Natural maps between CAT(0) boundaries

It is shown that certain natural maps between the ideal, Gromov, and end boundaries of a complete CAT(0) space can fail to be either injective or surjective. Additionally the natural map from the Gromov boundary to the end boundary of a complete CAT(-1) space can fail to be either injective or surjective.Comment: 8 page

Rough CAT(0) spaces

We investigate various notions of rough CAT(0). These conditions define classes of spaces that strictly include the union of all Gromov hyperbolic length spaces and all CAT(0) spaces.Comment: Corrected typos and updated Corollaries 3.22 and 3.2

Sobolev spaces: a short course

Notes for a course in Sobolev spaces given to the Department of Mathematical Analysis, University of La Laguna, Tenerife in May 2000

Pointwise multipliers for reverse Hölder spaces II

We classify weights which map strong reverse Hölder weight classes to weak reverse Hölder weight spaces under pointwise multiplication

Slice conditions and their applications

Slice-type conditions and their relatives have been used in a number of papers of the author and various collaborators to investigate a variety of problems in analysis including Sobolev-type imbedding theorems ([BuKo], [BuOS], [BuSt1], [Bu]), quasiconformal equivalence of domains ([BuSt2], [Bu]), and Gromov hyperbolicity ([BaBu]). Despite their usefulness, these conditions are still somewhat mysterious, but some recent research has helped to shed more light on them ([BaBu], [BuSt3], [BuDiSt]). There are quite a few varieties of slice conditions, partially reflecting the fact that different applications require dierent conditions. In this paper, we define some of these variants, discuss how they differ from each other, and summarize their applications. Along the way, we encounter quite a few examples, counterexamples, and open questions. We discuss (strong) slice conditions and Gromov hyperbolicity in Section 2, weak slice conditions and their applications in Section 3, and some examples and useful lemmas in Section 4

Sequences of reverse Minkowski type

We study sequences of positive numbers satisfying a reverse Minkowski condition. In particular, we classify those monotonic decreasing sequences which can be rearranged to satisfy such a condition

Mixed norms and analytic function spaces

We dene and investigate general mixed-norm type sequence spaces, and strengthen inequalities of Hardy-Littlewood, Hausdor-Young, and others

Space-filling curves and related functions

In this paper, we shall investigate several questions related to space-lling curves. We start with a question whose answer has been known (although not widely known, it would appear) for rather a long time

Nonpositive curvature and complex analysis

We discuss a few of the metrics that are used in complex analysis and potential theory, including the Poincaré, Carathéodory, Kobayashi, Hilbert, and quasihyperbolic metrics. An important feature of these metrics is that they are quite often negatively curved. We discuss what this means and when it occurs, and proceed to investigate some notions of nonpositive curvature, beginning with constant negative curvature (e.g. the unit disk with the Poincaré metric), and moving on to CAT(k) and Gromov hyperbolic spaces. We pay special attention to notions of the boundary at infinity