Inequalities for some maximal functions. I
Michael
Cowling;
Giancarlo
Mauceri
431-455
Abstract: This paper presents a new approach to maximal functions on ${{\mathbf{R}}^n}$. Our method is based on Fourier analysis, but is slightly sharper than the techniques based on square functions. In this paper, we reprove a theorem of E. M. Stein [16] on spherical maximal functions and improve marginally work of N. E. Aguilera [1] on the spherical maximal function in ${L^2}({{\mathbf{R}}^2})$. We prove results on the maximal function relative to rectangles of arbitrary direction and fixed eccentricity; as far as we know, these have not appeared in print for the case where $n \geqslant 3$, though they were certainly known to the experts. Finally, we obtain a best possible theorem on the pointwise convergence of singular integrals, answering a question of A. P. Calderón and A. Zygmund [3,3] to which N. E. Aguilera and E. O. Harboure [2] had provided a partial response.
Classification of semisimple rank one monoids
Lex E.
Renner
457-473
Abstract: Consider the classification problem for irreducible, normal, algebraic monoids with unit group $G$. We obtain complete results for the groups $ \operatorname{Sl}_2(K) \times {K^\ast}$, $ \operatorname{Gl}_2(K)$ and $\operatorname{PGl}_2(K) \times {K^\ast}$. If $ G$ is one of these groups let $ \mathcal{E}(G)$ denote the set of isomorphy types of normal, algebraic monoids with zero element and unit group $G$. Our main result establishes a canonical one-to-one correspondence $\mathcal{E}(G) \cong {{\mathbf{Q}}^ + }$, where ${{\mathbf{Q}}^ + }$ is the set of positive rational numbers. The classification is achieved in two steps. First, we construct a class of monoids from linear representations of $G$. That done, we show that any other $ E$ must already be one of those constructed. To do this, we devise an extension principle analogous to the big cell construction of algebraic group theory. This yields a birational comparison morphism $ \varphi :{E_r} \to E$, for some $ r \in {{\mathbf{Q}}^ + }$, which is ultimately an isomorphism because the monoid $ {E_r} \in \mathcal{E}(G)$ is regular. The relatively insignificant classification problem for normal monoids with group $ G$ and no zero element is also solved. For each $G$ there is only one such $E$ with $ G \subsetneqq E$.
Some sharp neighborhoods of univalent functions
Johnny E.
Brown
475-482
Abstract: For $\delta \geqslant 0$ and $f(z) = z + {a_2}{z^2} + \cdots$ analytic in $ \vert z\vert < 1$ let the $\delta$-neighborhood of $f$, ${N_\delta }(f)$, consist of those analytic functions $g(z) = z + {b_z}{z^2} + \cdots$ with $\sum\nolimits_{k = 2}^\infty {k\vert{a_k} - {b_k}\vert \leqslant \delta } $. We determine sufficient conditions guaranteeing which neighborhoods of certain classes of convex functions belong to certain classes of starlike functions. We extend some recent results of St. Ruscheweyh and R. Fournier and, at the same time, provide much simpler proofs. We also prove precisely how boundaries affect the value of $\delta$ for some general classes of functions.
Estimates for operators in mixed weighted $L\sp p$-spaces
Hans P.
Heinig
483-493
Abstract: A weighted Marcinkiewicz interpolation theorem is proved. If $T$ is simultaneously of weak type $ ({p_i},{q_i})$, $i = 0,1$; $1 \leqslant {p_0} < {p_1} \leqslant \infty$ and $u$, $v$ certain weight functions, then $ T$ is bounded from $ L_v^p$ to $L_u^q$ for $0 < q < p$, $ p \geqslant 1$. The result is applied to obtain weighted estimates for the Laplace and Fourier transform, as well as the Riesz potential.
Asymptotic expansions of ratios of coefficients of orthogonal polynomials with exponential weights
Attila
Máté;
Paul
Nevai;
Thomas
Zaslavsky
495-505
Abstract: Let ${p_n}(x) = {\gamma _n}{x^n} + \cdots$ denote the $n$th polynomial orthonormal with respect to the weight $ \exp ( - {x^\beta }/\beta )$ where $\beta > 0$ is an even integer. G. Freud conjectured and Al. Magnus proved that, writing $ {a_n} = {\gamma _{n - 1}}/{\gamma _n}$, the expression ${a_n}{n^{ - 1/\beta }}$ has a limit as $n \to \infty$. It is shown that this expression has an asymptotic expansion in terms of negative even powers of $n$. In the course of this, a combinatorial enumeration problem concerning one-dimensional lattice walk is solved and its relationship to a combinatorial identity of J. L. W. V. Jensen is explored.
Epimorphically closed permutative varieties
N. M.
Khan
507-528
Abstract: We show that for semigroups all permutation identities are preserved under epis and that all subvarieties of the permutative variety defined by any permutation identity $\displaystyle {x_1}{x_2} \cdots {x_n} = {x_{{i_1}}}{x_{{i_2}}} \cdots {x_{{i_n}}},$ with $n \geqslant 3$ and such that ${i_n} \ne n$ or $ {i_1} \ne 1$, are closed under epis. Finally we find some sufficient conditions that an identity be preserved under epis in conjunction with any nontrivial permutation identity.
Unstable towers in the odd primary homotopy groups of spheres
Martin
Bendersky
529-542
Abstract: The unstable elements in filtration $2$ of the unstable Novikov spectral sequence are computed. These elements are shown to survive to elements in the homotopy groups of spheres which are related to $ \operatorname{Im}\, J$. The computation is applied to determine the Hopf invariants of compositions of $\operatorname{Im}\, J$ and the exponent of certain sphere bundles over spheres.
Probabilities of first-order sentences about unary functions
James F.
Lynch
543-568
Abstract: Let $f$ be any fixed positive integer and $ \sigma$ a sentence in the first-order predicate calculus of $f$ unary functions. For positive integers $n$, an $n$-structure is a model with universe $\{ 0,1, \ldots ,n - 1\}$ and $f$ unary functions, and $\mu (n,\sigma )$ is the ratio of the number of $ n$-structures satisfying $ \sigma$ to ${n^{nf}}$, the number of $ n$-structures. We show that ${\lim _{n \to \infty }}\mu (n,\sigma )$ exists for all such $\sigma$, and its value is given by an expression consisting of integer constants and the operators $+ , - , \cdot ,/$, and ${e^x}$.
Differential operators and theta series
Solomon
Friedberg
569-589
Abstract: Let $f$ be a modular form on a congruence subgroup of $ {\text{SL}}(2,\mathbb{Z})$--not necessarily holomorphic, but an eigenfunction of the weight $k$ Casimir operator. Maass introduced differential operators (coming from the complexified universal enveloping algebra) which raise and lower by $2$ the weight of such a form and shift the eigenvalue. Here we introduce differential operators on hyperbolic $3$ space analogous to the Maass operators. These change by $2$ the weight of a modular form for an imaginary quadratic field. Theorem. The Maass operators and the hyperbolic space operators are intertwined by the imaginary quadratic Doi-Naganuma (base change) lifting. That is, the following diagram is commutative: \begin{displaymath}\begin{array}{*{20}{c}} & F & {\underset{{{\text{operators}}}... ...ext{Maass}}}}{\leftrightarrow}}} & {\tilde f} & \end{array} \end{displaymath} Using similar techniques for the dual pair $ ({\text{SL}}(2,\mathbb{R}),\;{\text{SO}}(2,1))$, we give a simple proof that the Shimura correspondences preserve holomorphicity (for weight $\geqslant 5/2$) and an explanation for this property directly in terms of the theta series (Weil representation) integral kernel. We also establish similar results for the real quadratic Doi-Naganuma lifting.
A property equivalent to the existence of scales
Howard
Becker
591-612
Abstract: Let ${\text{UNIF}}$ and $ {\text{SCALES}}$ be the propositions that every relation on ${\mathbf{R}}$ can be uniformized, and every subset of $ {\mathbf{R}}$ admits a scale, respectively. For $A \subset {\mathbf{R}}$, let $w(A)$ denote the Wadge ordinal of $ A$, and let $\delta _1^1(A)$ be the supremum of the ordinals realized in the pointclass $ {\Delta ^1}_1(A)$. Theorem $ {\text{(AD)}}$. The following are equivalent: (a) ${\text{SCALES}}$, (b) ${\text{UNIF}} +$ the set $\{ w(A):\delta _1^1(A) = {(w(A))^ + }\}$ contains an $\omega$-cub subset of $\Theta$. Using this theorem, Woodin has shown that if the theory $ {\text{(ZF}} + {\text{DC}} + {\text{AD}} + {\text{UNIF)}}$ is consistent, then the theory ${\text{(ZF}} + {\text{DC}} + {\text{AD}}_{\mathbf{R}} + {\text{SCALES)}}$ is also consistent. In this paper we give a proof of the above theorem and of a local version of it. We also study the ordinal $\delta _1^1(A)$ and give several characterizations of it.
An application of flows to time shift and time reversal in stochastic processes
E. B.
Dynkin
613-619
Abstract: A simple proposition (Theorem 1) on flows allows the investigation of random time shift and time reversal in Markov processes without assuming any regularity of paths. Theorem 5 is a generalization of Nagasawa's time reversal theorem and Theorem 4 generalizes a recent result of Getoor and Glover.
Bounded homotopy equivalences of Hilbert cube manifolds
C. Bruce
Hughes
621-643
Abstract: Let $M$ and $F$ be Hilbert cube manifolds with $ F$ compact. The purpose of this paper is to study homotopy equivalences $f:M \to {{\mathbf{R}}^m} \times F$ which have bounded control in the $ {{\mathbf{R}}^m}$-direction. Roughly, these homotopy equivalences form a semi-simplicial complex $ \mathcal{W}\mathcal{H}({{\mathbf{R}}^m} \times F)$, the controlled Whitehead space. Using results about approximate fibrations, $ \mathcal{W}\mathcal{H}({{\mathbf{R}}^m} \times F)$ is related to the semi-simplicial complex of bounded concordances on $ {{\mathbf{R}}^m} \times F$. Then the homotopy groups of $ \mathcal{W}\mathcal{H}({{\mathbf{R}}^m} \times F)$ are computed in terms of the lower algebraic $K$-theoretic functors $ {K_{ - i}}$.
Thrice-punctured spheres in hyperbolic $3$-manifolds
Colin C.
Adams
645-656
Abstract: The work of ${\text{W}}$. Thurston has stimulated much interest in the volumes of hyperbolic $3$-manifolds. In this paper, it is demonstrated that a $3$-manifold $M\prime$ obtained by cutting open an oriented finite volume hyperbolic $3$-manifold $M$ along an incompressible thrice-punctured sphere $S$ and then reidentifying the two copies of $S$ by any orientation-preserving homeomorphism of $S$ will also be a hyperbolic $3$-manifold with the same hyperbolic volume as $M$. It follows that an oriented finite volume hyperbolic $3$-manifold containing an incompressible thrice-punctured sphere shares its volume with a nonhomeomorphic hyperbolic $3$-manifold. In addition, it is shown that two orientable finite volume hyperbolic $3$-manifolds ${M_1}$ and ${M_2}$ containing incompressible thrice-punctured spheres ${S_1}$ and ${S_2}$, respectively, can be cut open along $ {S_1}$ and ${S_2}$ and then glued together along copies of ${S_1}$ and ${S_2}$ to yield a $3$-manifold which is hyperbolic with volume equal to the sum of the volumes of ${M_1}$ and ${M_2}$. Applications to link complements in $ {S^3}$ are included.
Singular behavior in nonlinear parabolic equations
Wei-Ming
Ni;
Paul
Sacks
657-671
Abstract: In this paper, we study the well-posedness of the initial-boundary value problems of some quasilinear parabolic equations, namely, nonlinear heat equations and the porous medium equation in the fast-diffusion case. We establish nonuniqueness (local in time) and/or nonregularizing effect of these equations in some critical cases. The key which leads to the resolution of these problems is to study some singular solutions of the elliptic counterparts of these parabolic problems (the so-called $ M$-solutions of the Lane-Emden equations in astrophysics).
A weighted inequality for the maximal Bochner-Riesz operator on ${\bf R}\sp 2$
Anthony
Carbery
673-680
Abstract: For $f \in \mathcal{S}({{\mathbf{R}}^2})$, let $(T_R^\alpha f)\hat \emptyset (\xi ) = (1 - \vert\xi {\vert^2}{R^2})_ + ^\alpha \hat f(\xi )$. It is a well-known theorem of Carleson and Sjölin that $T_1^\alpha$ defines a bounded operator on $ {L^4}$ if $\alpha > 0$. In this paper we obtain an explicit weighted inequality of the form $\displaystyle \int {\mathop {\sup }\limits_{0 < R < \infty } \vert T_R^\alpha f(x){\vert^2}w(x)\;dx \leqslant \int {\vert f{\vert^2}{P_\alpha }w(x)\;dx,} } $ with ${P_\alpha }$ bounded on ${L^2}$ if $ \alpha > 0$. This strengthens the above theorem of Carleson and Sjölin. The method gives information on the maximal operator associated to general suitably smooth radial Fourier multipliers of $ {{\mathbf{R}}^2}$.
Estimates of the harmonic measure of a continuum in the unit disk
Carl H.
FitzGerald;
Burton
Rodin;
Stefan E.
Warschawski
681-685
Abstract: The harmonic measure of a continuum in the unit disk is estimated from below in two ways. The first estimate is in terms of the angle subtended by the continuum as viewed from the origin. This result is a dual to the Milloux problem. The second estimate is in terms of the diameter of the continuum. This estimate was conjectured earlier as a strengthening of a theorem of D. Gaier. In preparation for the proofs several lemmas are developed. These lemmas describe some properties of the Riemann mapping function of a disk with radial incision onto a disk.
Finely harmonic functions with finite Dirichlet integral with respect to the Green measure
Bernt
Øksendal
687-700
Abstract: We consider finely harmonic functions $h$ on a fine, Greenian domain $V \subset {{\mathbf{R}}^d}$ with finite Dirichlet integral wrt $Gm$, i.e. $(\ast)$ $\displaystyle \int_V\vert\nabla h(y)\vert^2G(x,y)\,dm(y) < \infty \quad {\text{for}}\;x \in V,$ where $m$ denotes the Lebesgue measure, $G(x,y)$ the Green function. We use Brownian motion and stochastic calculus to prove that such functions $h$ always have boundary values ${h^\ast}$ along a.a. Brownian paths. This partially extends results by Doob, Brelot and Godefroid, who considered ordinary harmonic functions with finite Dirichlet integral wrt $m$ and Green lines instead of Brownian paths. As a consequence of Theorem 1 we obtain several properties equivalent to $( \ast )$, one of these being that $h$ is the harmonic extension to $ V$ of a random "boundary" function ${h^\ast}$ (of a certain type), i.e. $h(x) = {E^x}[{h^\ast}]$ for all $x \in V$. Another application is that the polar sets are removable singularity sets for finely harmonic functions satisfying $( \ast )$. This is in contrast with the situation for finely harmonic functions with finite Dirichlet integral wrt $m$.
Some estimates for nondivergence structure, second order elliptic equations
Lawrence C.
Evans
701-712
Abstract: We obtain various formal estimates for solutions of nondivergence structure, second order, uniformly elliptic ${\text{PDE}}$. These include interior lower bounds and also gradient estimates in ${L^p}$, for some $p < 0$.
The Mackey topology and complemented subspaces of Lorentz sequence spaces $d(w,p)$ for $0<p<1$
M.
Nawrocki;
A.
Ortyński
713-722
Abstract: In this paper we continue the study of Lorentz sequence spaces $ d(w,p)$, $0 < p < 1$, initiated by N. Popa [8]. First we show that the Mackey completion of $d(w,p)$ is equal to $d(v,1)$ for some sequence $v$. Next, we prove that if $d(w,p) \not\subset {l_1}$, then it contains a complemented subspace isomorphic to ${l_p}$. Finally we show that if $\lim {n^{ - 1}}\left(\sum\nolimits_{i = 1}^n {w_i}\right)^{1/p} = \infty$, then every complemented subspace of $d(w,p)$ with symmetric bases is isomorphic to $ d(w,p)$.
A kinetic approach to general first order quasilinear equations
Yoshikazu
Giga;
Tetsuro
Miyakawa;
Shinnosuke
Oharu
723-743
Abstract: This paper presents a new method for constructing entropy solutions of first order quasilinear equations of conservation type, which is illustrated in terms of the kinetic theory of gases. Regarding a quasilinear equation as a model of macroscopic conservation laws in gas dynamics, we introduce as the corresponding microscopic model an auxiliary linear equation involving a real parameter $\xi$ which plays the role of the velocity argument. Approximate solutions for the quasilinear equation are then obtained by integrating solutions of the linear equation with respect to the parameter $ \xi$. All of these equations are treated in the Fréchet space $ L_{{\text{loc}}}^1({R^n})$, and a convergence theorem for such approximate solutions to the entropy solutions is established with the aid of nonlinear semigroup theory.
On the Gauss-Bonnet theorem for complete manifolds
Steven
Rosenberg
745-753
Abstract: For a manifold diffeomorphic to the interior of a compact manifold with boundary, several classes of complete metrics are given for which the Gauss-Bonnet Theorem is valid.
Pull-backs of $C\sp \ast$-algebras and crossed products by certain diagonal actions
Iain
Raeburn;
Dana P.
Williams
755-777
Abstract: Let $G$ be a locally compact group and $p:\Omega \to T$ a principal $G$-bundle. If $A$ is a ${C^\ast}$-algebra with primitive ideal space $ T$, the pull-back $ {p^\ast}A$ of $ A$ along $p$ is the balanced tensor product $ {C_0}(\Omega ){ \otimes _{C(T)}}A$. If $\beta :G \to \operatorname{Aut}\,A$ consists of $C(T)$-module automorphisms, and $\gamma :G \to \operatorname{Aut}\,{C_0}(\Omega )$ is the natural action, then the automorphism group $ \gamma \otimes \beta$ of $ {C_0}(\Omega ) \otimes A$ respects the balancing and induces the diagonal action ${p^\ast}\beta $ of $G$ on ${p^\ast}A$. We discuss some examples of such actions and study the crossed product ${p^\ast}A{ \times _{{p^\ast}\beta }}G$. We suggest a substitute $D$ for the fixed-point algebra, prove ${p^\ast}A \times G$ is strongly Morita equivalent to $D$, and investigate the structure of $D$ in various cases. In particular, we ask when $D$ is strongly Morita equivalent to $ A$--sometimes, but by no means always--and investigate the case where $ A$ has continuous trace.
Finite codimensional ideals in function algebras
Krzysztof
Jarosz
779-785
Abstract: Assume $ S$ is a compact, metric space and let $M$ be a finite codimensional closed subspace of a complex space $C(S)$. In this paper we prove that if each element from $M$ has at least $k$ zeros in $S$, then for some ${s_1}, \ldots ,{s_k} \in S,M \subseteq \{ f \in C(S):f({s_1}) = \cdots = f({s_k}) = 0\}$.
Essential dimension lowering mappings having dense deficiency set
Mladen
Bestvina
787-798
Abstract: Two classes of surjective maps $ f:{S^m} \to {S^n}$ that are one-to-one over the image of a dense set are constructed. We show that for $ m,n \geq 3$ there is a monotone surjection $ f:{S^m} \to {S^n}$ that is one-to-one over the image of a dense set; and for $3 \leq n \leq m \leq 2n - 3$, each element of $ {\pi _m}({S^n})$ can be represented as a monotone surjection $f:{S^m} \to {S^n}$ that is one-to-one over the image of a dense set.
Analytic operator algebras (factorization and an expectation)
Baruch
Solel
799-817
Abstract: Let $M$ be a $\sigma$-finite von Neumann algebra and ${\{ {\alpha _t}\} _{t \in T}}$ a periodic flow on $M$. The algebra of analytic operators in $ M$ is $\{ a \in M:{\text{sp}_\alpha }(a) \subseteq {{\mathbf{Z}}_ + }\} $ and is denoted ${H^\infty }(\alpha )$. We prove that every invertible operator $a \in {H^\infty }(\alpha )$ can be written as $a = ub$, where $u$ is unitary in $M$ and $b \in {H^\infty }(\alpha ) \cap {H^\infty }{(\alpha )^{ - 1}}$. We also prove inner-outer factorization results for $a \in {H^\infty }(\alpha )$. Another result represents $ {H^\infty }(\alpha )$ as the image of a certain nest subalgebra (of a von Neumann algebra that contains $M$) via a conditional expectation. As corollaries we prove a distance formula and an interpolation result for the case where $M$ is an injective von Neumann algebra.
Banach spaces with the $L\sp 1$-Banach-Stone property
Peter
Greim
819-828
Abstract: It has previously been shown that separable Banach spaces $V$ with trivial $L$-structure have the $ {L^1}$-Banach-Stone property, i.e. every surjective isometry between two Bochner spaces $ {L^1}({\mu _i},V)$ induces an isomorphism of the two measure algebras. We remove the separability restriction, employing the topology of the measure algebra's Stonean space. The result is achieved via a complete description of the $L$-structure of $ {L^1}(\mu ,V)$.
Pro-Lie groups
R. W.
Bagley;
T. S.
Wu;
J. S.
Yang
829-838
Abstract: A topological group $ G$ is pro-Lie if $ G$ has small compact normal subgroups $K$ such that $G/K$ is a Lie group. A locally compact group $ G$ is an $ L$-group if, for every neighborhood $U$ of the identity and compact set $C$, there is a neighborhood $V$ of the identity such that $ gH{g^{ - 1}} \cap C \subset U$ for every $g \in G$ and every subgroup $H \subset V$. We obtain characterizations of pro-Lie groups and make several applications. For example, every compactly generated $L$-group is pro-Lie and a compactly generated group which can be embedded (by a continuous isomorphism) in a pro-Lie group is pro-Lie. We obtain related results for factor groups, nilpotent groups, maximal compact normal subgroups, and generalize a theorem of Hofmann, Liukkonen, and Mislove [4].
On Chebyshev subspaces in the space of multivariate differentiable functions
András
Kroó
839-852
Abstract: In the present paper we give a characterization of Chebyshev sub-spaces in the space of (real or complex) continuously-differentiable functions of two variables. We also discuss various applications of the characterization theorem.
On the higher Whitehead groups of a Bieberbach group
Andrew J.
Nicas
853-859
Abstract: Let $\Gamma$ be a Bieberbach group, i.e. the fundamental group of a compact flat Riemannian manifold. In this paper we show that if $p > 2$ is a prime, then the $ p$-torsion subgroup of $ {\text{Wh}_i}(\Gamma )$ vanishes for $ 0 \leq i \leq 2p - 2$, where $ {\text{Wh}_i}(\Gamma )$ is the $i$th higher Whitehead group of $\Gamma$. The proof involves Farrell and Hsiang's structure theorem for Bieberbach groups, parametrized surgery, pseudoisotopy, and Waldhausen's algebraic $K$-theory of spaces.