Abstract. We investigate geometric conditions related to H older imbeddings, and show, among other things, that the only bounded Euclidean domains of the form U V that are quasiconformally equivalent to inner uniform domains are inner uniform domains. Inner uniform domains, as de ned by V ais al a V5], satisfy a uniformity condition with respect to the inner Euclidean metric. These domains form a class intermediate between uniform and John domains and, in particular, they include all Lipschitz domains see Section 1 for de nitions. We prove the following theorem which indicates that this class is well suited to the study of quasiconformal equivalence.

Abstract. We introduce weak slice conditions and investigate imbeddings of Sobolev spaces in various Lipschitz-type spaces. IntroductionBojarski [B] proved that Sobolev-Poincaré imbeddings are valid on all John domains; see also [Mto]. In [BK1], it is shown that John domains are essentially the right class for this imbedding, since a bounded domain G ⊂ R n is a John domain if and only if it supports a Sobolev-Poincaré imbedding and satisfies a certain separation condition. Corresponding results for the p = n (Trudinger) and p > n (Hölder) cases of the Sobolev Imbedding Theorem are given in [BK2], where it is shown that for domains satisfying a certain slice condition, each of these imbeddings is equivalent to a mean cigar condition dependent on p; see Section 1 for definitions of these concepts. For other results on Hölder imbeddings, we refer the reader to [A], [Mz], and [KR].In one way, the results of [BK2] are less satisfying than those of [BK1]. In [BK1], the strong geometric condition (John) is equivalent to the combination of the weak geometric condition (separation) and a Sobolev-Poincaré imbedding. However in [BK2], the weak geometric condition (slice) is not implied by the strong geometric condition (mean cigar) for any value of p ≥ n. For p = n, Buckley and O'Shea [BO] overcame this deficiency by showing that the strong geometric condition is equivalent to the combination of a so-called weak slice condition (which is implied by a slice condition) and the Trudinger imbedding. Here we prove the following analogous result for p > n; the terminology is explained in Sections 1 and 2.

Section 6.1: A Brief Account of the History of Rootfinding 1 An asterisk that precedes a section indicates that the section may be skipped without a significant loss of continuity to the main development of the text.

For a finitely connected planar domain Ω it is shown that the analytic-Poincaré inequalityholds uniformly for all holomorphic functions f on Ω (z 0 ∈ Ω fixed, K a p (Ω) an absolute constant) if and only if the Sobolev-holds for an absolute constant K p (Ω) and for all u ∈ C 1 (Ω) whose integral over Ω is zero. This paper extends a result of Hamilton (1986) who established this equivalence when 1 < p < ∞.

Abstract. The slice condition and the more general weak slice conditions are geometric conditions on Euclidean space domains which have evolved over the last several years as a tool in various areas of analysis. This paper explores some of the finer distinctive properties of the various weak slice conditions. IntroductionThe slice condition is a metric-geometric condition for domains in Euclidean spaces R n that was introduced by the first author and Koskela [BK2] to obtain a set of geometric classifications of a large class of domains in Euclidean spaces which support any of the Sobolev imbeddings, for integrability index p ≥ n. In later works ([BO], [BS1], [BS2]), variations of the slice condition were used to refine these results and also to obtain unrelated results in other areas of analysis. There were many variants of these conditions such as (inner) α-wslice and (inner) α-wslice + conditions, all of which were defined for a variety of purposes; see Section 2 for definitions. We refer to them generically as weak slice conditions, since α-wslice conditions are strictly weaker than the slice condition. The index α indicates which metric is employed; the applications to Sobolev imbeddings require different metrics for each value of p ≥ n.All slice-type conditions, whether the slice condition or any weak slice condition, are very weak. For instance, they are all satisfied by every simply connected planar domain. This follows by the Riemann mapping theorem from the more general fact that the quasiconformal images of inner uniform domains in R n satisfy the slice condition and all weak slice conditions (once a certain auxiliary parameter C is sufficiently large); see Theorem 3.1 of [BS2]. Inner uniform domains, introduced by Väisälä [V5], generalize the well-known class of uniform domains.In [BS2], the authors used weak slice conditions to obtain results regarding quasiconformal equivalence of product domains. One pleasing aspect of the main results of this work was that the weak slice conditions did not appear at all in the statement of the results, but rather as a tool in the proof. In [BO, Theorem 3.4], the slice condition is used to establish one-half of an equivalence in certain situations between a two-weighted variant of Trudinger's inequality and a certain global balance condition. Finally, in [BB], it is shown that under certain rather minimal assumptions, the metric spaces on which the associated quasihyperbolic metric is

We examine two related problems concerning a planar domain Cl. The first is whether Sobolev functions on Q can be approximated by global C 00 functions, and the second is whether approximation can be done by functions in C°°(n) which, together with all derivatives, are bounded on Q. We find necessary and sufficient conditions for certain types of domains, such as starshaped domains, and we construct several examples which show that the general problem is quite difficult, even in the simply connected case.

We generalize the heat polynomials for the heat equation to more general partial differential equations, of higher order with respect to both the time variable and the space variables. Whereas the heat equation requires only one family of polynomials, for an equation of the th order with respect to time we introduce families of polynomials. These families correspond to the initial conditions specified by the Cauchy problem.

Abstract.A domain O. C RN of finite /V-dimensional Lebesgue measure is a p-Poincare domain (1 < p < oo) if there exists a positive constant K such that the p-Poincare inequality ||u||/>>(£2)