Electronic Edition Vol. 30, No. 3, 2004

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen (Houston), J. A. Johnson (Houston), J. Nagata (Osaka), V. I. Paulsen (Houston), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics


Katrinak, Tibor and Zabka, Marek, Comenius University, 842 48 Bratislava, Slovakia (katrinak@fmph.uniba.sk), (zabka@fmph.uniba.sk).
A Weak Boolean Representation of Double Stone Algebras, pp. 615-628.
ABSTRACT. S. Burris and H. Werner in Trans. AMS 284(1979) introduced a (weak) Boolean product of algebras. It is a special subdirect product over a Boolean space. It was shown in the above paper that the (weak) Boolean product of algebras is equivalent to the formation of global sections of sheaves of algebras over Boolean spaces. In this paper we show that every non-trivial double Stone algebra can be characterized in terms of weak Boolean products of pure double Stone algebras. Using this technique a new characterization of the free (regular) double Stone algebras is given.

Mohammad Saleh, Mathematics Department, Birzeit University, Palestine (msaleh@birzeit.edu).
On q.f.d. Modules and q.f.d. Rings, pp. 629-636.
ABSTRACT. The purpose of this paper is to further the study of weakly injective and weakly tight modules a generalization of injective modules. A right R-module M is said to be weakly tight ( tight) if every finitely generated submodule N of its injective hull E(M) is embeddable in a direct sum of copies of M (is embeddable in M). For some classes K of modules , we study when direct sums of modules from K are weakly tight, tight , weakly injective. In particular, we get necessary and sufficient conditions for sum -weak tightness of the injective hull of a simple module.

Saunders, D. J., Faculty of Mathematics and Computing, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK mailto:david@symplectic.demon.co.uk.
Prolongations of Lie groupoids and Lie algebroids, pp. 637-655.
ABSTRACT. We describe a method of prolonging a Lie groupoid to higher-order groupoids in a way which, unlike the tangent functor, takes account of the double fibration. The usual Lie algebroid may be identified with the identity submanifold of the first prolonged groupoid. The Lie algebroid generated by the first prolonged groupoid is identified with the "prolonged algebroid" used recently in the study of Lagrangian dynamics.

M. Crampin and D.J. Saunders, Department of Applied Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK (Crampin@btinternet.com), (david@symplectic.demon.co.uk)
The Hilbert-Caratheodory and Poincare-Cartan forms for higher-order multiple-integral variational problems, pp. 657-689.
ABSTRACT. We consider higher-order homogeneous multiple-integral variational problems defined in the context of m-frame bundles, and construct an m-form (the Hilbert-Caratheodory form) with the same extremals as the Lagrangian. We show that for second-order problems, and for problems where m = 1, this m-form is projectable to the corresponding sphere bundle. Where a second-order problem of this kind has been obtained from a problem on an affine jet bundle, the projected m-form is a generalization of the Caratheodory form used in the study of first-order problems.

Yi, Inhyeop, University of Victoria, Victoria, BC V8W 3P4, Cananda (yih@math.uvic.ca)
Bratteli-Vershik systems for one-dimensional generalized solenoids, pp. 691-704.
ABSTRACT. Let ƒ : XX be an edge-wrapping rule which presents a one-dimensional generalized solenoid X, and let M be the adjacency matrix of ƒ. When X is a wedge of circles, ƒ leaves the unique branch point fixed, and ƒ is orientation preserving, we show that the Bratteli-Vershik map naturally associated to ƒ is topologically conjugate to the return map of the flow on X to a certain cross section.

Tetsuya Hosaka, Department of Mathematics, Utsunomiya University, Mine-machi, Utsunomiya, 321-8505, Japan (hosaka@cc.utsunomiya-u.ac.jp).
The interior of the limit set of groups, pp. 705-721.
ABSTRACT. In this paper, we investigate the interior of the limit set of a group acting on a hyperbolic or CAT(0) space. We show that the interior of the limit set of a group acting on a hyperbolic space is empty, if the interior does not coincide the boundary of the hyperbolic space. Also we show that the interior of the limit set of a convex-cocompact group acting on an almost extendible CAT(0) space is empty, if the interior does not coincide the boundary of the CAT(0) space.

E. D. Tymchatyn, Department of Mathematics and Statistics, Room 142, MacLean Hall, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK S7N 5E6, Canada, (tymchatyn@math.usask.ca) and Chang-Cheng Yang, Department of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China (chang-cheng.yang@risk.sungard.com).
Characterizing Spaces by Disconnection Properties, pp 723-744.
ABSTRACT. We consider a number of very strong separation properties for connected spaces. A space X is finitely separated provided each separator between two points of X contains a finite separator between those two points. If (X, T) is a connected, Hausdorff, finitely separated space then there is a weaker topology W for X such that (X,W) embeds in a finitely separated continuum. We give several characterizations of finitely separated continua and show that under mild conditions finitely separated spaces are ANRs.

D. Daniel, Lamar University, Department of Mathematics, Beaumont, Tx 77710 (daniel@math.lamar.edu), J. Nikiel, American University of Beirut, Department of Mathematics, Beirut, Lebanon (nikiel@aub.edu.lb), L. B. Treybig, Texas A&M University, Department of Mathematics, College Station, TX 77843 (treybig@math.tamu.edu), M. Tuncali, Nipissing University, Faculty of Arts and Sciences, North Bay, Ontario P1B 8L7 (muratt@dns3.nipissingu.ca), and E.D. Tymchatyn, University of Saskatchewan, Department of Mathematics, Saskatoon, Saskatchewan S7N 0W0 (tymchat@snoopy.usask.ca).
Concerning Continua That Contain No Metric Subcontinua, pp. 745-750.
ABSTRACT. In earlier work the authors raised the following question. Let X denote a locally connected continuum such that X is rim-metric and such that X contains no nondegenerate metric subcontinuum. Is X rim-finite and therefore the continuous image of a compact ordered space? Herein we study this question. In so doing, we obtain analogues of a classical result of Whyburn.

Druzhinina, Irina, Departamento de Matemáticas, Universidad Autónoma Metropolitana, C.P. 09340, México D.F. mailto: mich@xanum.uam.mx
Condensations onto connected metrizable spaces, pp. 751-766.
ABSTRACT. We study when a metrizable space X has a weaker connected metrizable topology and prove that:
(a) if the weight of X is less than or equal to the continuuum, then X admits a weaker connected separable metrizable topology whenever X contains a closed subspace which condenses onto a connected non-compact metrizable space;
(b) if the weight of X is greater than or equal to the continuum, then X admits a weaker connected metrizable topology whenever  X contains a closed discrete subset of cardinality equal to its weight. It is also established that if Y is a sigma-discrete or a closed subset of a connected metrizable space X, then the complement X\Y condenses onto a connected metrizable space.

Earl Berkson, Department of Mathematics, University of Illinois, 1409 W.Green Street, Urbana, IL 61801 U.S.A. (berkson@math.uiuc.edu) and T.A. Gillespie, Department of Mathematics and Statistics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3JZ, Scotland (t.a.gillespie@ed.ac.uk).
Operator Means and Spectral Integration of Fourier Multipliers, pp. 767-814.
ABSTRACT. Let (Y,µ) be a sigma-finite measure space. Suppose that p is a real number greater than 1, and T is an absolutely mean-bounded, invertible, separation-preserving linear mapping of Lp(µ) onto itself. In this setting Martín-Reyes and de la Torre established the existence of a uniform Ap weight estimate in terms of discrete weights generated by the pointwise action on Y of certain measurable functions canonically associated with T. By proceeding from this fact we have previously used the interaction between real analysis methods and Ap weights to show that the spectral structure of T transfers to the space Lp(µ) the multiplier actions of the class M1(C), consisting of the classical Marcinkiewicz multiplier functions defined on the unit circle C in the complex plane. The purpose of the present article is to study the overall state of affairs for the function classes Mq(C), where q > 1. Mq(C) consists of the bounded complex-valued functions on C whose q-variations on the dyadic arcs are uniformly bounded, and the source of the multiplier properties of Mq(C) in weighted Lp-spaces can ultimately be found in Rubio de Francia’s weighted Littlewood-Paley inequality for arbitrary intervals, which requires p to exceed 2, and treats weights belonging to A(p/2). Accordingly, we show here that the relevant operator class for achieving the transference of appropriate Mq(C) multiplier classes to Lp(µ), p at least 2, consists of the mean2-bounded operators on Lp(µ) which are separation-preserving. When p > 2 the spectral structure of such operators also transfers to Lp(µ) the aforementioned weighted Littlewood-Paley Inequality for Arbitrary Intervals.

Steven M. Seubert, Bowling Green State University, Bowling Green, OH, 43403-0221 (sseuber@bgnet.bgsu.edu)
Semigroups of compressed Toeplitz operators and Nevalinna - Pick interpolation, pp. 815-827.
ABSTRACT. The purpose of this paper is to determine conditions for an operator commuting with the compression S of the standard unilateral shift on the Hardy space to a shift coinvariant subspace H(B) to embed in a semigroup of operators commuting with S. For B an interpolating Blaschke product, a necessary and sufficient condition is that a symbol of the operator and B have no common zeroes. The condition is shown to be necessary for an arbitrary inner function, but not sufficient for any interpolating Blaschke product. For B an interpolating Blaschke product, necessary and sufficient conditions are given for an operator commuting with S to embed in a semigroup of operators commuting with S consisting entirely of contraction operators using the Nevanlinna-Pick Interpolation Theorem.

Miao, Changxing, Institute of Applied Physics and Computational Mathematics, Beijing, China (miao_changxing@mail.iapcm.ac.cn), and Zhang, Bo, Coventry University, Coventry, UK (b.zhang@coventry.ac.uk).
The Cauchy Problem for Semilinear Parabolic Equations in Besov Spaces, pp. 829-878.
ABSTRACT. In this paper we first give a unified method by introducing the concept of admissible triplets to study local and global Cauchy problems for semi-linear parabolic equations with a general nonlinear term in different Sobolev spaces. In particular, we establish the local well-posedness and small global well-posedness of the Cauchy problem for semi-linear parabolic equations without the homogeneous condition on the nonlinear term. Our results improve the previously known ones, whilst the proofs are simpler compared with previous ones. Secondly, we establish the local well-posedness and small global well-posedness in Besov spaces of the Cauchy problem for semi-linear parabolic equations under suitable conditions. Finally, we study the local well-posedness and small global well-posedness in the critical Besov spaces of the Cauchy problem by means of the improved Sobolev inequality established by Nakamura and Ozawa (J. Funct. Anal. Vol. 127, 1995, pp. 259-269; Vol. 155, 1998, pp. 365-380).

Matos, Julia, Département de Mathématiques, Université d'Evry Val-Essonne, Boulevard François Mitterrand, 91025 Evry Cedex, France,(jmatos@maths.univ-evry.fr) and Souplet, Philippe,  Département de Mathématiques, INSSET, Université de Picardie, 02109 St-Quentin, France, and Laboratoire de Mathématiques Appliquées,  Université de Versailles, 45 avenue des Etats-Unis, 78035 Versailles France (souplet@math.uvsq.fr).
Instantaneous smoothing estimates for the Hermite semigroup in uniformly local spaces and related nonlinear equations, pp. 879-890.
ABSTRACT. We consider the Hermite semigroup, generated by the operator Δ−½y.∇ in  RN.
We establish instantaneous smoothing estimates for the Hermite semigroup in the uniformly
local Lebesgue spaces introduced by Kato [1].
It was known from the classical works of Nelson [3] and Gross [2] that such smooting properties fail in usual weighted Lebesgue spaces. This linear result enables us to prove instantaneous smoothing estimates for some related
nonlinear parabolic problems.
[1] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal., 58 (1975), 181-205.
[2] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1976), 1061-1083.
[3] E. Nelson, The free Markoff field, J. Funct Anal., 12 (1973), 211-227.

Christopher Winfield, Western Kentucky University, Bowling Green, KY 42101 (christopher.winfield@wku.edu)
Local Solvability of Some Partial Differential Operators with Polynomial Coefficients, pp. 891-927.
ABSTRACT. We study the local solvability of partial differential operators which can be expressed as certain polynomials in the vector fields X=Dx and Y=Dy+xkDw for odd integers k greater than or equal to 1. More specifically, we study operators of the form P(X,Y), where P is a homogenous polynomial in X and Y of degree n greater than or equal to 2 where, for complex z, P(z,0) = zn and where P(iz,1) has distinct roots. We apply asymptotic estimates by F.M. Christ (1993) to form a constructive proof of local solvability; to do so, we assume the absence of non-trivial Schwartz-class functions in the kernels of the operators P(-iDx,+x)* and P(-iDx,-x))* (* denoting adjoint) and we assume a certain technical condition on the kernels of the operators P(-iDx,+(z-xk)) and P(-iDx,-(z-xk)) with parameter z>0. The novelty of our results lie in those for k greater than or equal to 3, odd. As a special case, these results prove the local solvability of the operators X2 + Y2 + ia[X,Y] for any complex constant, a, not an odd integer.