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In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim ⁡ A 2 + dim ⁡ A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .

In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove the stability of the isomorphism classes of the nerves of good coverings in the non-collapsing case. In the proof, we need a version of Perelman's fibration theorem, which is also proved in this paper., Minor change basically on the proof of Theorem 1.2 in Section 5

In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $\mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.

This is the third in a series of papers in which we prove a conjecture of Boston and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.

This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$ "ceilings" and $d$ "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.

In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of C is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem. This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of C cannot tend to zero uniformly on an arc of the unit circle. Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let n be so large that v satisfies δn≤ 1. We show that any polynomial in has at least zeros in any disk with center on the unit circle and radius αn.

This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green’s functions for the equations are then random variables. Regularity properties for expectation values of Green’s functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green’s function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares $(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lema\'nczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function., Comment: 35 pages. We added an Appendix concerning upper densities of subsequences of automatic sequences

We prove new results on the stability of the absolutely cont;inuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate (1 + x) 3 -e or if it is continuously differentiable with derivative from the H6lder space C, (R), with any ae > 0. 0. INTRODUCTION In this paper, we study the stability of the absolutely continuous spect;rum of one-dimensional Stark operators under various classes of perturbations. Stark Schr6dinger operators describe behavior of the charged particle in the constant electric field. The absolutely continuous spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather fast rate (see, e.g. [2], [12]). It is therefore interesting to describe the classes of perturbations which preserve the absolutely continuous spectrum of the Stark operators. In the first part of this work, we study perturbations of Stark operators by decaying potentials. This part is inspired by the recent work of Naboko and Pushnitski [14]. The general picture that we prove is very similar to the case of perturbatiorns of free Schr6dinger operators [9]. In accordance with physical intuition, however, the absolutely continuous spectrum is stable under stronger perturbations than in the free case. If in the free case the short range potentials preserving purely absolutely continuous spectrum of the free operator are given by condition (on the power scale) lq(x)l < C(1 + xKl)'E, in the Stark operator case the corresponding condition reads lq(x)l < C(1 + IXD)-2-E. If c is allowed to be zero in the above bounds, imbedded eigenvalues may occur in both cases (see, e.g. [14], [151). Moreover, in both cases if we allow potential to decay slower by an arbitrary function growing to infinity, very rich singular spectrum, such as a dense set of eigenvalues, -may occur (see [13] for the free case and [141 for the Stark case for precise formulation and proofs of these results). The first part of this work draws the parallel further, showing that the absolutely continuous spectrum of Stark operators is preserved under perturbations satisfying lq(x)l < C(1 + Jxl)-3-E, in particular even in the regimes where a dense set of eigenvalues occurs; hence in such cases these eigenvalues are genuinely imbedded. Similar results for the free case were proven in [9], Received by the editors April 14, 1997. 1991 Mathematics Subject Classification. Primary 34L40, 81Q10. ?)1999 American Mathematical Society 243 This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:57:39 UTC All use subject to http://about.jstor.org/terms 244 ALEXANDER KISELEV [10]. Our main strategy of the proof here is similar to that in [9] and [10]: we study the asymptotics of the generalized eigenfunctions and then apply Gilbert-Pearson theory [7] to derive spectral consequences. While the main new tool we introduce in our treatment of Stark operators is the same as in the free case, namely the a.e. convergence of the Fourier-type integral operators, there are some major differences. First of all, the spectral parameter enters the final equations that we study in a different way and this makes analysis more complicated. Secondly, we employ a different method to analyze the asymptotics. Instead of Harris-Lutz asymptotic method we study appropriate Prilfer transform variables, simplifying the overall consideration. In the second part of the work we discuss perturbations by potentials having some additional smoothness properties, but without decay. It turns out that for Stark operators the effects of decay or of additional smoothness of potential on the spectral properties are somewhat similar. It was known for a long time that if a potential perturbing Stark operator has two bounded derivatives the spectrum remains purely absolutely continuous (actually, certain growth of derivatives is also allowed, see Section 2 for details or Walter [21] for the original result). We note that the results similar to Walter's on the preservation of absolutely continuous spectrum were also obtained in [4] by applying different types of technique (Mourre method instead of studying asymptotics of solutions). On the other hand, if the perturbing potential is a sequence of derivatives of 6 functions in integer points on R with certain couplings, the spectrum may turn pure point [3], [5], [1]. In some sense, the 6' interaction is the most singular and least "differentiable" among all available natural perturbations of one-dimensional Schrbdinger operators [11]. Hence we have very different spectral properties on the very opposite sides of the smoothness scale. This work closes part of the gap. We improve the well-known results of Walter [21] concerning the minimal smoothness required for the preservation of the absolutely continuous spectrum and show that in fact existence and minimal smoothness of the first derivative is sufficient to imply absolute continuity of the spectrum. After submitting this paper, the author learned about the work of J. Sahbani [17], where, in particular, the results related to Theorem 2.1 of the present work are proven. Sahbani's results are slightly stronger than Theorem 2.1: the derivative of potential V'(x) is required to be bounded and Dini continuous in order for the absolutely continuous spectrum to be preserved. In addition, he shows that the imbedded singular spectrum in this case may only consist of isolated eigenvalues. The approach employed in [17] is an extension of conjugate operator method. 1. DECAYING PERTURBATIONS Consider a self-adjoint operator Hq defined by the differential expression Hqu =-u" -xu + q(x)u on the L2(-oo, oo). Let us introduce some notation. For the function f C L2 we denote by bf its Fourier transform

In this paper we prove the algorithmic solvability of finite systems of equations over the Q-completion of a torsion-free hyperbolic group. It was recently proved in [1] that finite systems of equations over the Q–completion of a finitely generated free group are algorithmically solvable. In this paper we generalize the results of [1] to the case of the Q–completion G of an arbitrary torsion–free hyperbolic group G with generators d1, . . . , dN . (A detailed definition of the Q–completion of a group can be found in [1].) Main Theorem. Let G be a torsion–free hyperbolic group. Then there exists an algorithm that decides if a given finite system of equations over the Q–completion of G has a solution, and if it does, finds a solution. By a triangular equation we mean an equation with at most three terms. Given a system S of equations in G, we may assume that all equations in S are triangular. Indeed, an arbitrary equation x1 1 x e2 2 . . . x en n = 1 (where xi stands either for a variable or for a constant, ei = ±1) is equivalent to a finite system of triangular equations: x1 1 x e2 2 y −1 1 = 1, y1x e3 3 y −1 2 = 1, . . . , yn−2x en−1 n−1 x en n = 1. Adding a finite number of new variables and equations, we may also assume that S contains only equations with coefficients in G. To achieve this, we replace every constant of the form d m n by a new variable z satisfying the equation z = d. From now on, we fix a finite system S of triangular equations over G with coefficients in G. We will reduce the system S to a finite set of systems in a specific hyperbolic group G ∗ 〈t1, . . . , tψ(m)〉, where the number ψ(m) can be determined effectively given the number m of equations in the original system S. The resulting systems are accompanied by the restriction that some of the variables belong to a certain subgroup G ∗ 〈t1, . . . , ti〉 of G ∗ 〈t1, . . . , tψ(m)〉, i.e. do not contain certain t’s. To each of the systems we can apply the (slightly modified) method of Rips and Sela [4] to see if they are decidable. If none of them is consistent, then the original system S has no solution; if at least one has a solution, then it is possible to find a corresponding solution to the system S. Suppose that the system has a solution in G; let {X1, . . . , XL} be a solution with the minimal possible number of roots. Since the solution belongs to G, it is contained in a certain group K, obtained from G by adding finitely many roots. It is the union of a chain of subgroups Hi defined as follows. Let G = H0. Received by the editors August 14, 1996. 1991 Mathematics Subject Classification. Primary 20E05, 20F10. The first author was supported by grants from NSERC and FCAR. The third author was supported by the NSF grant DMS-9103098. c ©1999 American Mathematical Society 2961 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2962 O. KHARLAMPOVICH, E. LIOUTIKOVA, AND A. MYASNIKOV Step 1. Consider pairwise nonconjugated cyclically minimal primitive elements u1, . . . , uk1 ∈ G, |u1| ≤ . . . ≤ |uk1 | (here |u| denote the length of u in G), and add roots t1, . . . , tk1 , such that uj = t sj j . (Notice that ui+1 does not become a proper power after we add roots t1, . . . , ti.) The corresponding groups are denoted by H1, . . . , Hk1 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Step 2. Consider pairwise nonconjugated primitive elements uk1+1, . . . , uk2 ∈ H1, cyclically reduced in the amalgamated product, each having the reduced form u = t1 1 c1 . . . t αk 1 ck, where αi 6= 0, αi ∈ Z, ci ∈ G, |uk1+1|H1 ≤ . . . ≤ |uk2 |H1 ; and add roots tk1+1, . . . , tk2 , such that uj = t sj j , to the group Hk1 . The corresponding groups are denoted by Hk1+1, . . . , Hk2 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Step i+1. Suppose that H1, . . . , Hki have been constructed. Consider pairwise nonconjugated primitive elements uki+1, . . . , uki+1 ∈ Hi, cyclically reduced in the amalgamated product, each having the reduced form t1 i c1 . . . tk i ck, where αi 6= 0, αi ∈ Z, ci ∈ Hi−1 ( ck is not a power of ui, because the elements are cyclically reduced), |uki+1|Hi ≤ . . . ≤ |uki+1 |Hi , and add roots tki+1, . . . , tki+1 , such that uj = t sj j , to the group Hki . The corresponding groups are denoted by Hki+1, . . . , Hki+1 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Finally, for some number i one has K = Hki+1 . The group Hki+1 is called the group at level i, corresponding to the sequence u1, . . . , uki+1 . The group Hi will be called the group of rank i. We also order the set of tj ’s: tk < tl if k < l. Let Fi be the free group with the same generating set as Hi, i = 1, . . . , k1, and let β : Hi → Fi be a section, i.e. a mapping of sets such that π ◦ β = id. For our purposes, β has to be of specific form; we will define it explicitly in section 2. For X ∈ Hi we will call β(X) the canonical representative of X . Definition 1. By an equational diagram over a group Hi we mean an equation X1 . . . Xn = 1 together with a solution A1, . . . , An and a diagram over Hi having β(A1) . . . β(An) as its boundary label. An equational triangle is an equational diagram corresponding to a triangular equation. A system of equational diagrams is a system of equations together with the system of diagrams such that the solution associated to each equational diagram is a solution of the whole system. A free equational diagram is an equational diagram with no cells; the corresponding equations are called free equations. If the level i of the group Hki+1 is greater than zero, we can use the method of [1] to reduce each equation of the system S to a bounded number of free equations and at most one equation over the group in the previous rank. We can proceed in the same way until we reach level 0 (i.e. the group Hk1), for which we need to develop a separate method. In what follows we will denote Hk1 by H . 1. Some properties of the Cayley graph of H We will frequently use the following lemma: Lemma 1. ([3, Lemma 1.5]) For each geodesic triangle [x1, x2, x3] in a δ-space, there are points yi ∈ [xi−1, xi+1] (indices are considered modulo 3) such that d(xi, yi−1) = d(xi, yi+1) = (xi−1 · xi+1)xi , License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE Q–COMPLETION OF A TORSION–FREE HYPERBOLIC GROUP 2963

Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain tree-like obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. We prove that if H has tree duality then the H-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the X-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the H-colouring problem is NP -complete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial H-colouring problems. If P 6= NP , then no oriented triad H with an NP -complete H-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NP -complete Hcolouring problems given in the companion paper has tree duality.

Consider a semisimple connected Lie group G with an aiSine symmetric space X. We study abstractly the intertwining operators from the discrete series of X into representations with reproducing kernel and in particular, into the discrete series of G; each such is given by a convolution with an analytic function. For X of Hermitian type, we consider the holomorphic discrete series of X and here derive very explicit formulas for the intertwining operators. As a corollary we get a multiplicity one result for the series in question. 0. INTRODUCTION The purpose of this work is twofold. The first is to study abstractly the intertwining operators between discrete series representations of an affine symmetric space X and discrete series of the corresponding semisimple group. The second is to obtain very explicit formulas for certain discrete representations on X in effect giving the analogue of Harish-Chandra's original construction of the holomorphic discrete series in the group case. Not surprisingly, these two problems are highly connected. Thus Theorem 5.4 below gives the formula for the lowest K-type (the Flensted-Jensen function) on X, whereas in Theorem 7.3 essentially the same function gives the intertwining operator. Much of our analysis builds on the framework in [14], but otherwise we have tried to make the paper reasonably self-contained. This is, in particular, the case in §1, §2, and §4, where we abstract the properties of our intertwining operators with an eye towards later applications. We shall always let G be a connected, noncompact semisimple Lie group such that the associated Riemannian symmetric space is a bounded symmetric domain D c Cn . The best known and most interesting discrete series representations of G are the holomorphic discrete series representations. As explained above, one of our aims in this paper is to generalize this construction to affine symmetric spaces of Hermitian type and relate those with other well-known representations. Received by the editors September 23, 1988 and, in revised form, June 27, 1989. 1980 Mathematics Subject Classiffcation (1985 Rerision). Primary 22E46; Secondary 43A85. (D 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

In this paper the Kobayashi metric on a domain in Cn is used to define a new function space. Elements of this space belong to a nonisotropic Lipschitz class. It is proved that if f is holomorphic on the domain and in the classical Lipschitz space A. then in fact f is in the new function space. The result contains the original result of Stein on this subject and provides the optimal result adapted to any domain. In particular, it recovers the Hartogs extension phenomenon in the category of Lipschitz spaces. This paper is part of a program to work out the function theory-especially the harmonic analysis-of domains in Cn in terms of invariant metrics and related constructs. The point is that, whereas in one complex dimension the principal objects of study are functions, it turns out that in several complex dimensions the principal objects of study should be domains. The correct context in which to perceive the differences between domains seems to be that of metric geometry. The present paper is concerned with Lipschitz spaces, and draws its inspiration from [ST]. In [ST], Stein announced that holomorphic Lipschitz a functions on bounded C2 domains in Cn, n > 1, are in fact Lipschitz 2a in complex tangential directions; a detailed proof appears, for instance, in [KR3]. It turns out that this result is optimal only for strongly pseudoconvex domains (see [KR6] for improved results on some other domains), and our purpose here is to find a language in which to formulate the best result for any domain. The Eisenman-Kobayashi metric and volume construction provides the most natural language. For related work using the Eisenman-Kobayashi ideas, see [KR1, KR2, KM, GK]. A second point of the present work is to eliminate the arbitrary distinction between normal and tangential directions. All directions should be treated equally, and the metric should read off the variation in smoothness. An interesting by-product of the work is that it rediscovers the Hartogs extension phenomenon (or "Kugelsatz"-see [KR3]) in the category of Lipschitz functions. This feature is explored in ??2 and 4. ? 1 contains basic definitions and terminology, including the definition of a new Lipschitz class on domains in Cn. It also contains the statement of our Received by the editors October 12, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 32A10; Secondary 32A30. Work supported in part by the National Science Foundation. ? 1990 American Mathematical Society 0002-9947/90 $1.00 + $.25 per page

Multiplication by the independent variable on H 2 ( R ) {H^2}(R) for R R a bounded open region in the complex plane C \mathbb {C} is a subnormal operator. This paper characterizes its minimal normal extension N N . Any normal operator is determined by a scalar-valued spectral measure and a multiplicity function. It is a consequence of some standard operator theory that a scalar-valued spectral measure for N N is harmonic measure for R R , ω \omega . This paper investigates the multiplicity function m m for N N . It is shown that m m is bounded above by two ω \omega -a.e., and necessary and sufficient conditions are given for m m to attain this upper bound on a set of positive harmonic measure. Examples are given which indicate the relationship between N N and the boundary of R R .

In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex. One consequence of our general result is the following: for any dimension there exists some ϵ > 0 \epsilon > 0 so that if the squeezing function on a smoothly bounded convex domain is greater than 1 − ϵ 1-\epsilon outside a compact set, then the domain is strongly pseudoconvex (and hence the squeezing function limits to one on the boundary). Another consequence is the following: for any dimension d d there exists some ϵ > 0 \epsilon > 0 so that if the holomorphic sectional curvature of the Bergman metric on a smoothly bounded convex domain is within ϵ \epsilon of − 4 / ( d + 1 ) -4/(d+1) outside a compact set, then the domain is strongly pseudoconvex (and hence the holomorphic sectional curvature limits to − 4 / ( d + 1 ) -4/(d+1) on the boundary).

Sidorenko’s conjecture states that for every bipartite graphH on {1, · · · , k} ∫ ∏ (i,j)∈E(H) h(xi, yj)dμ |V (H)| ≥ (∫ h(x, y) dμ )|E(H)| holds, where μ is the Lebesgue measure on [0, 1] and h is a bounded, nonnegative, symmetric, measurable function on [0, 1]2. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph H to a graph G is asymptotically at least the expected number of homomorphisms from H to the Erdős-Renyi random graph with the same expected edge density as G. In this paper, we present two approaches to the conjecture. First, we introduce the notion of tree-arrangeability, where a bipartite graph H with bipartition A∪B is tree-arrangeable if neighborhoods of vertices in A have a certain tree-like structure. We show that Sidorenko’s conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko’s conjecture holds if there are two vertices a1, a2 in A such that each vertex a ∈ A satisfies N(a) ⊆ N(a1) or N(a) ⊆ N(a2), and also implies a recent result of Conlon, Fox, and Sudakov [3]. Second, if T is a tree and H is a bipartite graph satisfying Sidorenko’s conjecture, then it is shown that the Cartesian product T H of T and H also satisfies Sidorenko’s conjecture. This result implies that, for all d ≥ 2, the d-dimensional grid with arbitrary side lengths satisfies Sidorenko’s conjecture.

The paper is a comprehensive study of Lp and Schauder estimates for higher-order divergence type parabolic systems with discontinuous coefficients on a half space and cylindrical domains with the conormal derivative boundary conditions. For the Lp estimates, we assume that the leading coefficients are only bounded and measurable in the t variable and have vanishing mean oscillations (VMOx) with respect to x. We also prove the Schauder estimates in two situations: the coefficients are Holder continuous only in the x variable; the coefficients are Holder continuous in the t variable as well on the lateral boundary.

The Szego projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of Diederich, Fornxess, and Catlin. It preserves local smoothness near boundary points of finite type. The Bergman projection, being linked to the 8 problem, plays an important role in several complex variables, as demonstrated by recent applications to the boundary behavior of holomorphic maps [7,16]. The Szego projection is interesting both by analogy and by its connection with the ab problem and hence with unsolvable equations of Lewy type. While in strictly pseudoconvex domains the Bergman and Szego projections are both well understood, in general much more is known about the Bergman projection. This is no surprise, for the theory of the 8 problem is further advanced than the theory of the 8b problem. Nonetheless I believe that the Szego projection ought not be any more mysterious than the Bergman projection. This paper is intended to support this contention. The most important domains for which the Bergman projection is known to be regular are domains with a good a-Neumann operator. Here it is shown that in such domains the Szego projection also is regular. More precisely, in smooth bounded pseudoconvex domains satisfying Catlin's property (P) (the most general condition presently known that guarantees global regularity of the Neumann operator) the Szego projection is shown to be globally regular (Theorem 4.1). Moreover at boundary points of finite type the Szego projection is locally regular (Theorem 5.1). In both cases the Szego projection exactly preserves the degree of differentiability, measured in Sobolev norms. The method of proof is not the obvious one of directly generalizing from the strictly peudoconvex case, for there are serious obstacles to such a generalization. For instance the approach of analysis on the Heisenberg group, used by Phong and Stein [29] to derive estimates for both the Bergman and Szego projections, cannot yet be extended because the theory of singular integrals on weakly pseudoconvex surfaces is still in its early stages [28,15]. Similarly one does not know how to carry over the Fourier integral operator approach of Boutet de Monvel and Sjostrand [10] to the weakly pseudoconvex case. Explicit construction of Cauchy-Fantappie forms, Received by the editors March 11, 1986. Paper presented April 11, 1986 to the 826th meeting of the American Mathematical Society, Indianapolis, special session on Several Complex Variables. 1980 Mathematics Subject Classification (1985 Revision). Primary 32A25, 32H10. Partially supported by National Science Foundation grants MCS-8201063 and DMS-8501758. ?1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

This paper introduces the notion of weak subordination: If F F and G G are meromorphic in the unit disc U \mathcal {U} , then F F is weakly subordinate to G G , written F > G F > G , provided there exist analytic functions ϕ \phi and ω : U → U \omega :\mathcal {U} \to \mathcal {U} , with ϕ \phi an inner function, so that F ∘ ϕ = G ∘ ω F \circ \phi = G \circ \omega . A class X \mathcal {X} of meromorphic functions is termed stable if F ≺ w G F \stackrel {w}{\prec } G and G ∈ X ⇒ F ∈ X G \in \mathcal {X} \Rightarrow F \in \mathcal {X} . The motivation is recent work of Burkholder which relates the growth of a function with its range and boundary values. Assume F F and G G are meromorphic and G G has nontangential limits, a.e. Assume further that F ( U ) ∩ G ( U ) ≠ ∅ F(\mathcal {U}) \cap G(\mathcal {U}) \ne \emptyset and G ( e i θ ) ∉ F ( U ) G({e^{i\theta }}) \notin F(\mathcal {U}) , a.e. This is denoted by F > G F > G . Burkholder proved for several classes X \mathcal {X} that ( ( ∗ ) ) F > G and G ∈ X ⇒ F ∈ X . \begin{equation}\tag {$(\ast )$}F > G \qquad {\text {and}}\quad G \in \mathcal {X} \Rightarrow F \in \mathcal {X}.\end{equation} The main result of this paper is the Theorem: F > G ⇒ F ≺ w G F > G \Rightarrow F{ \prec ^w}G . In particular, implication (*) holds for all stable classes X \mathcal {X} . The paper goes on to study various stable classes, which include BMOA, H p {H^p} , 0 > p ⩽ ∞ 0 > p \leqslant \infty , N ∗ {N_{\ast }} , the space of functions of bounded characteristic, and the M Φ {M^\Phi } spaces introduced by Burkholder. VMOA and the Bloch functions are examples of classes which are not stable.

This paper continues the study of the smoothness properties of (real) topological linear spaces. First, the smoothness results previously obtained about various important classes of locally convex spaces, such as Schwartz spaces, are improved. Then, following the ideas of Bonic and Frampton, we use these results to give sufficient conditions for the existence of smooth partitions of unity on manifolds modelled on topological linear spaces. 1. Preliminaries. In order to make this paper self-contained we include the definition of the two types of derivative used and the definition of an 8-category. We will employ the definitions of the derivative in topological linear spaces investigated in detail by Averbukh and Smolyanov [2], [31. See also [14], [15] and [191. Let TLS denote the class of all Hausdorff topological linear spaces over the real field R. Let 2 1(E, F) = 2(E, F) denote the set of all continuous linear maps from E into F, where E, F e TLS. We define by induction p(E, F) = 2(E, p 1(E, F)). Each ?1p(E, F) is given the topology of uniform convergence on bounded subsets of E. If X is a topological space, W(X) will denote the class of all open subsets of X. Let f: U -' V, where U e ((E), V E C(F), E e TLS and F c TLS. Then we say f is Fre6cbet differentiable at x e U, if there exists u e S(E, F) such that for each bounded subset B of E and for each o-neighbourhood W in F, there exists 8 > 0 such that f(x + th) f(x) u * th E tW, whenever b E B and ItL

In this paper we consider degenerate parabolic equations of the form ( ∗ ) ({\ast }) \[ β ( u ) t − div ⁡ A ( x , t , u , u x ) + B ( x , t , u , u x ) ∋ 0 \beta {(u)_t} - \operatorname {div} A(x,\,t,\,u,\,{u_x}) + B(x,\,t,\,u,\,{u_x}) \ni 0 \] where A A and B B are, respectively, vector and scalar valued Baire functions defined on U × R 1 × R n U \times {R^1} \times {R^n} , where U U is an open subset of R n + 1 ( x , t ) {R^{n + 1}}(x,\,t) . The functions A A and B B are subject to natural structural inequalities. Sufficiently general conditions are allowed on the relation β ⊂ R 1 × R 1 \beta \subset {R^1} \times {R^1} so that the porus medium equation and the model for the two-phase Stefan problem can be considered. The main result of the paper is that weak solutions of ( ∗ ) ({\ast }) are continuous throughout U U . In the event that U = Ω × ( 0 , T ) U = \Omega \times (0,\,T) where Ω \Omega is an open set of R n {R^n} , it is also shown that a weak solution is continuous at ( x 0 , t 0 ) ∈ ∂ Ω × ( 0 , T ) ({x_0},{t_0}) \in \partial \Omega \times (0,\,T) provided x 0 {x_0} is a regular point for the Laplacian on Ω \Omega .

We provide a general setting for studying admissible and singular translates of measures on linear spaces. We apply our results to measures on D(0, 1). Further, we show that in many cases convex, balanced, bounded, and complete subsets of the admissible translates are compact. In addition, we generalize Sudakov's theorem on the characterization of certain quasi-invariant sets to separable reflexive spaces which have the Central Limit Property. Introduction. It is the purpose of this paper to provide a general setting for studying admissible and singular translates of measures on linear spaces. In (9) Sudakov studied admissible translates of measures on a separable Hilbert space by exploiting Hilbert-Schmidt operators. These operators were also con- sidered by Dudley (1) but are not available in a more general context. However, by considering maps between spaces one can obtain analogues of those operators. Based on our results we would conjecture the following (cf. Corollary 1.8 below): Let AT be a locally convex linear topological space. Let p be a positive Borel measure on X for which the set of admissible translates forms a subspace of X. Then there exists auxiliary spaces Ya and "compact" maps Ta: Ya—+ X, such that the set of admissible translates of p is given by "lara(ya). The organization of the paper is as follows. The first section contains gen- eral theorems in which properties of certain auxiliary spaces and maps are dis- cussed. In §2 we show that in many cases a convex, balanced, complete, and bounded subset of the admissible translates is compact. In §3 we extend Sud- akov's theorem on the characterization of certain quasi-invariant sets to separable reflexive Banach spaces with the Central Limit Property. These include / and separable ?p(m)-spaces, 2 < p < °°. In §4 we apply the techniques of §§1 and 2 to obtain similar results in the nontopological vector space D(0, 1). To the best of our knowledge these are the first general results on admissible translates for measures on D(0, 1).

In this paper several results on analytic diameters and analytic centers are obtained. We show that the extremal function for analytic diameter is unique and that there exist compact sets with many analytic centers. We answer negatively several problems posed by F. Miinsker. Introduction. In the theory of functions in the complex plane, many capacitytype set functions are defined in order to measure the size of sets. The notions of analytic diameter and analytic center were first introduced by Vitushkin [13], [14] in the way of developing his beautiful theory of rational approximation. Recently F. Minsker [9] obtained their several properties and posed in the last part of his paper [9, p. 93] open problems on them. In this paper we investigate certain properties of the extremal function for the problem of analytic diameters and construct instructive examples, most of which offer negative answers to Minsker's problems. In ? 1 we list the definitions and the notation which we use throughout this paper. In ?2 we state preliminary known results as a series of lemmas. In ?3 we derive a property of the extremal function, and as a corollary we show that the extremal function is unique. In ?4 we consider the case where the compact set consists of finitely many continua. In ?5 we give examples, most of which serve as counterexamples to Minsker's problems. 1. Definition and notation. Let K be a compact set in the complex plane, and let D = D(K) denote the unbounded component of the complement KC of K with respect to the Riemann sphere. By AB(D) we denote the Banach space of all bounded analytic functions in D, normed by the uniform norm 1 in D, that is 0110 = sup{ff(z)1: z E D} (1) for anyf E AB(D). Let eg(K) = {f E AB(D): IItIOO < l,f(oo) = 0). Any function which belongs to Gi(K) is often called an admissible function for K. The analytic capacity y(K) of K is defined by y(K) = sup{If'(oo)I:f E zie(K)} (2) It is well known that there exists a unique extremal function fo J e6(K) with fo(oo) = y(K), which is called the Ahlfors function of K [1]-[4], [7], [8]. Received by the editors January 11, 1980. A MS (MOS) subject classifications (1970). Primary 30A34, 30A44. ( 1981 American Mathematical Society 0002-9947/81/0000-041 4/$03. 50

"M-subharmonic" functions are defined in the unit ball of cn. Their basic properties are developed, leading to the following generalization of a theorem of Littlewood: An M-subharmonic function such that its restrictions to spheres centered at the origin are bounded in L1 must have radial limits almost everywhere on the unit sphere. The purpose of the present paper is to define and develop the basic properties of the class of "M-subharmonic" functions in the unit ball of Cn (see Definition 1.15). This is the version of "subharmonic" associated with "M-harmonic" (as in [RU]). The paper is divided into four sections: §1 consists of preliminary definitions and a few basic results, concluding with a characterization of those M-subharmonic functions which possess " least M-harmonic majorants" (Theorems 1.22 and 1.23). §2 proves the Riesz decomposition theorem for M-subharmonic functions with a least M-harmonic majorant (Theorem 2.16). §3 contains our main theorem: An M-subharmonic function satisfying the appropriate growth condition has radial limits almost everywhere (Theorem 3.1). (In the case n = 1 this was proved by Littlewood [LW].) §4 shows, via a very simple example, that Theorem 3.1 does not extend to admissible limits, nor even to nontangential limits, nor to any sort of approach strictly more general than radial. Note however that an M-subharmonic function satisfying stronger growth conditions than those in Theorem 3.1 must have admissible limits a.e. (see [CS]). This paper is a revised version of the author's thesis written at the University of Wisconsin under the direction of Professor Walter Rudin. It is a pleasure to record here my gratitude for many stimulating discussions on this and various other topics. 1. Preliminaries. Notations are for the most part as in [RU]. Throughout this paper n will be a fixed positive integer, usually n > 1. (Our results are all standard when n = 1; including the case n = 1 would occasionally necessitate dividing our notation, if not our argument, into two cases.) For z, w E Cn set

This paper will prove that a bounded homogeneous domain is symmetric if and only if, in the Bergman metric, all sectional curvatures are nonpositive. Introduction. It is well known that a bounded symmetric domain has nonpositive sectional curvature in the Bergman metric. This paper is devoted to the converse, namely, that a bounded homogeneous domain (equivalently, homogeneous Siegel domain) which has nonpositive sectional curvature in the Bergman metric must be a symmetric domain. The proof uses the techniques of normal j-algebras [13], as well as results of Vinberg [16, 17], the first author [1-3], and, indirectly, Dorfmeister [5, 6]. The body of the paper is in two sections. In the first, we derive certain relations between the dimensions of the root spaces in an irreducible normal j-algebra, which, in the presence of the additional curvature assumption, show the equality of the dimensions of all root spaces corresponding to certain roots for which no multiple is a root and the equality of the dimensions of all root spaces corresponding to roots which are half of another root. In the second section, we show that the first dimension result implies that the cone of the corresponding Siegel domain is self-dual while the two dimension results together imply that the domain is quasi-symmetric. The proof is then finished because [3] a quasi-symmetric domain with nonpositive sectional curvature in the Bergman metric is known to be symmetric. We remark that some of the subsidiary results mentioned above are not essentially new. However, for consistency of presentation, it is necessary to have them in the language of normal jalgebras and for that reason they are included. There is also overlap here with work of Zelow (Lundquist) [21]. 1. Throughout this paper, (s, j) will denote a normal j-algebra with admissible form w). This means that s is a finite dimensional real split solvable Lie algebra with almost complex structure j: s -) s such that [X, Y] + j[jX, Y] + j[X, jY] = [jX, jY] and w is a linear form on s such that the bilinear form = o][jX, Y] is symmetric, positive-definite, and j-invariant (note that the assumption that s is real split was incorrectly omitted in [2]). Let n = [s, s] and let a be the orthogonal complement of n in s. By the basic structure theorem of Pyatetskii-Shapiro [13], a is a commutative Received by the editors July 13, 1981. 1980 Mathematics Subiect Classification. Primary 32MI0, 53C30. ' 1983 American Mathematical Society ? 0002-9947/82/000-0759/$09.00

Recently, in their remarkable paper Critical point theory for indefinite functionals, V. Benci and P. Rabinowitz gave a direct approach-avoiding finitedimensional approximationsto the existence theory for critical points of indefinite functionals. In this paper we develop under weaker hypotheses a simpler but more general theory for such problems. In the second part of the paper the abstract results are applied to a class of resonance problems of the Landesman and Lazer type, and moreover they are illustrated by an application to a wave equation problem. 0. Introduction. This paper is devoted to the critical point theory of strongly indefinite functionals in a real Hilbert space H. Here a strongly indefinite functional is a Cl-map H -* R which is neither bounded from above or below not even modulo a finite-dimensional subspace. Such functionals arise for example in the study of periodic solutions of the one-dimensional wave equation, in the study of periodic solutions of Hamiltonian systems of ordinary differential equations, in the existence theory for systems of elliptic equations, etc. To find the critical points of strongly indefinite functionals different methods were used. For example, Rabinowitz studied the restrictions of the functional to finite-dimensional subspaces, [1, 2], and then he tried to find the critical points of the original problem by passing to the limit for a suitable subsequence of the critical points of the restricted problems. Amann [3] reduced the indefinite functionals by means of a saddlepoint-reduction to a finite-dimensional problem. Of course, this is only possible under some strong hypotheses. Under certain assumptions the functional can be replaced by a dual variational problem, having some better properties. (See for example [4 or 5].) The first direct approach was given by Benci and Rabinowitz [6]. The essential in their paper is the observation that more detailed information about the special form of the deformations (a substitute for the gradient flow) associated to the functional is needed. They constructed deformations having special representations by solving appropriate differential equations approximately by time discretisation. This method has some disadvantages because one needs strong hypotheses in order to get the approximations uniformly on bounded sets. A main part of our paper will be the construction of flows and deformations having special representations by solving appropriate differential equations exactly Received by the editors March 30, 1981 and, in revised form, November 30, 1981. 1980 Mathematics Subject Classification. Primary 58E05; Secondary 47H25, 35L05.

In this paper we develop algebraic and geometric conditions which imply that a given proper Dehn map can be replaced by an embedding. The embedding, whose existence is implied by our theorem, retains most of the algebraic and geometric properties required in the original proper Dehn map. I. Preliminaries. In this paper all spaces will be simplicial complexes and all maps will be piecewise linear. The boundary, closure, and interior of a subset X of a space Y will be denoted by bd (X), cl (X) and int (X), respectively. A proper map f, taking a space X into a space Y, is a map such that f bd (X) = bd (Y) n f(X). A Dehn map f, taking a compact, bounded, two-manifold F into a 3-manifold M, is a map such that (1) bd (F)=f-lfbd (F); (2) fIbd (F) is a homeomorphism. All unlabeled homomorphisms are natural maps induced by inclusion. We shall consider the problem of replacing a proper Dehn map of a compact, connected, orientable, bounded surface F into a compact, (necessarily) bounded, irreducible, orientable 3-manifold M with an embedding. As will be pointed out later, the requirement of orientability may not be strictly necessary. The author would like to thank J. Gross for a number of helpful conversations. In [3] Papakyriakopoulos proved Dehn's lemma, that is, a proper Dehn map f of a disk 9 into a 3-manifold M can be replaced by a proper embedding g of 9 into M such that g bd (9) =f bd (9). A natural question to consider is the following: Let f: F -* M be a proper Dehn map of a compact, connected, bounded, orientable surface F into a compact 3-manifold M. Does there always exist a proper embedding g: F1 -? M such that (1) g(bd (F1))'f(bd (F)); (2) genus (F1)

Introduction. The concept of isotopy, recently introduced(') by A. A. Albert in connection with the theory of linear non-associative algebras, appears to have its value in the theory of quasigroups. Conversely, the author has been able to use quasigroups(2) in the study of linear non-associative algebras. The present paper is primarily intended as an illustration of the usefulness of isotopy in quasigroup-theory and as groundwork for a later paper on algebras, but is bounded by neither of these aspects. The first two sections are devoted to the basic definitions of quasigroup and isotopy, along with some elementary remarks and two fundamental theorems due to Albert. Then there is initiated a study of special types of quasigroup, beginning with quasigroups with the inverse property (I. P. quasigroups). A system Q of elements a, b, * * * is called an I. P. quasigroup if it possesses a single-valued binary operation ab and there exist two one-to-one reversible mappings L and R of Q on itself such that

Introduction. Let D be an arbitrary domain on the Riemann sphere (that is, D is an open connected set on the sphere) and let p be a point of D. Let H '(D) denote the space of bounded analytic functions on D. We consider the problem of maximizing If'(p)l for fin H-(D) with If(z)lj< 1 for all z in D, and f(p) =0. In the case when D is the unit disc U and p =0, this is Schwarz's lemma and the solution is classical: the maximum is one and if If'(0) = 1, then f(z) = eiOz for some real constant 0. In the general case, the theory of normal families assures us that there is at least one fo in H (D) with foll? 1 I and fo(p) ? f'(p)I for all f in H (D) bounded by 1 which vanish at p. We call any such function extremal. Theorem 1 of this paper provides an elementary proof that there is only one extremal function. We thus add to our problem the question of investigating the properties of the extremal function. In the case when D is bounded by a finite number of disjoint analytic simple closed curves, the properties of the extremal function have been studied extensively (see [1] and [7]). In this case, the extremal function is known to be analytic over the boundary and to have modulus one there. We show that this is a local property and carries over to all points in the boundary of D where the boundary is sufficiently well behaved, no matter what the remainder of the boundary of D is like. In ?1 we also show that the extremal function is closely related to the problem of removable singularities for bounded analytic functions. In ?2 we discuss a related extremal problem and show that the properties of its solution imply that on many planar domains D, each function analytic and bounded by 1 on D may be approximated uniformly on compact subsets of D by a sequence of inner functions. We also briefly discuss extensions of the results to open Riemann surfaces. Finally, some remarks on previous work on Schwarz's lemma are in order. Carleson [3, pp. 78-82] and Havinson [8, Theorem 9] have previously shown that the extremal function is unique. Our proof differs substantially from each of theirs and is considerably more elementary. Furthermore, it does not make use of any of the results from the finitely-connected case. Also, our Theorems 2 and 3 and a weaker version of Theorem 4 have previously been proved by Havinson [8, Theorems 22, 28, and 20]; however, as in the case of Theorem 1, the proofs given here are appreciably simpler and more elementary. Indeed, part of the motivation for this paper was to find simpler and more transparent proofs of these results. In particular, it is only in the proof of Theorem 5, which describes the behavior

Introduction. In a recent paper [17] we obtained a spectral representation for the bounded normal operator solutions of a certain functional equation whose special cases include the semigroup equation (U,,, = U,LUV) as well as many others. In this note we continue our study of the abstract functional equation expressed in terms of algebras of measures. We shall consider therefore a mapping z -* UZ of a locally compact space Z into the set of unbounded normal operators on a Hilbert space. Since the operators are not necessarily bounded, the conditions imposed on the mapping z -* Uz vary somewhat from the conditions in the bounded case. The main result (Theorem 1) which we obtain here is a spectral representation analogous to the result valid in the bounded case [17; 18, Theorem 2, p. 8]. The methods which we use are suggested by the papers [12] and [13] of C. Ionescu Tulcea. For the general theory and terminology of unbounded operators we refer the reader to the works of E. Hille and R. Phillips [10], C. Ionescu Tulcea [11], F. Riesz and B. Sz.-Nagy [21]. Spectral representations for semigroups of unbounded operators are also discussed in A. Devinatz [3], A. Devinatz and A. E. Nussbaum [4; 5], R. Getoor [7], C. Ionescu Tulcea and A. Simon [14] and A. E. Nussbaum [19]. Theorem 1 of this paper contains the main part of Theorem 1 in [14] (and henceforth certain of the results given in [3; 4; 5; 7; 19]). The setting of the problem in algebras of measures is given in ??1, 2, and 5. Spectral families of Radon measures and direct sums thereof are summarized in ??3 and 4. The main result is stated and proved in ?6. In ?8 we apply the main theorem to derive a corollary which in the bounded case extends a result of S. Kurepa [16].

Introduction. In the preceding paper [8] we proved that if all harmonic functions in n-dimensional bounded domain D satisfy the mean value property (MVP) with respect to some point P and a given density function ,t (volumedensity, surface-density, etc.) then, under some simple assumptions on ,, D must necessarily be a ball with center P. In the present paper we are interested in studying the MVP from a complementary point of view, namely, we are interested in finding conditions on ,u (u>?0) under which the MVP holds at most for a finite number of linearly independent functions. The MVP is meant to be

Introduction. Problems of great interest in real-variable theory are the analysis of the structure of a function whose derivative is finite, and the determination of a function when its derivative is given and is finite at each point. The study of these problems has led to a theory of non-absolutely convergent integrals.t The corresponding problems when the derivative in question is with respect to a function of bounded variation have received little attention, and it is with these that the present paper is mainly concerned. In this case we find that there is involved a theory of non-absolutely convergent integrals with respect to a function of bounded variation. Lebesguet has given these questions some consideration. His results depend, in part, on a transformation which changes integration with respect to a function into ordinary integration. By means of this transformation very general results are easily obtained. But for dealing with some particular situations it is not altogether suitable; for example, a discussion of integral equations which involve integration with respect to a function of bounded variation. For this reason we have, in the present work, adhered to methods which are direct. Incidental to our main purpose is a study of the derivative of a function F(x) with respect to a function of bounded variation a(x). Here, too, we have been preceded by others, chiefly Daniell.? In the Transactions paper Daniell concerns himself with the central derivative with respect to a

Suppose C is a compact subset of the plane having a piecewise smooth boundary AC. Let F(r, 0) be the Fourier transform, in polar coordinates, of the indicator function of the set C, where by the indicator function of C, we mean the function whose value on C is 1, and whose value on the complement of C is 0. In ?1 of this paper, we shall describe some relationships between geometric properties of C, and the asymptotic behavior of F(r, 0) as r -x- 00. In ?2, we shall give applications of the results of ?1 to some questions in the geometry of numbers. 1. If AC is sufficiently smooth, and has everywhere positive Gaussian curvature, it is known that the function 'D(6) = sup, r312IF(r, 0)1 is bounded on S' (cf. [1]). If AC has points of zero curvature, this need no longer be true (cf. [3]). The following, however, remains true: THEOREM 1. If AC is of class Cn + 3, for some integer n 1, and if the Gaussian curvature of AC is nonzero at all points of AC, with the possible exception of a finite set, at each point of which the tangent line has contact of order 1. Moreover, 'D(6) is always bounded, except in neighborhoods of those points of S' which, regarded as vectors, correspond to exterior or interior normals to AC at points of zero curvature. In a neighborhood of such a point 006 'D(6) is bounded by a multiple of [dist (6, 00)] -n - 1)/2n , where dist (6, 00) is the length of the smaller arc of S' connecting 0 and 6o, and nj is the largest order of contact which can occur between AC and its tangent line, at those points of AC at which the exterior normal is either 00 or- o REMARK. Theorem 1 has analogues in higher dimensions. I shall show in another paper, by different methods, that if C is a compact convex subset of Rn, whose boundary is analytic, and if F(r, 0) is the Fourier transform, in polar coordinates, of the indicator function of the set C, then supr r(n + 1)/21 F(r, 0)1 is of class LP on Sn1, for some p>2. If C is a polygon, the estimates are of a quite different character. THEOREM 2. Suppose C is a polygon. Then

The importance and usefulness of Taylor's theorem need not be dwelt upon here. We are interested in it for functions whose arguments and functional values belong to abstract spaces of the Frechet type. Consequently no Rolle's theorem can be even stated, and the proofs will be somewhat different from the usual proofs in the theory of numerically-valued functions. It is also to be expected that a slight strengthening of hypotheses will be required. However, it is not necessary to assume the existence of a uniformly continuous nth differential, as was done in the first announcement of these results.: It is sufficient that the function should have an nth variation (in the sense of Gateaux), with certain limitations on its discontinuities. The functions we discuss will be one-valued functions whose arguments x and functional values y belong to linear metric spaces X and ) respectively, briefly, functions F on X to S9. Linear metric spaces correspond to the spaces called by Frechet, "espaces (D) vectoriels." For the notations, postulates and fundamental propositions, we refer the reader to Part 1 of a paper by T. H. Hildebrandt and the author,? to avoid practically entire repetition of that section. No other parts of that paper are needed here. We shall consistently use the letters r, s, a, b, c to refer to real numbers, and the German t to refer to the real axis. The notation To will then refer to a set of open intervals of that axis. By the notation (ab) we shall mean the bounded closed interval of St with end points a and b. Whenever the space M) is required to be complete, this will be specifically mentioned. The form of remainder obtained in our Taylor's formula is a generalization of that given by Jordan|| and analogous at least to those obtained in

Introduction. Let G be a locally compact topological group which is unimodular, that is, the left and right Haar measures coincide. We propose, in this and subsequent papers, to investigate the structure of its "L2-system." By the "L2-system" we mean the (complex) Hilbert space formed from the complex-valued measurable functions f on G for which If| 2 is integrable with respect to the Haar measure, and with a product defined in terms of convolution. This product is not always defined (for that reason we call it an L2-system rather than an L2-algebra) but there are dense subsets (for example, L,nL2) on which multiplication always is defined so it is "almost" an algebra. Our theorems depend only on certain key properties of these L2-systems so we shall investigate general systems with these properties, calling them H-systems. We do this not only for generality but because when decomposing an L2-system into minimal parts we expect those parts to be H-systems though they need not be the L2-system of any group. This paper contains the definitions of an H-system and an L2-system; the proof that an L2-system is an H-system; theorems on idempotents and generation of ideals by idempotents; a functional calculus of self-adjoint elements in an H-system; a simple structure theorem for abelian H-systems; and an example of the L2-system of a special group, which has an interesting structure. Using the theorems of this paper we believe we shall be able to obtain the complete structure of H-systems--the ultimate theorems presumably asserting that every such system is a "direct integral" of simple systems, and that every simple system is a certain type of full matrix systembut with continuous rather than discrete matrices. At present these theorems are not proved but before even considering them carefully it seems necessary to develop the material of this paper as a foundation. One might ask why we choose to investigate this L2-system, in which multiplication is only partially defined, when there are other systems (for example, L1-the integrable functions) which are actually algebras and whose structure is still unknown. Our belief is that in spite of having only a partially defined multiplication our system may be easier to study, in somewhat the same way that certain questions are easier to answer about unbounded hypermaximal operators on Hilbert space than about bounded operators on a general Banach space. That there is a relation between that situation and ours is shown by the fact that left (or right) multiplication by a fixed element in an L2-system is, for many choices of the element, an unbounded hyper

without the assumption of the continuity of the derivative f'(Z) on the closed region R bounded by the curve of integration C, and thereby he has laid deeper foundations for the CAUCHY-RIEMANN theory of fuinctions of the complex variable. An abstract of these memoirs is to be found in the Bulletin of this Society for June, 1899, pp. 427-429. GOURSAT set out by a direct process t to evaluate the integral in question. In the present paper, by an indirect process, I prove that the integral has the value 0. The essential elements of the proof are those of GOURSAT'S first paper; by the modification indicated, and by the imposition on the curve C of a certain condition fulfilled by all the usual curves, one avoids the necessity of introducing the lenmma to which GOURSAT'S second paper is devoted. The necessary preliminary definitions and theorems are given in some detail in ? 1, in which connection I refer especially to JORDAN'S Cours d'Analyse, 2d ed., vol. 1, 1893, and to HURWITZ'S address at the Zurich Congress of 1897 entitled.: Uber die Entwickelung der (illyemeinen Theorie der analytischen Functionen in neuer er Zeit (Verhandlungen des iilathematiker Kongresses inZiirich . .; Teubner, 1898). Then in ?? 2 and 3 I state and prove the two

The matrix A may be chosen to be a Jacobi matrix, that is, Ajj=0 for i-jI >1; and An,n+l> 0 may be required. If I is {0, 1, 2, * }, A is then determined uniquely, assuming the convention that any invariant subspace of A orthogonal to the 0th coordinate subspace will be ignored. (Assuming, that is, that if Ak,k?l =Ak1, k = 0 then in A = ((A ij)) indices will be let run up to k only.) For finite I, A is not in general determined uniquely. This paper gives a convenient canonical form for A. There is little trouble in including in this result the generalized sort of moment problem introduced by Nagy [6], where un above are in 63, the set of bounded self-adjoint operators on a Hilbert space SC. One virtue of the generalized problem is its application to the classical problem; see Proposition 4 below. Accordingly some of the facts outlined in the preceding paragraph may as well be proved for the generalized problem; this is done in ?2, which in fact is essentially a recitation in the wider setting of the proof for the classical case. The main result is in ?3. The following sections mostly examine the classical problem in light of it. I have found the canonical form handy for getting numerical bounds on momenits, but will not discuss this use further. Throughout the paper I use the notation a--1 -a, where a may be a number or an operator. If it is an operator this requires some understanding as to the space on which it operates, but I think I have avoided amlbiguity.

Introduction. Martingale theory for a linearly ordered index set in a finite measure space has been systematically discussed by Doob [4]. As convergence theory and system theory are the two main topics in martingale theory, Helms [6] has extended Doob's results on mean convergence to martingales in a finite measure space, indexed by directed sets, and Krickeberg [8; 9] extended the results of pointwise convergence and stochastic convergence to martingales in a a-finite measure space, indexed by directed sets. Krickeberg uses measure algebra and lattice theoretic methods, and in [8] makes the assumption in pointwise convergence that the martingales have some covering property, which is called Vo in his paper. Dieudonne [3] has given a martingale which is not pointwise convergent and has non-negative, bounded functions in a finite measure space, indexed by a countable directed set. Therefore, some additional conditions like Krickeberg's Vo to the usual ones seem to be necessary for pointwise convergence. However, the condition Vo is too weak to be the only additional condition needed for pointwise convergence of martingales indexed by noncountable directed sets. Some conditions such as terminal separability are needed. For the system theory, Bochner [2] has given some extensions to martingales indexed by directed sets, but without proofs. Some of his results are, unfortunately, not true. Snell [14] has defined regularity (which is closely related to system theorems) of martingales in a finite measure space, indexed by positive integers, and has given some sufficient conditions. He has not discussed the necessary conditions, nor the relation between regularity and the system theorems. In this paper, ?1 gives preliminary definitions and notation. ?2 is devoted to the definition of conditional expectation in a a-finite measure space W, and ?3 to that of martingales indexed by directed sets in W as well as to individual system theorems and inequalities. In ?4, a general pointwise convergence theorem is proved, and a new kind of convergence theorem, based on a differentiation theorem of interval functions [13, p. 192] is given. We

One of the themes of Edwin Hewitt's fundamental and stimulating work [16] is that the Q-spaces (now called realcompact spaces) introduced there, although they are not in general compact, enjoy many attributes similar to those possessed by compact spaces; and that the canonical realcompactification vX associated with a given completely regular Hausdorff space X bears much the same relation to the ring C(X) of real-valued continuous functions on X as does the Stone-Cech compactification gX to the ring C*(X) of bounded elements of C(X). Much of the Gillman-Jerison textbook [12] may be considered to be an amplification of this point. Over and over again the reader is treated to a "/" theorem and then, some pages later, to its "v" analogue. One of the most elegant "/3" theorems whose "v" analogue remains unproved and even unstated is the following, given by Irving Glicksberg in [13] and reproved later by another method in [11] by Zdenek Frolik: For infinite spaces X and Y, the relation 3(X x Y) = gX x / Y holds if and only if X x Y is pseudocompact. The present paper is an outgrowth of the author's unsuccessful attempt to characterize those pairs of spaces (X, Y) for which v(X x Y) = vX x v Y. It is shown (Theorem 2.4) that, barring the existence of measurable cardinals, the relation holds whenever Y is a k-space and vX is locally compact; and more generally (Theorem 4.5) that the relation holds whenever the k-space Y and the locally compact space vX admit no compact subsets of measurable cardinal. The question arises naturally as to when it will occur that vX is locally compact. Our best result in this direction, a part of Theorem 4.8, is, again, not definitive: X is locally pseudocompact if and only if there is a locally compact space Y for which Xc Yc vX. Some of the questions treated here are susceptible to attack when thrown into the uniform space context. See Onuchic [18] and, more intensively, Isbell [17, especially Chapter 8]. I am indebted to Tony Hager for several references, and for allowing me to announce here that, in addition to the results of [15], he has achieved certain new theorems on the relation v(X x Y) = vX x v Y, which being " uniform " and not "topological" have negligible overlap with those of this paper.

Preface. For some time it has been recognized that the properties of symmetry and bounded closure are of cardinal importance in the theory of locally convex spaces. The purpose of this paper is to discuss symmetry and bounded closure with the aid of a new topology, and to discuss the new topology itself. The introduction includes the preliminaries to the rest of the paper. The account of the basic theory of locally convex spaces is fairly complete, but brief. We have given proofs only when they represent some contribution. As the references indicate, most of the theorems of the introduction can be found in the literature. In ?1 we introduce the new topology, called the GF topology, and develop the theory of the GF space. A space is a GF space if and only if it is symmetric and boundedly closed. Study of the GF topology on duals leads to information about interior maps and quotient spaces, information which is not obvious from direct consideration of symmetry and bounded closure, We conclude the section with several examples. In view of the general results of the first section it is important to know what spaces are GF spaces. In ?2 we take up this question for direct products and their duals, and also the questions of symmetry and bounded closure. ?3 serves the same purpose for continuous function spaces. Finally in ?4 we discuss metric and LF spaces and their duals. The influence of the works of G. W. Mackey, J. Dieudonne, and L. Schwartz is apparent throughout the paper, and it is to these authors that we owe our interest in locally convex spaces. The terminology used in the paper is almost exclusively that employed in the works of Dieudonne, Schwartz, and Bourbaki. For the convenience of the reader we list the exceptions. We distinguish between weak and weak-star topologies; our total bounded is the French precompact and corresponds to the definition given in [14]; we owe the term relatively strong to Mackey [13]; where the French authors use filters we make use of nets, as in Kelley [10]; and for a uniform structure topology the natural definition of a Cauchy net leads to a concept of completeness which is equivalent to the French term and stronger than that given in [14]. There are no French equivalents for symmetry and bounded closure.

The first chapter of this paper concerns itself with questions of uniform boundedness of sets of points in a Banach space and sets of functionals on a Banach space, as well as with a group of closely related resonance theorems. A well known example coming under this heading is the theorem of Toeplitz [36] t stating that supm Z I= a,.I < oo providing 77m =Zn-, -amntn converges whenever S, does. Another is the theorem of Hahn [12 ] stating that if an arbitrary continuous function has the partial sums of its Fourier expansion, with respect to an orthonormal sequence of bounded functions (n, essentially bounded, then the sequence f ZX= 1c^,=(x)co,(t) I dt is also essentially bounded. Still another is the theorem stating that if the adjoint of an everywhere defined transformation between Banach spaces is everywhere defined, then the transformation is continuous. This was proved, at least for Hilbert space, by von Neumann [21 ], Stone [34 ], Tamarkin ( [34], p. iv), and Stone and Tamarkin [35 ] and is probably not usually thought of as a theorem on uniform boundedness. Most of the results w-e have in the first chapter are new, others have been proved only in special cases, and some are well known but the proofs heretofore given have been different. Previous methods for discussing questions of uniform boundedness divide themselves into three groups (i) those associated with the names of Lebesgue [18], Banach [2], Hahn [11], and Hildebrandt [13 ]; (ii) those characterized by the elegant and direct use of the Baire category theorem as in the works of Banach [1], Saks [31], Saks and Tamarkin [32], and others; and (iii) those employed in a recent theorem of Gelfand [9] the proof of which is closely related to that of the category theorem. The second chapter of this paper is concerned with cases where a weak limiting process implies a strong one. These cases seem to be rather rare but probably more will come to light in the future. The theorem quoted above, stating that the existence of the adjoint implies the continuity of the function, belongs to the class of questions discussed in Chapter II as well as those discussed in Chapter I, while its analogue (Theorem 42) in the Boolean ring of

Introduction. By "the space of continuous functioins" we mean the Banach space C of real continuous functions on a compact Hausdorff space X. C, its first dual, and its second dual are not only Banach spaces, but also vector lattices, and this aspect of them plays a central role in our treatment. In ??1-3, we collect the properties of vector lattices which we need in the paper. In ?4 we add some properties of the first dual of C-the space of Radon measures on X-denoted by L in the present paper. In ?5 we imbed C in its second dual, denoted by M, and then identify the space of all bounded real functions on X with a quotient space of M, in fact with a topological direct summand. Thus each bounded function on X represents an entire class of elements of M. The value of an element f of M on an element ,i of L is denoted as usual by f(,u). If f lies in C then of course f(,u) =,u(f) (usually written f fd,u), and thus the multiplication between M and L is an extension of the multiplication between L and C. Now in integration theory, for each ,uEL, there is a standard "extension" of ffd,u to certain of the bounded functions on X (we confine ourselves to bounded functions in this paper), which are consequently called ,i-integrable. The question arises: how is this "extension" related to the multiplication between M and L. ??6-8 (and ?12) are devoted to this question. We show that given a bounded function f on X the equivalence class of M represented by f contains two distinguished elements, which we denote by f* and f*, such that for every Radon measure ,u, f*(,u) = f*fd,u and f*(,u) = f*fdM, where f* and f* are the lower and upper integrals of f with respect to ,u. Moreover, if f is integrable for every Radon measure, then f* and f* coincide, and we have a unique element whose value on each ,u is ffd4. In ?9 we study order-convergence in M. It is in this area that Af offers its most striking immediate gain in clarity over consideration of only the bounded functions oil X. We cite two examples. The first example: The characteristic functions of finite subsets of the real initerval 0< x < 1 form a directed system which order-converges to the constant function 1, while their Lebesgue integrals, being all zero, do not converge to the Lebesgue integral of 1. In M, however, this directed system order-converges to an element different from 1, and the above unsatisfactory situation disappears. In point of fact, order-convergence is perfectly well-behaved in M: if a directed system order converges, its values on every Radon measure converge

1. A mean on a semi-group 2 is a positive linear functional of norm one on the space m(z) of bounded, real-valued functions on 1. A bounded semigroup S of linear operators from a Banach space B to itself is called ergodic if there exists a system cd of averages A such that for every S in s limA (AS-A)=limA (SA -A) =0; we have three strengths of ergodicity of S according as uniform, strong, or weak convergence is used in the operator algebra. The first part of this paper deals with the relationship between existence of invariant means and ergodicity of bounded representations. In Theorem 2 it is shown that weak ergodicity of every bounded representation of I is equivalent to weak ergodicity of the right and left representations of f by right and left translations on m(2), and equivalent to the existence of a mean on m(z) invariant under right and left translations. These conditions, in turn, are equivalent to existence of a directed system of finite means on mn(z) converging weakly to two-sided invariance under all right and left translations of m(z). Uniform ergodicity is similarly related to existence of finite means converging in the norm of m(z2)* to two-sided invariance (Theorem 4). The second part of this paper gives some sufficient conditions for existence of invariant means or of finite means converging in norm to invariance. For the former, the Markoff method of proof by fixed-point arguments is applied (?5) to "solvable" semi-groups and groups, and to semi-groups which are the union of expanding directed systems of sub-semi-groups with means. (For example, a group G such that every finite subset generates a finite subgroup has an invariant mean.) It is also proved by a direct construction (Theorem 6) that if G is a normal subgroup of a group H and if G and H/G have two-sided invariant means, so has H. In ?6 a parallel result is proved for finite means converging in norm to invariance. These results greatly increase the family of groups known to have invariant means. Solvable groups formed the only such class previously known; ?6 now shows that a solvable group satisfies a stronger property; it has a system of finite means converging in norm to invariance.

It is shown here that the mixed initial-boundary value problem for the Euler equations for ideal flow in bounded domains of R3 has a unique solution for a small time interval. The existence of a solution is shown by converting the equations to an equivalent system involving the vorticity and applying Schauder's fixed point theorem to an appropriate mapping. Introduction. The purpose of this paper is to prove the existence and uniqueness of classical solutions to the mixed initial-boundary value problem for the Euler equations for an ideal incompressible fluid in bounded domains of R3 for a small time interval. This has been shown by Ebin and Marsden [3] for compact Riemannian manifolds possibly with boundary. In this paper we obtain a new proof by converting the equations to an equivalent system involving the vorticity and solving this by using the Schauder fixed point theorem. The technique is similar to that of Kato [4] where he proved existence of ideal flow in bounded domains of the plane for an arbitrary time interval. The author also used a similar method to obtain an existence result for ideal flow in all R3 by showing that a solution is the limit of Navier-Stokes flow as the viscosity goes to zero [7]. Existence of a solution for a short time in all R3 was originally shown by Lichtenstein [5]. The Euler equations for ideal incompressible flow in a domain D in R3 with boundary bD are (Et) ar/at + (r grad)r =-grad P + B, V * U = O, with constraints U n = 0 on bD and r(0) = A, where x is a point in D U bD; t E [O, T], Iv(X, t) is the velocity vector, P(x, t) is the (scalar) pressure, B(x, t) is the external force field vector, n(x) is the outward normal vector to bD and A(x) is the initial velocity vector. By formally computing the curl of (E') we get the system Presented to the Society, January 18, 1972 under the title The existence and uniqeness of nonstationary idealflow in bounded domains in R3; received by the editors October 12, 1971 and, in revised form, March 20, 1972. A J1S (MOS) subject classifications (1970). Primary 35F25, 35F30, 35Q05, 76C05.

Introduction. The first proof of uniqueness of a plane subsonic flow of a compressible fluid past an obstacle was given by Bers [1]. This proof utilizes an elaborate mathematical apparatus encompassing some of the most advanced tools of modern function theory. A conceptually simpler proof under somewhat weaker hypotheses and a proof of the Joukowsky force formula, due to the authors [3], make essential use of a general existence theorem and hence cannot properly be called elementary. In this note we show that the uniqueness of a compressible flow and the Joukowsky force formula can be obtained directly from a simple geometrical property of the velocity field defined by the flow. Specifically, we base our proof on the facts that the velocity components u, v of the flow define a quasi-conformal(') mapping of the (x, y) plane, and that a quasi-conformal mapping with dilatation ratio not exceeding K = 1/M satisfies at each point a Hilder condition with exponent ,. Until recently we would have considered this proof more difficult than our original one. We are, however, now able to refer to the preceding paper [5 ], in which a simple demonstration is given for the needed lemma. We note, further, that a simpler form of the arguments in [5] suffices for the present paper. The problem of proving uniqueness of a flow past an obstacle P can be divided into two parts; to prove the uniqueness if the velocity at infinity and circulation are both prescribed, and to prove that if P has a corner or cusp T the circulation is uniquely determined by the condition that the speed is finite at T. We shall settle both points here, but for the second we need the additional assumption, not used in [1] or [3], that the velocity components have bounded derivatives up to P. (This assumption can be avoided by a discussion analogous to that in [3, ?6.2].) By the same method we shall also obtain a new proof, simpler than that of [3], for the force formula of gas dynamics. 1. Uniqueness with prescribed circulation. As in [3 ] we define the velocity potential of a flow past P to be a solution k(x, y) of

This paper adapts the method of F. Riesz to the proof of certain general ergodic theorems for Abelian semi-groups of operators on a Banach space to itself. The main features of the method are that no measurability conditions are imposed on the semi-group under consideration and that consistent use of the second conjugate space and its compactness properties make it possible to replace the compactness conditions often imposed by a more natural restriction on the transforms of points. Theorem 1 and the various supplementary results include as special cases theorems of Lorch [10], Dunford [7], Yosida [15], F. Riesz [12], and Cohen [6]. It overlaps the work of Alaoglu and Birkhoff [3] at those points where they consider Abelian cases; for example, Corollary 8 is a great generalization of their Theorem 5. Section 1 contains some introductory material on conjugate spaces and adjoint operations. Section 2 introduces bounded Abelian semi-groups of operators and near invariance of a system of set functions on such a semi-group; this section also contains the principal theorem (Theorem 1) of the paper. The form of this theorem raises three questions ((A) to (C) at the beginning of ?3). The answer to (A) shows, among other things, that every Abelian semi-group has a property much like "ergodicity" in the sense of Alaoglu and Birkhoff; Theorem 3 is the main result here. The answer to (B) again indicates the importance of reflexivity in theorems of this type; Corollary 8 is one example. Two special cases of (C) give a generalization of Dunford's theorem (Theorem 5) and a theorem on bounded Abelian semi-groups of projections (Theorem 6) which has not, so far as I know, been considered before. 1. Some properties of Banach spaces. If B is a Banach space(2), let B* be the set of all linear-that is, additive and continuous-real-valued functions on B. If, for A in B*, |I|3 =SUpjjbJJ 0, k a positive integer, and I1, . , f3k in B*. With this topology B is a linear topological

Introduction. This paper is mainly concerned with the characterization problem for bounded spectral operators (see [5]). We restrict the discussion to scalar type operators with real spectrum, which we call pseudo-hermitian operators (p. h.). The criteria are given in terms of certain properties of the exponential group generated by the operator. The simplest criterion of this kind is for Hilbert space: a bounded linear operator S is p.h. if and only if || eitS| ? M < oo for all real t. This statement is false for Banach spaces, and even for reflexive Banach spaces. In the latter case, we give additional necessary conditions on the group eitS, which together with its uniform boundedness are both necessary and sufficient for S to be p.h. Another useful criterion is the following: a bounded linear operator S on a reflexive Banach space is p.h. if and only if for everyf in L1(R) (R the real line), we have II fRf(t)e itsdt || _ M || f l,,, where the norm on the left is the operator norm, f is the Fourier transform off and || * is the sup norm. Unlike the general theorems in [5] about the spectrality of an operator, the characterizations obtained in this paper are easily applied to analytically given operators on concrete Banach spaces. An example is discussed in ?4. Applications are discussed at the end of ?5. ?2 deals with an improvement of a theorem of Foguel [7] about the resolutions of the identity of sums and products of commuting spectral operators. ?1 contains preliminaries. The characterization problem for spectral operators is discussed in ??3 and 5.

Several authors have given definitions of an end point of a continuum, making use of properties which are possessed by an end point of a straight line interval, but which are not possessed by any interior point of the interval. We shall show in the present paper that the properties used in these so-called "definitions" are not logically equivalent, and shall determine the logical relations which do exist among these properties under various conditions. In Part 2, we shall show the equivalence of a number of these properties in the case of a continuous curve. In Part 3 is shown the equivalence with the first set, of two additional properties in the case of a continuous curve of a special type. In Parts 4 and 5, we determine the logical relations existing among certain of these properties in the case of a bounded continuum. Finally, in Part 6, some theorems are proved concerning points having one or more of the given properties. In this paper we shall consider only plane point sets, although in a number of cases it is obvious that our results are true in space of any number of dimensions. In regard to the use of the word "end point," we intend hereafter to use this word only in the sense of a point of a continuous curve satisfying Wilder's definition or one of the other definitions equivalent to it, in other words, a point having any one of the properties 1-7 given in Part 2. We shall not use the word "end point" in referring to a point of a continuum which is not a continuous curve. The examples of Part 4 show that a point of a continuum may have certain of the given properties and yet be so placed with respect to that continuum as hardly to deserve the name of "end point."

on the range space, the class of semigroups which permit the two types of invariant extensions is contained in the class of functions which have invariant means definable on the associated Banach space of bounded, real-valued functions defined on these semigroups. Further it was shown that every semigroup known to have an invariant mean also permitted the two types of extensions. When structure in addition to the minimal required structure is placed on the range space, e.g., when the space is an ordered subspace of the conjugate space of an ordered linear space with reproducing cone, a semi-group which has an invariant mean also has the two extension properties relative to these particular range spaces. Many of the standard function spaces satisfy these conditions. In addition this paper will exhibit conditions which guarantee the continuity of these extensions in the case when linear topological spaces are considered and make some application to real-valued functions. ?11 outlines definitions and results preceding this paper. ?III contains theorems on the existence of invariant extensions for a given semigroup G with an invariant mean and a given range space V. Various extra conditions added to the known necessary condition that V be a boundedly complete vector lattice are sufficient for these extensions. Some converse results are also given under which the existence of some sort of invariant extension implies that G has an invariant mean. ?IV contains examples of spaces satisfying the conditions imposed on the space V. ?V discusses continuity of the extensions when order and topology are both present. ?VI presents applications of the theorems of ??III and V to some of the examples of ?IV.

Introduction. This paper is concerned with extremal problems in the family of bounded analytic functions in a multiply-connected domain D, and it is concerned with extremal problems for the mean modulus f cIf Ids of meromorphic functions f in D taken over the boundary C of D. These two types of problems are shown to be closely related, and solutions are obtained simultaneously for both types by a method of contour integration. An altogether analogous method was exploited by Grunsky in his thesis [8](2) for the investigation of schlicht functions in a multiply-connected domain. Thus it is interesting to remark that the fundamental distortion theorems of schlicht conformal mapping theory have been developed by Grunsky by a method which we are able to apply here to obtain the fundamental distortion theorems for bounded functions. It will appear, then, that the generalization of Schwarz's lemma to multiply-connected domains and the generalization of the Koebe distortion theorem can be carried out by a unified technique(3). We shall find in addition to this that, while the recent papers of Bergman and Schiffer [5, 16] have developed a close relationship between the theory of schlicht canonical mappings and the Bergman kernel function [3], we are able to develop here a relationship between the theory of bounded functions and the Szego kernel function [19]. Thus the Szego kernel function does for the theory of distortion of bounded functions what the Bergman kernel function does for the theory of distortion of schlicht functions. We point out that both these kernel functions are actually differentials, and that in the Szego case one is dealing with length and in the Bergman case one is dealing with area. Thus the mean modulus fc f I ds, or ffD IfI 2dxdy, should be thought of as a length, or area, and not as a mean modulus. All these remarkable relationships are brought to light by using the simple boundary relations satisfied by the classical domain functions, such as