An Embedding into an Orlicz Space for Irregular John Domains
- PDF / 327,389 Bytes
- 21 Pages / 439.37 x 666.142 pts Page_size
- 37 Downloads / 235 Views
An Embedding into an Orlicz Space for Irregular John Domains Petteri Harjulehto · Ritva Hurri-Syrjänen · Juha Kapulainen
Received: 18 September 2013 / Revised: 19 November 2013 / Accepted: 16 December 2013 © Springer-Verlag Berlin Heidelberg 2014
Abstract We prove an embedding into an Orlicz space for irregular John domains when the irregularity is controlled by a logarithmic type function. By constructing an example we show that the embedding is essentially sharp. Keywords
Orlicz embedding · Irregular domain · ϕ-John domain
Mathematics Subject Classification (2010)
46E35 · 26D10 · 46E30
1 Introduction We are interested in sharp embeddings into Orlicz spaces for bounded irregular domains in Euclidean n-space Rn , n ≥ 2. We study ϕ-John domains with the function ϕ(t) =
t , t > 0. log(e + t −1 )
Communicated by Olli Martio. To the memory of Fred Gehring and with deep gratitude to Fred and Lois for all their mentoring and encouragement. P. Harjulehto Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland e-mail: [email protected] R. Hurri-Syrjänen (B) · J. Kapulainen Department of Mathematics and Statistics, University of Helsinki, 00014 Helsinki, Finland e-mail: [email protected] J. Kapulainen e-mail: [email protected]
123
P. Harjulehto et al.
This means a domain D where any point x in D can be joined to a given point x0 in D by a rectifiable curve γ parametrized by its arc length, such that for any point γ (t) which is on the image of the curve, the inequality ϕ(t) ≤ β dist γ (t), ∂ D holds. Here, the constant β = β(D, x0 ) depends on the domain D and the chosen point x0 only. Lipschitz domains and classical John domains are ϕ-John domains, since for them the function ϕ is replaced by an identity function. But there are ϕ-John domains which are not classical John domains; for example, a mushroom-type domain is given in Sect. 6. On the other hand, the ϕ-John domains are examples of s-John domains, s > 1, where ϕ is replaced by a power function t → t s . Other examples of s-John domains are quasihyperbolic boundary condition domains which were introduced by Gehring and Martio, [7, 3.6]. We refer to the discussion after Definition 3.1. It is well known that there is an embedding W 1, p (D) → L np/(n− p) (D) for a classical John domain D with 1 ≤ p < n, [2, Lem. 3, Lem. 4, (6)] and [26, Thm.]. For an s-John domain D with 1 ≤ p < n there is an embedding W 1, p (D) → L np/((n−1)s+1− p) (D) for sufficiently small s > 1 [16, Thm. 2.3], and hence also for a domain satisfying the quasihyperbolic boundary condition. Thus, our target space should be between these two Lebesgue spaces L np/(n− p) (D) and L np/((n−1)s+1− p) (D). Let 1 < p < n. Let us define : [0, ∞) → R, (t) =
t logn−1 (m + t)
np n− p
,
where m = m(n, p) ≥ e. We study the Orlicz space L (D) equipped with its Luxemburg norm. For every bounded domain D, 1 ≤ p < n and s > 1, the strict inclusions np
np
L n− p (D) L (D) L (n−1)s+1− p (D) hold. The following theorem i
Data Loading...
