*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

*Contents*

** 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 : **X** → **X**
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 L^{p}(µ) onto itself. In this
setting Martín-Reyes and de la Torre established the existence of a uniform A_{p}
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 A_{p} weights to show that the spectral structure
of T transfers to the space L^{p}(µ) the multiplier actions of the class
M_{1}(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 M_{q}(C),
where q > 1. M_{q}(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 M_{q}(C) in weighted L^{p}-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 M_{q}(C)
multiplier classes to L^{p}(µ), p at least 2, consists of the mean_{2}-bounded
operators on L^{p}(µ) which are separation-preserving. When p > 2 the
spectral structure of such operators also transfers to L^{p}(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 R^{N}.

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=D_{x} and Y=D_{y}+x^{k}D_{w}
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) = z^{n}
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(-iD_{x},+x)^{*} and P(-iD_{x},-x))^{*}
(* denoting adjoint) and we assume a certain technical condition on the kernels
of the operators P(-iD_{x},+(z-x^{k})) and P(-iD_{x},-(z-x^{k}))
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 X^{2} + Y^{2} + ia[X,Y] for any complex
constant, a, not an odd integer.