Representation theorems for nonlinear disjointly additive functionals and operators on Sobolev spaces
Moshe
Marcus;
Victor J.
Mizel
1-45
Abstract: An abstract characterization is obtained for a class of nonlinear differential operators defined on the subspace $S = {\ring{W}}_k^p[a,b]$ of the kth order Sobolev space $W_k^p[a,b], 1 \leqslant k, 1 \leqslant p \leqslant \infty$. It is shown that every mapping $T:S \to {L^1}[a,b]$ which is local, continuous and ${D^k}$-disjointly additive has the form $ (Tu)(t) = H(t,{D^k}u(t))$, where $ H:[a,b] \times R \to R$ is a function obeying Carathéodory conditions as well as $ (\ast )H( \cdot ,0) = 0$. Here ${D^k}$-disjoint additivity means $T(u + v) = Tu + Tv$ whenever $({D^k}u)({D^k}v) = 0$. Likewise, every real functional N on S which is continuous and ${D^k}$-disjointly additive has the form $N(u) = \smallint Tu$, with T as above. Liapunov's theorem on vector measures plays a crucial role, and the analysis suggests new questions about such measures. Likewise, a new type of Radon-Nikodým theorem is employed in an essential way.
Hall-Higman type theorems. III
T. R.
Berger
47-83
Abstract: This paper continues the investigations of this series. Suppose that $G = ANS$ where S and NS are normal subgroups of G. Suppose that $ (\vert A\vert,\vert NS\vert) = 1$, S is extraspecial, and $ S/Z(S)$ is a faithful minimal module for the subgroup AN of G. Assume that k is a field of characteristic prime to $ \vert G\vert$ and V is a faithful irreducible ${\mathbf{k}}[G]$-module. The structure of G is discussed in the minimal situation where N is cyclic, A is nilpotent, and $ V{\vert _A}$ does not have a regular A-direct summand.
Hopf invariants and Browder's work on the Kervaire invariant problem
Warren M.
Krueger
85-97
Abstract: In this paper we calculate certain functional differentials in the Adams spectral sequence converging to Wu cobordism whose values may be thought of as Hopf invariants. These results are applied to reobtain Browder's characterization: if $q + 1 = {2^k}$, there is a 2q dimensional manifold of Kervaire invariant one if and only if $h_k^2$ survives to ${E_\infty }({S^0})$.
Systems of nonlinear Volterra equations with positive definite kernels
Olof J.
Staffans
99-116
Abstract: We study the boundedness and the asymptotic behavior of the solutions of a nonlinear, $ {{\mathbf{R}}^n}$-valued Volterra equation with a positive definite kernel, generalizing earlier scalar results.
Essential central range and selfadjoint commutators in properly infinite von Neumann algebras
Herbert
Halpern
117-146
Abstract: The essential central range of an element A of a von Neumann algebra with respect to a central ideal is characterized as those elements arbitrarily close to the compression of A to a subspace large with respect to the ideal. The selfadjoint commutators in a properly infinite algebra are shown to be the elements whose essential central ranges with respect to the strong radical contain 0.
The asymptotic behavior of a Volterra-renewal equation
Peter
Ney
147-155
Abstract: Theorem. Assume that the functions $x( \cdot ),h( \cdot ),G( \cdot )$ satisfy: (i) $0 \leqslant x(t),t \in [0,\infty );x(t) \to 0$ as $t \to \infty ;x$ bounded, measurable; (ii) $0 \leqslant h(s);h(s)$ Lipschitz continuous for $s \in I$, where I is a closed interval containing the range of $(0,\infty )$ having nontrivial absolutely continuous component and finite second moment. Let $Hx(t) = \smallint _0^th[x(t - y)]dG(y)$. If $0 \leqslant (x - Hx)(t) = o({t^{ - 2}})$, with strict inequality on the left on a set of positive measure, then $x(t) \sim \gamma /t,t \to \infty$, where $\gamma$ is a constant depending only on h and G. The condition $o({t^{ - 2}})$ is close to best possible, and cannot, e.g., be replaced by $ O({t^{ - 2}})$.
Cone bundles
Clint
McCrory
157-163
Abstract: A theory of normal bundles for locally knotted codimension two embeddings of PL manifolds is developed. The classifying space for this theory is Cappell and Shaneson's space $ BR{N_2}$.
The structure of local integral orthogonal groups
D. G.
James
165-186
Abstract: Let M be a lattice on a regular quadratic space over a nondyadic local field. The normal subgroups of the integral orthogonal group $O(M)$ are determined.
Hausdorff content and rational approximation in fractional Lipschitz norms
Anthony G.
O’Farrell
187-206
Abstract: For $0 < \alpha < 1$, we characterise those compact sets X in the plane with the property that each function in the class ${\text{lip}}(\alpha ,X)$ that is analytic at all interior points of X is the limit in ${\text{Lip}}(\alpha ,X)$ norm of a sequence of rational functions. The characterisation is in terms of Hausdorff content.
Resolvents and bounds for linear and nonlinear Volterra equations
J. J.
Levin
207-222
Abstract: The asymptotic behavior of the resolvent of a linear Volterra equation is investigated without the assumption that the kernel of the equation is in $ {L^1}(0,\infty )$. A lower bound is obtained on the solutions of a related nonlinear Volterra equation. A special case of the latter result is employed in the proof of the former result.
Examples for the nonuniqueness of the equilibrium state
Franz
Hofbauer
223-241
Abstract: In this paper equilibrium states on shift spaces are considered. A uniqueness theorem for equilibrium states is proved. Then we study a particular class of continuous functions. We characterize the functions of this class which satisfy Ruelle's Perron-Frobenius condition, those which admit a measure determined by a homogeneity condition, and those which have unique equilibrium state. In particular, we get examples for the nonuniqueness of the equilibrium state.
The generalized Green's function for an $n$th order linear differential operator
John
Locker
243-268
Abstract: The generalized Green's function $K(t,s)$ for an nth order linear differential operator L is characterized in terms of the 2nth order differential operators $ L{L^\ast}$ and $ {L^\ast}L$. The development is operator oriented and takes place in the Hilbert space ${L^2}[a,b]$. Two features of the characterization are a determination of the jumps occurring in the derivatives of orders n, $n + 1, \ldots ,2n - 1$ at $t = s$ and a determination of the boundary conditions satisfied by the functions $K(a, \cdot )$ and $ K(b,\cdot)$. Several examples are given to illustrate the properties of the generalized Green's function.
Module structure of certain induced representations of compact Lie groups
E. James
Funderburk
269-285
Abstract: Let G be a compact connected Lie group and assume a choice of maximal torus and positive roots has been made. Given a dominant weight $\lambda$, the Borel-Weil Theorem shows how to construct a holomorphic line bundle on whose sections G acts so that the holomorphic sections provide a realization of the irreducible representation of G with highest weight $\lambda$. This paper studies the G-module structure of the space $\Gamma$ of square integrable sections of the Borel-Weil line bundle. It is found that $ \Gamma = {\lim _{n \to \infty }}\Gamma (n)$, where $\Gamma (n) \subset \Gamma (n + 1) \subset \Gamma$ and $ \Gamma (n)$ is isomorphic, as G-module, to $\displaystyle V(\lambda + n\lambda ) \otimes V(n{\lambda ^\ast}),$ where $V(\mu )$ denotes the irreducible representation of highest weight $\mu$, '+' is the Cartan semigroup operation, and '$ ^\ast$' is the contragredient operation. Similar formulas hold for powers of the Borel-Weil line bundle.
A new characterization of Ces\`aro-Perron integrals using Peano derivates
J. A.
Bergin
287-305
Abstract: The $ {Z_n}$-integrals are defined according to the method of Perron using Peano derivates. The properties of the integrals are given including the essential integration by parts theorem. The integrals are then shown to be equivalent to the Cesàro-Perron integrals of Burkill.
Bochner identities for Fourier transforms
Robert S.
Strichartz
307-327
Abstract: Let G be a compact Lie group and R an orthogonal representation of G acting on ${{\mathbf{R}}^n}$. For any irreducible unitary representation $\pi$ of G and vector v in the representation space of $\pi$ define $ \mathcal{S}(\pi ,v)$ to be those functions in $ \mathcal{S}({{\mathbf{R}}^n})$ which transform (under the action R) according to the vector v. The Fourier transform $\mathcal{F}$ preserves the class $\mathcal{S}(\pi ,v)$. A Bochner identity asserts that for different choices of G, R, $\pi ,v$ the Fourier transform is the same (up to a constant multiple). It is proved here that for G, R, $\pi ,v$ and $ G',R',\pi ',v'$ and a map $T:\mathcal{S}(\pi ,v) \to \mathcal{S}(\pi ',v')$ which has the form: restriction to a subspace followed by multiplication by a fixed function, a Bochner identity $f \in \mathcal{S}(\pi ,v)$ holds if and only if $ f \in \mathcal{S}(\pi ,v)$. From this result all known Bochner identities follow (due to Harish-Chandra, Herz and Gelbart), as well as some new ones.
Analysis on the Heisenberg manifold
Richard
Tolimieri
329-343
Abstract: A study of the function theory on the Heisenberg manifold in terms of theta functions. Subject to an explicit error, a ${C^\infty }$-function is written as an infinite sum, with theta functions of different degrees and characteristics playing the same role as exponentials do in the abelian theory.
On composite abstract homogeneous polynomials
Neyamat
Zaheer
345-358
Abstract: We study the null-sets of composite abstract homogeneous polynomials obtained from a pair of abstract homogeneous polynomials defined on a vector space over an algebraically closed field of characteristic zero. First such study for ordinary polynomials in the complex plane was made by Szegö, Cohn, and Egerváry and Szegö's theorem was later generalized to fields and vector spaces, respectively, by Zervos and Marden. Our main theorem in this paper further generalizes their results and, in the complex plane, improves upon Szegö's theorem and some other classical results. The method of proof is purely algebraic and utilizes the author's vector space analogue [Trans. Amer. Math. Soc. 218 (1976), 115-131] of Grace's theorem on apolar polynomials. We also show that our results cannot be further generalized in certain directions.
Extensions of Haar measure to relatively large nonmeasurable subgroups
H. Leroy
Peterson
359-370
Abstract: Let G be a locally compact group, with $\lambda$ a left Haar measure on G. A subgroup is large if it has finite index; a relatively large subgroup of G is a large subgroup of an open subgroup. In §1 we have an existence theorem for relatively large nonopen subgroups, and we observe that such subgroups are not $ \lambda$-measurable. This motivates the development, in §2, of a left-invariant countably additive extension ${\lambda ^ + }$ of $\lambda$, which includes in its domain all unions of left translates of a given relatively large subgroup K. For an arbitrarily chosen family ${\mathcal{K}_I}$ of relatively large subgroups of G, we define (in §3) a finitely additive measure $\lambda _I^ + $ such that, for any $K \in {\mathcal{K}_I},\lambda _I^ +$ is an extension of the corresponding ${\lambda ^ + }$ defined in §2. An example shows that $\lambda _I^ + $ need not be countably additive. Finally, in §4, we observe some aspects of the relationship between $ {\lambda ^ + }$-measurable and $\lambda$-measurable functions, in the context of existing literature on extensions of Haar measure. In particular, we generalize the well-known proposition that $\lambda$-measurable characters are continuous.