 H f 164

H7 164 1lm 33 1l(n) 1 1l+(n) 9 Hf 164

H~ 164 hy 90 lj 6 If 24 If,xo,T 6 II' 6 II' 22 II',xO,T 6 Jy,m 37 K (Poisson kernel) 22 KXO,T 6 )..' 18,24 Au,E 241 Lu 101 m(o)(E) 156 M(p)(E) 156

(0) mf lim 284 mq,(E) 156 M~p)(E) 156

J-ls (Riesz measure) 101 J-l. (harmonic measure) 172 J-l? 172 M(x,y) 234,236

329

330 Symbol Index

VK 134 v: 274 fj 237 ilM 237 c/> 69 <P f 164 c/>n(X) 69 <p7 164 Pm 33 p!nA) 53

tjJ'I 164 tjJF 164 Qm 33 R~ 129, 241 ~E

Ru 129,241

r(K) 151 'R(il) 165 U 35 S(il) 60 u# 203 Up. 99 U(il) 60 U+(il) 129 Uy 2,68 VN 75 W(Yia,b) 51 x' 8 X 19 Xo (Martin boundary reference point)

234

Index

accessible 62 analytic set 143 axial harmonic 38

balayage 129 barrier 180 best harmonic approximant boundary Harnack principle Bacher's theorem 57

capacitable 137 capacitary distribution 134 capacitary potential 134 capacity 134, 137 Cartan's theorem 141 Chebyshev polynomial 54

58,87 268

Choquet's capacitability theorem 145 Choquet's lemma 312 classical solution of the Dirichlet

problem 163 compactification 235 concave function 72 contraction 142 convex function 72 convex function of'1f;(t) 75 countable subadditivity 139 covering lemma 112

Dini's theorem 147 Dirichlet problem 6 Dirichlet problem for a ball 8 Dirichlet problem for a half-space 26,

43 distributional Laplacian 101 down-directed 15

equicontinuous 16 equilibrium measure 151

Fatou's theorem 117,298 Fatou-Naim-Doob theorem 289

331

fine boundary minimum principle 215 fine topology 197 fine topology on compactified space

215 finite logarithmic length 223 fundamental convergence theorem

146 fundamental harmonic function 1

generalized solution of the Dirichlet problem 165

Green function 90 Green function for a ball 91 Green function for a half-space 92 Green function for the complement of a

ball 96 Green function with pole at 00 151 Greenian open set 89

h-resolutive 277 harmonic approximation 47,226 harmonic function 1 harmonic measure 172 Harnack chain 261 Harnack chain principle 261 Harnack metric 31 Harnack's inequalities 13 Hausdorff dimension 157 Hausdorff cp-measure 156 homogeneous polynomial 29, 33 homogeneous polynomial expansion

42 hyperharmonic function 164 hypoharmonic function 164

inner capacity 137 inner logarithmic capacity 153 inverse of a point 19 inverse of a set 19 irregular boundary point 179

332 Index

Jacobi polynomial 55 Jensen's formula 108 Jensen's inequality 11,309

Kelvin transform 19 kernel function 270

Laplace's equation 1 Laurent expansion 46 least harmonic major ant 79 Lebesgue spine 187 Liouville 5 Lipschitz constant 260 Lipschitz domain 259 Lipschitz function 259 Littlewood's theorem 115,287,300 local Lipschitz constant 260 locally Weierstrass convergent 41 log-capacitable 154 logarithmic capacity 151 logarithmic potential 99 lower PWB solution 164 lower semicontinuous 60 lower semicontinuous regularization

83 Lusin-Privalov theorem 303

majorant 79 Maria-Frostman domination principle

126 Martin boundary 237 Martin compactification 237 Martin kernel 234.. Martin metric 235 Martin representation 250 Martin topology 237 maximum principle for harmonic

functions 5 maximum principle for subharmonic

functions 61 mean value property 3 measure function 156 Mergelyan's theorem 315 metric compactification 236 minimal fine limit 284 minimal fine topology 280 minimal harmonic function 242 minimally thin 279 minimum principle for harmonic

functions 5 minimum principle for superharmonic

functions 62 minor ant 79

Mittag-Leffier theorem 50 multi-index 26 Myrberg's theorem 133

negligible set 177 Newtonian capacity 134 Newtonian potential 96 non-minimal harmonic function 242 non-tangential limit 116,297 normal family 17

outer capacity 137 outer logarithmic capacity 153

peak 203 Phragmen-Lindelof theorems 62 Poincare exterior cone condition 186 Poisson integral 6, 22 Poisson kernel 6, 22 Poisson's equation 111 polar set 123 pole 243 potential 96 potential which determines thinness

204 PWB solution of the Dirichlet problem

165

q.e. 126 quasi-bounded harmonic function 32 quasi-everywhere 126 quasi-Lindelof property 205

Rad6-Stout theorem 313 Radon transform 52 real-analytic 27 reciprocity theorem 126 reduced function 129 reduite 129 reference point 234 reflection principle 8 regular boundary point 179 regular set 179 regularized reduced function 129 removable singularity 8, 50 resolutive boundary function 165 Riesz decomposition theorem 105 Riesz measure 102 Riesz-Herglotz theorem 9 Robin constant 151 Runge's theorem 310

saturated family 79

spherical reflection 31 strong subadditivity 136 subharmonic function 60 subharmonic mean value property 60 superharmonic function 60 superharmonic mean value property

60 swept measure 274

Taylor series 27 thin at 00 215 thin set 199 three circles theorem 311 tract 48 transfer of smallness 31

Index 333

ultraspherical polynomial uniformly equicontinuous universal harmonic function up-directed 15 upper PWB solution 164 upper semicontinuous 60

53 16

58

upper semicontinuous regularization 83

Weierstrass convergent 41 Weyl's lemma 312 Wiener's criterion 217

y-axial 37 y-axial harmonic of degree m 38