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Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.

The purpose of this paper is to prove a taming theorem for a codimension one manifold that is locally flat at some point and has 1-ULC complement. We also prove that any two sufficiently close locally flat embeddings of a codimension one manifold are ambient isotopic. Since this paper was first submitted, R. Daverman has shown that, given any point on a codimension one manifold with 1-ULC complement, some neighborhood of that point lies on a codimension one sphere that is locally flat at some points and has 1-ULC complement. Hence the two papers combined prove that a codimension one manifold is locally flat if and only if its complement is 1-ULC.

Let E, F be real analytic vector bundles over a real analytic compact manifold M and D: F(E)-+r(F) a real analytic polynomial differential operator satisfying D(0) = 0. Let s(i) be a parametrized family of cross sections of E, where t moves in some neighborhood of 0 in a euclidean space. We say that s(i) is a deformation of the solution 0 if s(0) = 0 and D(s(0) = 0. In the present paper, we will show the existence of deformations of the solution 0 under some conditions. Namely, let L be the linearized differential operator of D at 0 and assume: (1) the equation D(s) = 0 is involutive, (2) L is elliptic, (3) H(M9 0) = 0, where 0 is the solution sheaf of the equation L(s) = 0. Then we can prove that there is a deformation 5(0 which is complete at t = 0 in an appropriate sense. (Theorem 1, 2) We would like to point out the analogy between the above result and a theorem in [3] on the existence of deformations of complex structures. In fact, the arguments proceed along almost the same line as in [3]. In § 1, we prove some propositions which are needed in the later sections. In §2, we construct the deformation s(0 and prove its completeness in §3. Finally the author wishes to express his hearty gratitude to Professor N. Tanaka for his constant encouragement and valuable suggestions during the preparation of this paper.

In a previously published paper [9], the authors proved an approximation theorem for geodesically convex functions on Riemannian manifolds (Theorem 2 of [91). Applications of this theorem to complex function theory on noncompact K~ihler manifolds have been announced in [8, II and III] . The main result of the present paper is an application of this approximation theorem to Riemannian geometry proper. To describe this principal theorem, let M be a Riemannian manifold and let Z(M) be the closure of the set of all C ~176 subharmonic functions in C~ the algebra of continuous functions on M equipped with the compact open topology. (A C ~ function f is subharmonic if Af>O, where the sign

In this paper smooth maps /: Mn —> Nn with a zerodimensional critical set are considered. The singularities of these maps in the case n — 2 are known to be points where / is locally topologically equivalent to z -»zd {d~ 2,3, •••)• Originally these singularities were studied in connection with the regularity of Douglas' solution to Plateau's problem. In § 1, an Euler characteristic formula is developed which generalizes both the Riemann-Hurwitz equation from complex analysis and the usual Euler characteristic formula for covering maps. Section 2 is devoted to several technical lemmas while § 3 applies these lemmas to the case where M is the disc (with holes) and N is a compact orientable 2-manifold. It is shown that for the existence of such a map there is a lower limit depending upon the genus of N and on the number of holes of M. The singularities of these maps have been characterized by Church and Timourian [2]. In the case n — 2 these maps are locally topologically equivalent to z —-> zd (d = 2, 3, ) and for n > 2, these maps are covering projections. For n — 2, the singularities and maps are special cases of branch points and branched immersions. In this case, the Euler characteristic formula represents a generalization of the classical Riemann-Hurwitz equation for light interior transformations on 2-manifolds. When f'^dN) = dM, the formula produced here reduces to the Riemann-Hurwitz equation. For n > 2, the maps are covering projections and the Euler characteristic formula reduces to the usual equation. The mappings considered in this paper are not, in general, interior transformations on the boundary of M. These considered here, however, appear to have more applications with regard to questions which have arisen from Plateau's problem. Heinz [6] and Gulliver, Osserman, and Royden [5] have shown that these maps are of much more interest than solely in the context of minimal surfaces. A very readable account of the Riemann-Hurwitz formula may be found in Whyburn [10]. The authors wish to express their gratitude to the referee for his many helpful suggestions.

In this paper we give necessary conditions for the local solubility of linear differential and pseudodifferential equations with smooth coefficients. Our results generalize those of H. Lewi, Hormander, Nirenberg, Treves, Egorov, and others, described in the survey paper [16] (see also [17], [18]).

In this paper we are concerned with certain nonlinear type optimization problems related to the estimation of the universal constant KG appearing in Grothendieck's fundamental Theorem in the metric theory of tensor products. Let KG,k be the smallest constant such that Grothendieck's Theorem remains valid for m x m matrices, m≤k. In this paper we show KG,2=KG,3=√2.

In this paper a study is made of continuous collections of decomposable continua on a spherical surface. Properties of the decomposition spaces of such collections filling up continua are established, and a characterization of the decomposition spaces of such collections filling up a spherical surface is obtained. The results of the present paper are related to certain results obtained by R. D. Anderson. He has shown [2 ] that there is a continuous collection of nondegenerate continua filling up the plane which is with respect to its elements homeomorphic to the plane, and that there is a continuous collection of continua filling up a plane onedimensional continuous curve which is with respect to its elements homeomorphic to the plane. On the other hand, he has obtained [1 I a characterization of the plane decomposition spaces of continuous collections of nondegenerate continuous curves filling up a plane continuum. Such decomposition spaces are special types of hereditary continuous curves. The results of the present paper show that the decomposition spaces of continuous collections of decomposable continua filling up compact plane continua are special types of hereditary continuous curves, and consequently that even under this weaker hypothesis such collections are dimension reducing if they fill up twodimensional continua and are dimension preserving if they fill up one-dimensional continua. It might be noted that there do exist such collections filling up one-dimensional continua, but that Theorem 2 of this paper gives a strong restriction about the nature of such collections. Theorems about continuous collections of continua in a compact metric space can be stated equivalently in terms of monotone interior transformations (p. 130 of reference [10]). It is understood throughout this paper that space is compact and metric. DEFINITIONS. If H is a point set and e is a positive number, V(e, H) denotes the set of all points P such that some point of H is at a distance from P of less than e. If H and K are point sets, d(H, K) will be used to denote the greatest lower bound of the set of all positive numbers e such that K intersects V(e, H). For two

There exist real Banach spaces E such that the norm in E is of class C' away from zero; however, for any p, I < p < oo, the norm in the Lebesgue-Bochner function space LP(E, ,u) is not even twice differentiable away from zero. The main objective of this paper is to give a complete determination of the order of differentiability of the norm function in this class of Banach spaces. Introduction. The class of Lebesgue-Bochner function spaces, introduced by Bochner and Taylor [4] in 1938, has been found to be of considerable importance in various branches of mathematics, and is discussed at length in Dinculeanu [11], Dunford and Schwartz [12], and Edwards [13]. The study of the geometric properties of the Lebesgue-Bochner function spaces dates back about three decades: Day [8] and McShane [17], respectively, characterized uniform convexity and smoothness of these spaces. In fact, the only known result concerning the smoothness of the Lebesgue-Bochner function spaces is due to McShane, and his result concerns only the directional derivative (Gateaux derivative) of the norm in this class of Banach spaces. Even the Frechet differentiability of the norm has not been considered anywhere. It might be mentioned in this connection that the first systematic study of higher-order differentiability of the norm in a Banach space was made by Kurzweil [15] in 1954. Subsequently, in 1965, Bonic and Frampton [5a] extended Kurzweil's results, and in 1966 [5b] they discussed various categories of smooth Banach manifolds. In 1967, Sundaresan [18] extended some of Kurzweil's results independently. In [5b] and [18], the order of differentiability of the norm in the classical Lp spaces, 1 < p < oo, is obtained; while in Sundaresan [20], the smoothness of the norm in C(X, E) is discussed. For an elegant up-to-date account of smooth Banach spaces, and related concepts one might refer to the lecture notes by S. Yamamuro [23]. This paper contains the first systematic investigation of the higher-order differentiability of the norm function in the LebesgueBochner function spaces Received by the editors April 28, 1973. AMS (MOS) subject classifications (1970). Primary 46E40, 28A45; Secondary 58C20, 28A15.

The purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.

Introduction. In a recent paper M. Weiss and A. Zygmund [7] have studied the pointwise convergence of a trigonometric series a einx when the multipliers An= Injti (y real) are applied to it. The proof of their result makes use of Peano derivatives in LP, which bear a close connection with the tP classes of A. P. Calderon and A. Zygmund [1]. In this paper we prove that conditions of Marcinkiewicz type for a multiplier are enough to preserve the tP classes (Theorems 1 and 2). As a consequence we obtain results on pointwise convergence for multipliers which satisfy a variational condition of Marcinkiewicz type (Theorem 3).

Using the techniques developed by P. D. Lax and R. S. Phillips, qualitative results on the location of the poles of the scattering matrix for symmetric, hyperbolic systems are obtained. The restrictions placed on the system are that the coefficient matrices be of Gevrey type and that the bicharacteristic rays tend to infinity. Introduction. In a series of papers and a mongraph, P. D. Lax and R. S. Phillips [5a, b, c] have developed a scattering theory for symmetric hyperbolic systems of partial differential equations. Of primary concern in all scattering work is the location of the poles of the scattering matrix, and in particular, how their location depends on the scatterer. Using the techniques developed by Lax and Phillips, it will be shown that if the bicharacteristic rays tend to -o, then for coefficient matrices of Gevrey type,, the poles of the scattering matrix are bounded away from the real axis with the possible exception of a finite segment centered at the origin. A similar result holds for the acoustic equation with a potential of Gevrey type. The author wishes to express his gratitude to Professor P. D. Lax for his guidance and advice during the period in which this paper was researched. I. Review of scattering theory for symmetric hyperbolic systems; statement of results. The "perturbed" system under consideration is the following one: k a -u(x, t) = E Ai(x) u(x, t) + B(x)u(x, t) Gu, (1.1) at j=l ax. U(x, 0) = /(x), where u(x, t) is an n-vector depending on x = (xl, * , x^) and t. The n x n matrices A'(x), B(x) are assumed to satisfy the conditions below: Received by the editors July 24, 1972. AMS (MOS) sub/ect classifications (1970). Primary 47A40, 35P25, 35L45; Secondary 81A48.

In a recent paper (5) the author gave an alternative proof of the result due to Hille(2) for the representation of the infinitesimal generator of the Poisson operator. In this paper we obtain the n-dimensional analogue of the result by using arguments similar to those employed in (5).

A spectral theory for certain operators on a finite direct sum of Hilbert spaces was established in 1966 by N. Dunford [4]. In this paper, we shall show by using the approach of reduction theory of von Neumann (cf. [2, 7]) that his Theorem 2.7, 1-4] is true for a wider class of operators. In fact, we proved that any closed psectral operator in a Hilbert space can be decomposed into a direct integral of closed irreducible spectral operators. In Sections 1 and 3, we shall include some already developed results about the reduction theory of bounded and unbounded operators. Our main results are in Sections 2 and 4. A method of decomposition of closed spectral operator into irreducible part is included in the last section, in which we also indicate a problem of structure theory that we shall investigate in the future. The various applications of our results will appear elsewhere. Throughout this paper, the Hilbert space will be over the complex field. The author is grateful to Professor N. Suzuki for his suggestion of the problem and many valuable sessions of discussion with him.

In the case of n functions of a single variable, the vanishing of the wronskian is the most familiar criterion for their linear dependence. The wronskian also plays an important part in connection with the theory of a single ordinary homogeneous linear differential equation of the nth order, even in questions which do not concern the linear dependence of solutions of the equation. It is the purpose of the present paper to generalize the fundamental facts connected with these topics, to the case of functions of several variables. The characteristic property of an ordinary homogeneous linear differential equation is that any solution is expressible linearly, with constant coefficients, in terms of a fundamental set of solutions. The natural generalization to several independent variables is afforded by the completely integrable system of homogeneous linear partial differential equations, in one dependent variable, any solution of which is likewise linearly dependent on a finite number of solutions of the system. It is such systems which are discussed in the latter part of this paper; the subject of the linear dependence of functions of several variables is developed in the first part, chiefly with a view to its subsequent application to completely integrable systems. In our discussion of linear dependence, we consider throughout n functions Yl, Y2, Y n,, of p independent variables u1, u2, * up . Functions and variables may be either real or complex; if any of the variables be complex, we suppose the functions to be analytic in those variables. All of the main theorems, however, are stated and proved under the supposition that the independent variables are real; the modifications usually simplifications and omissions-are easily supplied for cases in which one or more of the independent variables is complex. The n functions, then, of the real variables

Let L be a linear elliptic differential operator in n-space of order m. Let M denote the adjoint operator to L. An integrable function u is called a weak solution of the equation L[u] = f in a domain D, if for every w of class C ∞, which vanishes outside a compact subset of D, the relation $$ \int_D {(uM[w]) - fw)d{x_1}...d{x_n}} $$ holds; u will be called a strict solution, if u is of class C m and satisfies L[u] =f in the ordinary sense. One of the remarkable facts concerning elliptic equations is that under suitable regularity assumptions on f and the coefficients of L a weak solution can be differentiated any number of times and is a strict solution. In a recent paper F. E. Browder1 gives the theorem: a weak solution which is square integrable on every compact subset of D is almost everywhere equal to a strict solution, provided f is in C 1 and the coefficients of the j-th derivatives in the operator L are in C m+i . Browder’s proof of this generalization of “Weyl’s lemma” makes use of the fundamental solution of elliptic equations with analytic coefficients.2 Results contained in a paper by K. O. Friedrichs,3 which appears in the same issue, imply the theorem that a weak solution is a strict solution, provided f is in C (n+1)/2 and the coefficients of L are in C m/2.

Few, if any, of the properties enjoyed by ergodic measure-preserving transformations defined on a finite measure space generalize in a natural way to those defined on an infinite measure space. Concrete examples of ergodic transformations which preserve a finite measure and ones which preserve an infinite measure exist in the literature, see [1]. It is not difficult to see that ergodic transformations never admit wandering sets of positive measure. In [2 ] it was shown that a basic difference exists between ergodic transformations which preserve a finite measure and those which preserve an infinite measure; namely, an ergodic measure-preserving transformation defined on an infinite measure space always admits weakly wandering sets of positive measure (Theorem 2 of [2]). Unlike wandering sets, it is not true in general that the union of two weakly wandering sets is again a weakly wandering set even if we require that a class of mutually disjoint images of one weakly wandering set does not intersect a class of mutually disjoint images of the other. One may ask then if there are any ergodic measure-preserving transformations defined on an infinite measure space which admit only weakly wandering sets of finite measure. In this paper we show that this is not the case. We prove that there always exist weakly wandering sets of infinite measure for any ergodic measure-preserving transformation defined on an infinite measure space (Theorem 3). In [2] the existence of a weakly wandering set of positive measure for an ergodic measure-preserving transformation defined on an infinite measure space was discovered while studying the necessary and sufficient conditions for the existence of a finite, invariant, and equivalent measure for a given measurable and nonsingular transformation. In this paper we construct the weakly wandering sets in a different way. Using the pointwise ergodic theorem we prove a simple and yet a useful fact about ergodic measure-preserving transformations defined on an infinite measure space (Theorem 2). It states that given two sets A and B both of finite measure, it is possible to find an image of A under some power of the transformation T which has small intersection with the set B. This fact is basic in proving Lemma 1 which shows the existence of a

The theorv of invariants of the linear ordinary differential equatioll dates from 1869, when Sir James Cocklet obtained certain seminvariaiits. In later papers he made progress in the theory, while other writers,j notablv Laguerre, Brioschi, Halphen and Forsyth, extended his results. The theory thus developed presents a striking analogy to the theory of algebraic invariants. One point of similarity, with which this paper has much to do, is the fact that by means of a transformation of the type y = X (x) -, the linear differential equation

This paper deals with the following problem: Can an arbitrary continuous function on [0, 1], which vanishes at the origin, be represented in some sense as a series of constant multiples of indefinite integrals of a complete orthonormal set of functions on [0,1]? Four contexts in which this problem arises naturally will be given in the introduction and the remainder of the paper will be devoted to giving a partial answer to the specific problem formulated in one of these contexts.

This paper gives a discussion of a system of infinitely many linear equations in infinitely many unknowns. The character of the system and of the results differs widely from that of other papers on this subject. It appears that most of the work on such systems is restricted to solutions in Hilbert space: i.e., where Z= X, converges, (xi) being a solution. The paper of Walsht may be cited as an exception. He obtains a unique solution, however. The present work may be described as follows: In ?1 we start with a system of linear equations which reduces to a system of difference equations yielding an infinity of solutions. These solutions are shown to be solutions of the original system, after suitable restrictions are laid on the coefficients of the system. We thus get a one-parameter family of solutions. Then we turn, in ?II, to a more general system of equations, whose solution can be effected by a method of approximations based on the results of ?I. There is shown to exist at least a one-parameter family of solutions (xi) satisfying the inequality l xi I

In a recent paper, Petryshyn and Tucker [10] introduced the concept of a P~-compact ("generalized P-compact ') operator, acting on some subset of a Banach space with projectionally complete system, and they proved general existence theorems for fixed points of such operators. The purpose of our paper is to establish conditions which ensure that the set of fixed points of a P~-compact mapping is a continuum, i.e. a compact and connected set. If it cannot be assured that a given operator has a unique fixed point, the general idea consists in approximating this operator by more "regular" ones, in a sense which will be made precise below. In case of completely continuous or proper mappings on a general Banach space this question had been investigated e.g. by Aronszajn [1], Stampacchia [11], Krasnosel'skii and Sobolevskii [7], Browder and Gupta [2] and Vidossich [12]. Some applications to Volterra integraland hyperbolic partial differential equations are given in [5] and [6], while [7] includes some results for parabolic equations. In w 1 we introduce some basic definitions to be used in this paper. w 2 includes a relation between P~-compact and demicompact mappings that can be used in particular to simplify the proofs of some propositions in [10]. In w 3 we prove our first main result which is concerned with the set of fixed points of a P~-compact operator on a bounded closed convex set, while in the last paragraph this theorem is extended to an arbitrary bounded region by means of the degree of an A-proper mapping recently developped by Browder and Petryshyn [3]. In the main, our notation is like that in [10]. The author is indebted to Prof. W.V. Petryshyn who called his attention to A-proper mappings and to Prof. G. Stampacchia for having made a preliminary draft of [12] available to him.

In a previous paper (l) in this journal L. J. Slater gave expansions of the generalized Whittaker functions . She gave this name to the generalized hypergeometric function in the sense that it is a generalization of the well-known Whittaker function . In this paper series of products of generalized Whittaker functions will be evaluated in terms of such functions or in terms of generalized hypergeometric functions pFp(x). These expansions are These formulae will be proved in § 2 and particular cases will be given in § 3.

In this paper we introduce the notion of “generalized” interpolation space, and state and prove a “generalized” interpolation theorem. This apparently provides a foundation for an axiomatic treatment of interpolation space theory, for subsequently we show that the “mean” interpolation spaces of Lions-Peetre, the “complex” interpolation spaces of A. P. Calderón, and the “complex” interpolation spaces of M. Schechter are all generalized interpolation spaces. Furthermore, we prove that each of the interpolation theorems established respectively for the above-mentioned interpolation spaces is indeed a special case of our generalized interpolation theorem. In §III of this paper we use the generalized interpolation space concept to state and prove a “generalized” duality theorem. The very elegant duality theorems proved by Calderón, Lions-Peetre and Schechter, respectively, are shown to be special cases of our generalized duality theorem. Of special interest here is the isolation by the general theorem of the need in each of the separate theorems for certain “base” spaces to be duals of others. At the close of §II of this paper we employ our generalized interpolation theorem “structure” to construct new interpolation spaces which are neither complex nor mean spaces.

We investigate the asymptotic relationship between the solutions of a linear differential system and its perturbed system. Our results depend upon a known result of F. Brauer on asymptotic equilibrium. We also study the asymptotic manifold of solutions of the nonlinear system generated by the solutions of the corresponding linear system. 1. Recently, Brauer and Wong [2] and Onuchic [5] obtained general results on the asymptotic relationship between the solutions of a linear differential system and its perturbed system. Toroselidze [6] considered the problem of perturbing the asymptotic manifold of a nonlinear scalar equation. Hallam and Heidel [3] extended the results of Toroselidze to nonlinear systems and also discussed the asymptotic relationships of solutions. We, in the present paper, wish to investigate these problems further. Our results depend upon a known basic result [1 ] on asymptotic equilibrium and consequently the proofs are short and fundamental. Our approach has thrown considerable light on the problems and at the same time improved the results of [2] and [3] by weakening the hypotheses. This is explained in detail in a remark at the end of the paper. 2. Let J denote the half-line 0t< oo and Rn, the euclidean nspace. Let |11 denote any convenient norm of a vector and the corresponding norm of a matrix. We consider the nonlinear differential system (2.1) xi = A(t)x + f(t, x), x(to) = xo, to 6 J, where A (t) is a continuous n X n matrix for t CJ andfE C[ JXRn, Rn]. Received by the editors April 19, 1970. AMS 1969 subject classifications. Primary 3440, 3442, 3450, 3453.

It is shown in this paper that all 16th powers of unoriented cobordism classes can be represented by stably almost symplectic manifolds. The generators are coefficients in the expan- sion of a Conner-Floyd characteristic class of the symplectification of a tensor product of symplectic line bundles. In this paper it will be shown that all 16th powers of unoriented cobordism classes are in the image of symplectic cobordism. The method will be similar to those of Ray (4). Some alternative, more geometric constructions will be given without proof, because the computations involved are considerably longer. Previously Stong (6) proved that in dimensions 20 and lower this image is zero, Segal (5) found that the first nonzero class in the image is (RP2)16, and Floyd (2) showed that the image must lie in the ring generated by certain 8th and 16th powers. In Ray's methods one considers a pair of spaces for which the spectral sequence from ordinary cohomology to symplectic cobordism collapses, so that they are "free" (allowing infinite sums) over the symplectic cobordism of a point. Bases are found for the symplectic cobordism of the two spaces, frequently involving the characteristic classes of Conner and Floyd (1, Corollary 8.2). A map is chosen from one space to the other, and the image of one basis is expanded in terms of the other. The coefficients are the desired classes in the symplectic cobordism of a point. Here we will consider the space HP^xHP™. Let K and K' denote the canonical symplectic line bundles over the two quaternionic projective space factors, and let EX(K) and EX(K') denote their Conner-Floyd characteristic classes in QiSp(HPx'xHP°°). Then E^IO'E^K')' is a basis in the sense mentioned above, for

1. In some recent papers I have developed the theory of what I have called harmonic integrals associated with an algebraic variety. My object has been to provide an apparatus with which to investigate the properties of the classical integralsof total differentials which are finite everywhere on an algebraic variety Vm of m dimensionsThese integrals are referred to in what follows as “Abelian integrals”, as it is desirable to have a single name to denote all integrals of this type. The usual qualification “of the first kind” is omitted, and is always to be understood. In the present paper it is shown how the theory of harmonic integrals can be applied to deduce certain results concerning Abelian integrals, and a number of interesting problems are suggested by these results.

*Presented to the Society in the form of two papers February 22, 1902, and (San Francisco) May 3, 1902. Received for publication March 14, 1902. t To facilitate references to my previous papers on this subject, I shall quote them in the text by the initial words of their titles, viz.: Invariants, Geometry, Reciprocal Systems They have all been published in these T ra n sacti o n s: Invariants, vol. 2, p. 1 ; Geometry, vol. 2, p. 343; Reciprocal Systems, vol. 3, p. 60. 423

In this paper we study the problem of existence and uniqueness to solutions of the nonlinear Volterra integral equation x = f + a 1 g 1 ( x ) + ⋯ + a n g n ( x ) x = f + {a_1}{g_1}(x) + \cdots + {a_n}{g_n}(x) , where the a i {a_i} are continuous linear operators mapping a Fréchet space F \mathcal {F} into itself and the g i {g_i} are nonlinear operators in that space. Solutions are sought which lie in a Banach subspace of F \mathcal {F} having a stronger topology. The equations are studied first when the g i {g_i} are of the form g i ( x ) = x + h i ( x ) {g_i}(x) = x + {h_i}(x) where h i ( x ) {h_i}(x) is “small", and then when the g i {g_i} are slope restricted. This generalizes certain results in recent papers by Miller, Nohel, Wong, Sandberg, and Beneš.

It is the purpose of the present paper to give a natural method of realizing bounded symmetric domains as matrix-spaces. Our method yields, as special cases, the well-known bounded models of irreducible bounded symmetric domains of classical type (I)-(IV), as were already described in the original paper of E. Cartan [1] (see §3; we follow in this paper the classification table in [14], not in [1]). A direct application of this method will be to determine the canonical bounded models of the irreducible bounded symmetric domains of exceptional type; it will be published in another paper (see [6], [7] for the summary of the results).

1. The main purpose of this paper is to prove some theorems about conjugate Fourier series, and similar orthogonal expansions, of a positive definite function. At the end of the paper there is given a slight generalization of the well-known continuity theorem for characteristic functions. It was proved several years ago by Dugue [1 ] that if X is a measurable positive-definite function then the Fourier series of 4) is uniformly convergent. Dugue's Theorem was actually stated in terms of characteristic functions, but it is equivalent to the present formulation on account of the Bochner-Khintchine Representation Theorem. More precisely one can state

In the fragment Zwei alg emeine Sit e iiber lineare D/ferentialgleichiungen nit algebraischen Coe]Jicienten numbered XXI in his collected works, RIEMANN laid the foundatioll of the moderni theory of linear differential equations by regarding the solutions as a linear family and studying the substitution group of such a family. The importance of his results will perhaps lend interest to an investigation of theorems of a converse, character. The present paper is not, however, concerned with the two theorems from which RIEMANN's fragment takes its name, but with two others from which they are deduced. Although the paper referred to discusses only differential equations all of whose singular points are regular, we add no new difficulties to our problem if we generalize the results there obtained to the case of n + I linear fanilies of the nth order, y() ** y(2), , y(f+l), all analytic in the same m-tuply connected region T , and having the same substitution group (Monodromiegruppe) in Tm. The term basis will be used here to designate any system of n linearly independent mnembers of a family. If bases chosen from different families have the same grouip of substitutions they will be said to corre spond, and homologous members of such bases will be called corresponding branches. With this terminology the two theorems of RIEMANN which we here consider may be stated as follows: A. If in the matrix

problem. The weak solution of [3] is a classical solution of the appropriate parabolic equation in every open set where u> 0 and a(u) >0 (or where uO and a(u) ? - a). However the set W where u = 0 and - a 1) W stays away from the fixed boundaries. It is not known, for instance, whether or not W has a positive measure or interior points. For the one-phase problem, however, it was proved in [3] that W is contained in the boundary of measurable point and, consequently, has no interior points. Physically, Wrepresents the "weak" free boundary between the solid and the liquid of the problem. The first purpose of the present paper is to prove that, in the case n = 1, W is actually a continuous curve x=s(t) (which is then the free boundary in the usual sense) and that u(x, t) = 0 at all the points where x = s(t). This result is stated in Theorem 3 below, and is proved in ??2, 3. The second purpose of the paper is to extend the results from two-phase problems to k-phase problems for any k. This is done in ??5-7. The lines t=a where some of the phases degenerate cause some difficulty. There may occur perhaps some intervals on t = ai where u(x, ai)_ 0 (see ?6) and regions lying above such intervals where u 0 (see ?7). But in any case we prove that W consists of a finite number of curves x = x(t) which are continuous in the intervals (at, ai + 1) As far as we know, global existence theorems of classical solutions for the twophase Stefan problem were considered only by Rubinstein [4]. However, there is a fundamental gap in his proof of existence. In ?8 we explain the nature of this gap. Thus it is not known, at this point, whether classical solutions do in fact exist globally.

Symmetric (Riemannian) spaces were introduced and developed by Cartan [1, 2] which led to the discovery of projectively symmetric (Riemannian) spaces by Soos [9]. Recently the theory of symmetric spaces has been extended to Finsler geometry by the present author [5]. The current paper deals with that class of Finsler spaces throughout which their projective curvature tensors possess vanishing covariant derivatives. Following Soos' terminology such spaces are calledprojectively symmetric Finsler spaces. Examples, conditions for a symmetric Finsler space to be projectively symmetric, reduction of various identities, and the discussion of a decomposed projectively symmetric Finsler space form the skeleton of the paper.

The classification of isolated singularities of minimal varieties leads to the study of minimal manifolds in the n-sphere. The object of this paper is to show that a minimal 2-sphere in SI with trivial normal bundle is the standard 2-sphere. In [1] Almgren showed that the only minimal 2-sphere in S3 is the standard S2. For minimal spheres in S4 an additional assumption is needed to rule out counterexamples. The Veronese surface, a minimal imbedding of the projective plane in S4 iS such an example. For details see [3]. Some of the ideas in this paper are related to a paper by H. Hopf [2 ] in which he proves that a 2-sphere immersed with constant mean curvature in E3 is the standard S2. This connection is further emphasized by Theorem 1, a generalization of Hopf's theorem. 2. Definitions and results. Throughout the paper, [4, Chapter VII] is used as a standard reference. For convenience we recall a few definitions here: The mean curvature H, a cross section of the normal bundle, is defined to be the trace of the second fundamental form. A surface is called minimal in Sn-l if the mean curvature vector H is parallel to the normal vector of Sn-1 in En. For technical reasons it is convenient to consider the minimal surfaces in SnIas surfaces in En. Of course the surfaces are no longer minimal, but the mean curvature vector H is still parallel; i.e., the covariant derivative of H in the normal bundle is zero. However, for spheres nothing new is added as the following theorem shows. THEOREM I. If I: S2->En is an immersion of S2 into En with parallel mean curvature vector of length one, then the image I(S2) is a minimal surface in Sn-1. The next theorem is the main result of this paper. THEOREM II. If I: S2->S4 is an immersion of S2 into S4 as a minimal Received by the editors February 7, 1970. AMS 1969 subject classifications. Primary 5304, 5375.

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

1. The fundamental ideas of the generalized Fourier transform have been presented by S. Bochner [l; 2], for functions which are either locally integrable or locally square integrable. A parallel line of thought has been followed by N. Wiener [7] and H. R. Pitt [4] in their theory of generalized harmonic analysis. While the parts of the Wiener theory which are connected with the idea of the generalized Fourier transform do not treat functions which are as large at infinity as those treated by Bochner, they do come closer to giving a theory for Lp integrability than does Bochner (see [4]). More recently L. Schwartz [5] has given a treatment of the generalized Fourier transform by quite different methods. One of the weak points of the theory has been the lack of satisfactory inversion formulas, particularly for the generalized transform of order greater than one. The main part of this paper is devoted to showing that by the use of Abelian summability, the generalized Fourier transform may be inverted in the metric of the proper function space. The last part of the paper attempts to unify some of the ideas of generalized harmonic analysis and the generalized Fourier transform and to present a generalization of a well known theorem of Titchmarsh ^6, Theorem 74] The function space Lj, l^p< oo, consists of those functions for which the norm

The theory of stochastic differential equations in a differentiate manifold has been established by many authors from different view-points, especially by R Lévy [2], F. Perrin [1], A. Kolmogoroff [1] [2] and K. Yosida [1] [2]. It is the purpose of the present paper to discuss it by making use of stochastic integrals.

In this paper we give a sufficient colldition that the characteristic numbers of an unsymmetric kernel exist and be real, and prove the expansibility of arbitrary functions in terms of the corresponding characteristic functions. This sufficient condition is stated in terms of a functional transformationt T(f) defined by eertain general properties (§1), and for the special case T(f) _f we obtain the known theory of the orthogonal integral equations. The method employed is that of infinitely many variables and is based to some extent on an earlier paper.t However, the present paper, with the exeeption of a few footnotes, can be read independently of (I) if Theorems 20 and 8 which are here stated in §1 are accepted; and if the suggestion in Remark 1 of §1 is carried out, the only reference to (I) that is necessary is Theorem 20.

The objective of this paper is to establish a Gronwall inequality for the weighted refinement integral [ F , ( w 1 , w 2 , w 3 ) ] ∫ a b f ( t ) d g ( t ) [F,({w_1},{w_2},{w_3})]\smallint _a^bf(t)dg(t) . This result generalizes a recent result by W. W. Schmaedeke and G. R. Sell for the mean sigma integral and the interior refinement integral. The proof in this paper is based on the one given by Schmaedeke and Sell but is shorter and simpler. B. W. Helton has established a product integral representation for a Gronwall inequality for the refinement integral ( L R ) ∫ a b ( f H + f G ) (LR)\smallint _a^b(fH + fG) . Helton’s result contains the result here for the special case where w 1 {w_1} and w 3 {w_3} are nonnegative real numbers such that w 1 + w 3 = 1 {w_1} + {w_3} = 1 . The ideas used here are considerably less complicated than those used by Helton. J. V. Herod has established a Gronwall inequality for linear Stieltjes integrals working with a linear function J [ f ] J[f] that is more general than the linear function J [ f ] = ( L R ) ∫ a b ( f H + f G ) J[f] = (LR)\smallint _a^b(fH + fG) considered by Helton.

In a paper published in 1889, Hurwitz [3] proved that the holomorphic automorphic forms and the meromorphic automorphic functions associated with groups arising from the Riemann surfaces of algebraic functions satisfy algebraic differential equations of order at most three; cf. [5], [8], and [11].1 More precisely, given such a form or functionf, there is a polynomial Pf (EC [Xo0 Xl, X2, X3] which is zero for the specialization (X0, Xl, X2, X3)-(fJ', f', f"1, fi). To generalize this theorem, let Z be analytically equivalent to an irreducible bounded symmetric domain in the sense of it. Cartan, and suppose that the (real) dimension of the Shilov boundary of Z is equal to dimcZ, the complex dimension of Z. Suppose further that r is a discrete group of holomorphic automorphisms of Z which is pseud-oconcave or such that Z/r has a compactification as a connected complex space whose meromorphic functions extend all of the meromorphic functions on Z/r. The main result of this paper is the following

This paper examines the possibility of extending the Sobolev Imbedding Theorem to certain classes of domains which fail to have the "cone property" normally required for that theorem. It is shown that no extension is possible for certain types of domains (e.g. those with exponentially sharp cusps or which are unbounded and have finite volume), while extensions are obtained for other types (domains with less sharp cusps). These results are developed via certain integral inequalities which generalize inequalities due to Hardy and to Sobolev, and are of some interest in their own right. The paper is divided into two parts. Part I establishes the integral inequalities; Part II deals with extensions of the imbedding theorem. Further introductory information may be found in the first section of each part. PART I. INTEGRAL INEQUALITIES 1.

This paper gives a sufficient condition for a submanifold of a Riemannian manifold of nonnegative constant curvature to be totally umbilical. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature. Introduction. In a previous paper [4] the present author proved the following theorem. Theorem A. Let M be an n-dimensional, connected, compact hypersurface with constant mean curvature immersed in an (n + l)-dimensional Riemannian manifold of nonnegative constant curvature. If the second fundamental tensor H satisfies (n 1) trace H2 < (trace H)2, then M is a totally umbilical hypersurface and consequently a sphere. In this theorem, if the length of the second fundamental tensor is constant, then we can replace the condition that M is compact by requiring that M is complete. In this case, the estimation in Theorem A is the best possible since Sn~ x E1 in F" + I satisfies (22 l) trace H2 = (trace H)2. In this paper, the author tries to generalize this theorem to a submanifold of any codimension which is immersed in a Riemannian manifold of nonnegative constant curvature. 1. Preliminaries. Let M be an 72-dimensional submanifold of an (22 +p)-dimensional Riemannian manifold M of constant curvature c. The Riemannian connections of M and M ate denoted by V and V respectively, whereas the connection in the normal bundle of M in M is denoted by D. Let N., • ■ • , N be mutually orthogonal unit normal vectors at a point P £ M and extend them to vector fields in a neighborhood of P. We define H . X (A = 1, 2, • • • , p) to be the tangential component of ^x^ A ^or ^ £ "^f/^ an

Introduction. Let M be a connected smooth (i.e., CI) manifold of dimension n, and B the total space of the frame bundle over M. Let a linear connection be given on M, and let B [zO] be the submanifold of B consisting of all the points which can be joined to a given point zo in B by sectionally smooth horizontal curves. The purpose of this paper is to elaborate on a natural correspondence set up by S. S. Chern [2, pp. 78-79] between tensors of type (r, s) on M and sets of nr+, functions of a particular type on B, and use it to prove the theorem that a tensor S on M is recurrent (i.e., S is not a zero tensor and its covariant derivative is equal to the tensor product of a covariant vector and S itself) if the restrictions to B [zo] of its corresponding functions on B have no common zero and are proportional to a set of constants. We give several applications of this theorem, obtaining among others the following results: (i) A recurrent tensor on M has no zero (Theorem 3.8); (ii) A property obtained by K. Nomizu [4, p. 73] characterizing linear connections with covariantly constant curvature tensor or covariantly constant torsion tensor (Theorem 4.3); (iii) A property characterizing linear connections with recurrent curvature or recurrent torsion similar to (ii) above (Theorem 4.2); (iv) The holonomy group of a linear connection with recurrent curvature is at most of dimension n(n1)/2 (Corollary 4.4). The author is deeply indebted to Professor S. S. Chern for his generous help during the first half of the year 1959 at the University of Chicago when the author tried to acquaint himself with the modern theory of linear connections. The author is also grateful to the referee for pointing out a gap in the original proof of our main theorem (3.9) and for his valuable suggestions which have resulted in several improvements in this paper. 1.1. Bundle of frames. In ? ?1.1-1.5, we summarize some of the results on linear connections which are needed for our later work. We assume as known the classical theory of linear connections and the elementary properties of n-dimensional smooth manifolds (class Cand with countable base). All the vector fields, tensor fields, q-forms, etc. defined on a smooth manifold or on part of it are assumed to be of class Co unless stated otherwise. Each of the

Let π : X → S \pi :X \to S be a flat proper morphism of analytic spaces. π \pi may be thought of as providing a family of compact analytic spaces, X s {X_s} , parametrized by the space S S . Let F \mathcal {F} be a coherent sheaf on X X flat over S S . F \mathcal {F} may be thought of as a family of coherent sheaves, F s {\mathcal {F}_s} , on the family of spaces X s {X_s} . Let o ∈ S o \in S be a fixed point, ξ o ∈ H q ( X o , F o ) {\xi _o} \in Hq({X_o},{\mathcal {F}_o}) . In this paper, we consider the problem of extending ξ o {\xi _o} to a cohomology class ξ ∈ H q ( π − 1 ( U ) , F ) \xi \in Hq({\pi ^{ - 1}}(U),\mathcal {F}) where U U is some neighborhood of o o in S S . Extension problems of this type were first considered by P. A. Griffiths who obtained some results in the case in which the morphism π \pi is simple and the sheaf F \mathcal {F} is locally free. We obtain generalizations of these results without the restrictions. Among the applications of these results is a necessary and sufficient condition for the existence of a space of moduli for a compact manifold. This application was discussed in an earlier paper by the author. We use the Grauert “direct image” theorem, the theory of Stein compacta, and a generalization of a result of M. Artin on solutions of analytic equations to reduce the problem to an algebraic problem. In §2 we discuss obstructions to deforming ξ o {\xi _o} ; in §3 we show that if no obstructions exist, ξ o {\xi _o} may be extended; in §4 we give a useful criterion for no obstructions; and in §5 we discuss some examples.

The subject of this paper is uniqueness and non-uniqueness in the Cauchy problem at simple characteristic points of the initial hypersurface. We only consider linear partial differential equations with analytic coefficients. We use the notation from the book [1] by H6rmander with the exception that D~ = t3/~xj, 1 0 in a neighbourhood of x °, and if H is of a type covered by [6, Theorem 3] then there is a neighbourhood V of x ° and a function u e C ~ ( V ) such that P ( x , D ) u = O , x ° ~ suppu C {x; ~0(x) =>0, x ~ V}." When we want to construct characteristic hypersurfaces that are not hyperplanes we have to rely on integration of Hamilton's equations. So we restrict ourselves to operators with real principal part and to simple characteristic points on the initial hypersurface. The three points of departure of this paper is the method to prove non-uniqueness by Malgrange [3], the uniqueness theorem [2, Theorem 8.1] by H6rmander, see Theorem 3 below, and the technique of proving uniqueness taken from Persson [5]. The part played by Theorem 3 is a passive one leaving to us to study the case when a priori the bicharacteristic curve does not leave the support of the solution. In [9] Zachmanoglou proves the following theorem.

1. The basic idea of the application of integral operators to the Weierstrass-Hadamard direction. In order to generate and investigate solutions of differential equations, operators p (defined as the integral operators of the first kind) have been introduced in [2; 6](2). p transforms analytic functions of one and two variables into solutions of linear elliptic differential equations of two and three variables, respectively. It has been shown in the abovementioned papers that p (as well as some other operators connected with p) preserves many properties of the functions to which the operator is applied. This situation permits us to use theorems in the theory of functions to obtain theorems not merely on harmonic functions in two variables, but on solutions of other linear differential equations as well(3). In the present paper the above-mentioned method is used to prove connections between the properties in the large of solutions i1 of certain linear differential equations, see (1.1) and (1.3), on one side and the structure of certain subsequences of the coefficients of the series development of V/ at the origin on the other. Let us formulate these procedures in a somewhat more concrete manner, at first for equations in two variables. Let i1 be a (real) solution of the differential equation

Introduction. A topological group (D is said to act on a topological space R if the elements of (M are homeomorphisms of R onto itself and if the mapping (o, p)->of(p) of (5XR onto R is continuous. Familiar examples include the rotation group acting on the Cartesian plane and the Euclidean group acting on Euclidean space. The set @(p) (that is, the set of all a(p) where orEz) is called the orbit of p. If p and q are two points of R, then (i(p) and @(q) are either identical or disjoint, hence R is partitioned by the orbits. The topological structure of the partition becomes an interesting question. In the case of the rotations of the Cartesian plane we find that, excising the singularity at the origin, the remainder of space is fibered as a direct product. A similar result is easily established for a compact Lie group acting analytically on an analytic manifold. In this paper we make use of Haar measure to extend this result to the case of a compact Lie group acting on any completely regular space. The exact theorem is given in ?3. ??1 and 2 are preliminaries, while in ?4 we apply the main result to the study of the structure of topological groups. These applications form the principal motivation of the entire study,' and the author hopes to develop them in greater detail in a subsequent paper.