## HOUSTON JOURNALOF MATHEMATICS

### Electronic Edition Vol. 27, No. 3, 2001

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), B. H. Neumann (Canberra), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Contents

Youngho Jang and Hoil Kim, T.G.R.C., Kyungpook National University, Taegu 702-701, Korea, (janggane@chollian.net), (hikim@gauss.kyungpook.ac.kr).
A series whose terms are products of two q-Bernoulli Numbers in the p-adic Case, pp. 495--510.
ABSTRACT. The symmetric property Am,n=An-1,m+1 of C. F. Woodcock is connected with the well-known Euler formula for the finite sum of products of two Bernoulli numbers. In this paper, we give a q-analog of the symmetric property in the p-adic case. In particular, putting m=0 into our q-analog, we obtain a q-analog of the Euler formula in the p-adic case.

Alireza Abdollahi, Department of Mathematics, University of Isfahan, Isfahan 81744, Iran, (abdolahi@math.ui.ac.ir).
Some Engel Conditions on Finite Subsets of Certain Groups, pp. 511-522.
ABSTRACT. Engel-like conditions with respect to nilpotency of groups are studied.

Keith Whittington, Department of Mathematics, University of the Pacific, Stockton, CA 95211, (kwhittin@uop.edu).
Open Connected Sets in Homogeneous Spaces, pp. 523-531.
ABSTRACT. It is shown that if a polish topological group acts transitively on a locally compact polish space, then for "most" pairs of points in the space, every continuum containing these points (if any) has arbitrarily small open connected neighborhoods. Characterizations are then given of local connectedness.

Louis Zulli, Department of Mathematics, Lafayette College, Easton, PA 18042-1781, (zullil@lafayette.edu)
Seifert 3-Manifolds that are Bundles and Semi-Bundles, pp. 533-540.
ABSTRACT. We identify the closed, connected, orientable Seifert 3-manifolds that are both surface bundles over the circle and semi-bundles---unions of twisted I-bundles over non-orientable surfaces. We show that there is a single infinite family (indexed by the integers) of closed, connected, oriented Seifert 3-manifolds that admit both structures.

Seokku Ko . Konkuk University, Choongjusi Choongbuk, Korea 380-701. (seokko@apmath.kku.edu), (seokko@kku.ac.kr).
Embedding Compact Riemann Surfaces in Riemannian Manifolds, pp. 541-577.
ABSTRACT. Any compact Riemann surface has a conformal model in any orientable Riemannian manifold. Precisely, we will prove that, given any compact Riemann surface, there is a conformally equivalent model in a prespecified orientable Riemannian manifold. The techniques we use include Garsia's Continuity Lemma, Brouwer's Fixed Point Theorem along with techniques from Teichmuller theory.

C. E. Gutierrez, Department of Mathematics,Temple University, Philadelphia, PA 19122 (gutierrez@math.temple.ed), A. Incognito, School of Science and Engineering, Al Akhawayn University in Ifrane, Ifrane, Morocco 53000, P.O. Box 1862 (A.Incognito@alakhawayn.ma) and J. L. Torrea, Departamento de Matematicas, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain (joseluis.torrea@uam.es).
Riesz Transforms, g-Functions, and Multipliers for the Laguerre Semigroup, pp. 579-592.
ABSTRACT. We study a version of Riesz transforms and show that they are bounded in Lp spaces.

Dutrieux, Yves, Université Pierre et Marie Curie, Paris 6, Boite 186, 4 place Jussieu, 75252 Paris Cedex 05, France (dutrieux@ccr.moka.jussieu.fr).
Quotients of c0 and Lipschitz-homeomorphisms, pp. 593-599.
ABSTRACT. In this paper it is shown that a Banach space whose dual has the approximation property and which is Lipschitz homeomorphic to a quotient of c0 is isomorphic to some quotient of c0. The nearly isometric case is also investigated.

C. E. M. Pearce, Applied Mathematics Dept, The University of Adelaide, Adelaide, SA 5005, Australia (cpearce@maths.adelaide.edu.au), J. Pevcaric, Applied Mathematics Dept, The University of Adelaide, Adelaide, SA 5005, Australia (jpecaric@maths.adelaide.edu.au), and S. Varovsanec, Department of Mathematics, The University of Zagreb, 10000 Zagreb, Croatia (varosans@math.hr).
Polya--type Inequalities for Arbitrary Functions, pp. 601-612.
ABSTRACT. Generalizations of the Polya integral inequality for nonnegative, nondecreasing functions are given involving real and operator functions which satisfy certain subordination and boundary conditions but are otherwise arbitrary. Both discrete and integral versions of the results are addressed. Applications are made to generalized quasiarithmetic means, symmetric means and to operator means.

M. Nakamura, Graduate School of Information Sciences(GSIS), Tohoku University, Sendai 980-8577, Japan (m-nakamu@math.is.tohoku.ac.jp) and T. Ozawa, Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan.
Small Solutions to Nonlinear Wave Equations in the Sobolev Spaces, pp. 613-632.
ABSTRACT. The local and global well-posedness for the Cauchy problem for a class of nonlinear wave equations is studied. The global well-posedness of the problem is proved in the homogeneous Sobolev space Hs of fractional order s>n/2 under the following assumptions: (1) The norm of the Cauchy data in H1/2 is relatively small with respect to the norm in Ha for any fixed a with n/2< a< s. (2) The nonlinearity, f(u), behaves as a power of 1+4/(n-1) near zero and has an arbitrary growth rate at infinity.

Leszek Skrzypczak, Faculty of Mathematics and Computer Science, A.Mickiewicz University Matejki 48-49 60-769 Poznan, Poland (lskrzyp@amu.edu.pl) and Bernadeta Tomasz, Faculty of Mathematics and Computer Science, A.Mickiewicz University Matejki 48-49 60-769 Poznan, Poland (betom@amu.edu.pl).
Compactness of Embeddings of the Trudinger-Strichartz Type for Rotation Invariant Functions, pp. 633-647.
ABSTRACT. Let H be a close subgroup of the group O(n) of rotations of the n-dimensional Euclidean space. Subspaces of functions of fractional Sobolev spaces and the Besov spaces invariant with respect to the natural action of the group H are investigated. Sufficient and necessary conditions for the compactness of the Trudinger-Strichartz type embeddings of the above subspaces into the exponential'' Orlicz classes are given. In particular it is proved that the compactness implies that any orbit of the action of the group H on the unit sphere is infinite.

Marino Badiale, Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa Italy (badiale@bibsns.sns.it) and Alessandro Duci, Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa Italy (duci@cibs.sns).
Concentrated Solutions for a Non Variational Semilinear Elliptic Equation , pp. 649-682.
ABSTRACT. We study a non-variational semilinear elliptic equation and use a reduction method for perturbation problems and find solutions that look like spikes concentrated near suitable points.

Annalisa Baldi, Dipartimento di Matematica, Universita di Bologna, Piazza di Porta S. Donato, 5, I-40127 Bologna, Italy (baldi@dm.unibo.it)
Weighted BV Functions , pp. 683-705.
ABSTRACT. We provide a definition of weighted function of bounded variation when the weight function \omega belongs to a certain subclass of Muckenhoupt's A1 weight class. We obtain Poincaré and isoperimetric inequalities in this space and, as an application, we prove existence of minimal surfaces.

Young-Bok Chung, Department of Mathematics, Chonnam National University, Kwangju 500-757, Korea (ybchung@chonnam.chonnam.ac.kr).
Higher Order Extremal Problem and Proper Holomorphic Mapping , pp. 707-718.
ABSTRACT. Higher order problems with respect to the Riemann mapping theorem are studied.