Embeddings into Orlicz Spaces for Functions from Unbounded Irregular Domains
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Abstract
We study Sobolev functions defined in unbounded irregular domains in the Euclidean nspace. We show that there exist embeddings into suitable Orlicz spaces from the space \(L^1_p\), \(1\le p <n\). It turns out that the corresponding Orlicz function depends on the geometry of the domain. The results are sharp for \(L^1_1\)functions.
Keywords
Riesz potential Pointwise estimate Orlicz space Unbounded convex domain Nonsmooth domain Sobolev inequality Poincaré inequalityMathematics Subject Classification
31C15 42B20 26D10 46E30 46E351 Introduction
Although embeddings for functions defined in bounded irregular domains have been studied systematically, see for example [13, 16], unbounded irregular domains seem to have been studied less, we refer to [10, 13].
A classical example of an embedding into an Orlicz space for Sobolev functions from the Sobolev space \(W^{1,n}\) is in [18]. But also, if the domain is irregular then an Orlicz space can be a natural target space for functions defined in \(L^1_p\) as in [6, 8]. For papers where an Orlicz space is a target space when the functions come from another Orlicz space we refer to [3, 4].
The following theorem tells which kind of Nfunctions we are interested in. These Nfunctions can encode and reveal the geometry of the domain.
Theorem 1.1
By Theorem 1.1 we prove as an intermediate step the Sobolevtype inequality (1.1) for functions defined in bounded \(\varphi \)John domains \(D_i\), in Theorem 4.1 (\(1<p<n\)) and Theorem 4.2 (\(p=1\)). These results seem to be new and they recover some known results when \(p=1\). By using these bounded domains’ results we obtain our main result for unbounded domains.
Theorem 1.2
 (a)
\(D = \cup _{i=1}^\infty D_i\), where \(D_1>0\);
 (b)
\(\overline{D}_i \subset D_{i+1}\) for each i;
 (c)
each \(D_i\) is a bounded \(\varphi \)cigar John domain with a constant \(c_J\).
We give examples in Example 4.5. Finally in Sect. 5 we show that the target space cannot be a Lebesgue space in general.
2 John Domains
 (1)
\(\varphi \) is continuous,
 (2)
\(\varphi \) is strictly increasing,
 (3)
\(\varphi (0) =0\),
 (4)there exists a constant \(C_\varphi \ge 1\) such thatwhenever \(0<t_1\le t_2\),$$\begin{aligned} \frac{\varphi (t_1)}{t_1}\le C_\varphi \frac{\varphi (t_2)}{t_2} \end{aligned}$$
 (5)
\(\varphi \) satisfies the \(\Delta _2\)condition i.e. there exists a constant \(C_\varphi ^{\Delta _2} \ge 1\) such that \(\varphi (2t) \le C_\varphi ^{\Delta _2} \varphi (t)\) for every \(t>0\).
The definition of a bounded John domain goes back to John [12, Definition, p. 402] who defined an inner radius and an outer radius domain, and later this domain was renamed as a John domain in [14, 2.1].
Definition 2.1
The set \(\mathrm{Cig}E(a,b)\) is called a cigar with core E joining a and b. We point out that if D is a \(\varphi \)cigar John domain with \(\varphi (t) = t^p\), \(p \ge 1\), then it is a \(\varphi \)cigar John domain with \(\varphi (t) = t^q\) for every \(q \ge p\). For the case \(\psi (t)=\varphi (t)=t\) for all \(t\ge 0\), in Definition 2.1, we refer to [17, 2.1] and [15, 2.11 and 2.13]. Note that it is crucial that the length of the curve does not depend on the distance between the end points a and b. In bounded uniform domains the length of the cigar depends on \(ab\) but they are much more regular than our cigar John domains, see [15].
If D is a bounded domain then the following definition from [7, Definition 4.1] for a \(\psi \)John domain gives an equivalent definition to a bounded \(\varphi \)cigar John domain.
Definition 2.2
Remark 2.3
(1) If the function \(\psi \) is defined as in (2.1) with the function \(\varphi \), then a bounded domain is a \(\psi \)John domain if and only if it is a \(\varphi \)John domain.
(2) If \(\psi (t) =t\), then our definition for bounded \(\psi \)John domains coincides with the definition of the classical John domains. If \(\psi (t) = t^{\lambda }\), \(\lambda \ge 1\) then our definition for bounded \(\psi \)John domains coincides with the definition of the flexible cone condition in [2].
Theorem 2.4
Note that when \(\mathrm{diam}(D) \rightarrow \infty \), then \(\alpha \rightarrow \infty \) with the same speed as \(\mathrm{diam}(D)\).
Proof
If \(a \in B(x_0, 2r)\), then it can be clearly joint to \(x_0\) by a line segment and the claim is clear.
Let \(z_0 \in E\) be the first point from a that satisfies \(z_0 \in \partial B(x_0, r)\). We denote by \(\gamma \) the function so that E is parametrized by its arc length such that \(\gamma (0) = a\), \(\gamma (t_0) = z_0\) and \(\gamma (\ell (E) ) = x_0\). We replace \(E[z_0, x_0]\) by the radius of the ball \(B(x_0, r)\), if needed. This new arc is written as \(E'\). Note that \(\ell (E') \le \ell (E)\).
3 Pointwise Estimates
We proceed to prove pointwise estimates for domains which are not classical John domains.
Lemma 3.1

\(B_0 = B\Big (x_0, \frac{1}{2} \mathrm{dist}(x_0, \partial D)\Big )\);

\(\psi (\mathrm{dist}(x, B_i))\le K r_i\), and \(r_i \rightarrow 0 \) as \(i\rightarrow \infty \);

no point of the domain D belongs to more than N balls \(B(x_i, r_i)\); and

\(B(x_i, r_i) \cup B(x_{i+1}, r_{i+1}) \le M B(x_i, r_i) \cap B(x_{i+1}, r_{i+1})\).
Proof
The proof is in [7, Lemma 4.3]. We recall only the proof of the inequality \(\psi (\mathrm{dist}(x, B_i))\le K r_i\), since we have to show that constant K does not blow up when \(\mathrm{diam}(D) \rightarrow \infty \).
Remark 3.2
(1) The constant K in the previous lemma can be chosen to be \( K=\max \{\frac{c \beta }{\mathrm{dist}(x_0,\partial D)}, 2C_\varphi \varphi (1), \frac{2\beta }{\alpha }\}.\)
We recall the following definitions. Let G be an open set of \({\mathbb {R}^n}\). We denote the Lebesgue space by \(L^{p}(G)\), \(1\le p < \infty \). By \(L^1_p(G)\), \(1\le p < \infty \), we denote those locally integrable functions whose first weak distributional derivatives belongs to \(L^p(G)\), that is, \(L^1_p(G) =\left\{ u \in L^1_{loc }(G): \nabla u \in L^p(G) \right\} \). By \(W^{1,p}(G)\), \(1\le p < \infty \), we denote those functions from \(L^p(G)\) whose first weak distributional derivatives belongs to \(L^p(G)\), that is, \(W^{1,p}(G) =\left\{ u \in L^p(G): \nabla u \in L^p(G) \right\} \).
Theorem 2.4 and Lemma 3.1 give the following pointwise estimate which we recall from [7, Theorem 4.4].
Theorem 3.3
We recall the definitions of Nfunctions and Orlicz spaces.
Definition 3.4
 (N1)
H is continuous,
 (N2)
H is convex,
 (N3)
\(\lim _{t \rightarrow 0^+}\frac{H(t)}{t} =0\) and \(\lim _{t \rightarrow \infty }\frac{H(t)}{t} =\infty \).
Continuity and \(\lim _{t \rightarrow 0^+}\frac{H(t)}{t} =0\) yield that \(H(0)=0\).
Convexity yields that \(\frac{H(t)}{t} \le \frac{H(s)}{s}\) for \(0<t<s\) and thus H is a strictly increasing function.
By the notation \(f\lesssim g\) we mean that there exists a constant \(C>0\) such that \(f(x)\le C g(x)\) for all x. The notation \(f\approx g\) means that \(f\lesssim g\lesssim f\).
Two Nfunctions H and K are equivalent, which is written as \(H\simeq K\), if there exists \(m\ge 1\) such that \(H(t/m)\le K(t)\le H(mt)\) for all \(t>0\). Equivalent Nfunctions give the same space with comparable norms. We point out that \(H\simeq K\) if and only if for the inverse functions \(H^{1}\approx K^{1}\).
Let G in \({\mathbb {R}^n}\) be an open set.
Theorem 3.5
Our goal is to find a formula which would give all suitable functions H. Examples of some of these functions were given in [7, Section 6].
Proof of Theorem 1.1
This yields that the function F exists and is strictly increasing.
Hästö has shown in [11, Proposition 3.1] that if \(f:[0, \infty ) \rightarrow [0, \infty )\) is leftcontinuous, \(f(0)=\lim _{s \rightarrow 0^+} f(s) =0\), \(\lim _{s \rightarrow \infty } f(s)= \infty \) and \(x \mapsto {f(x)}/{x}\) is increasing, then f is equivalent to a convex function. We obtain that F is equivalent to a convex function H. Here the implicit constant depends only on the constant in the \(\Delta _2\)condition, that is, it depends only on n and p.
Remark 3.6
Example 3.7
Functions \(\varphi (t)=t^{\alpha }/\log ^{\beta }(e+1/t)\), \(\alpha \in [1, \frac{n}{n1})\) and \(\beta \ge 0\), satisfy the assumptions of Theorem 1.1.
Theorems 1.1 and 3.5 yield the following result.
Theorem 3.8
As a corollary we obtain from Theorems 3.3 and 3.8:
Corollary 3.9
4 On Embeddings
Corollary 3.9 is essential in the proofs of the following Theorems 4.1 and 4.2.
Theorem 4.1
Proof
Theorem 4.2
The term \(\min \{1, \mathrm{diam}(D)\}\) means that the constant depends on the diameter only for small diameters. For large diameters the constant is independent of the diameter. \(\square \)
Proof
Remark 4.3
In Theorem 4.2 the Nfunction H is the best possible in a sense that it cannot be replaced by any Nfunction K that satisfies the \(\Delta _2\)condition and \(\lim _{t \rightarrow \infty } \frac{K(t)}{H(t)} = \infty \).
Remark 4.4
We refer to the detailed discussion in [6, 7] for the fact that our result is optimal when \(p=1\).
Next we prove our main theorem.
Proof of Theorem 1.2
Example 4.5
 (a)
\({\mathbb {R}^n}\), \(n\ge 2\).
 (b)
\(\left\{ (x', x_n)\in {\mathbb {R}^n}: x_n \ge 0 \quad \text {and} \quad x'< \psi (x_n) \right\} \).
 (c)
\(\mathbb {R}^2{\setminus } \left( \{(x,\varphi (x)) \in \mathbb {R}^2: 0\le x \le 1\} \cup \{(x,\varphi (x)) \in \mathbb {R}^2: 0\le x \le 1\} \right) \).
 (d)
5 Lebesgue Space Cannot be a Target Space
In this section we give an example which shows that for certain unbounded \(\varphi \)cigar John domains the target space cannot be a Lebesgue space. The idea is that at near the infinity the target space should be \(L^{np/(np)}\) but local structure of the domain may not allow so good integrability. We assume a priori that the function \(\varphi \) has the properties (1)–(5). Later on we give extra conditions to the function \(\varphi \).
We point out that with our assumptions the case \(\lim _{t \rightarrow 0^+} t/\varphi (t)=0\) is not possible. Namely if \(\lim _{t \rightarrow 0^+} t/\varphi (t)=0\), then \(\lim _{t \rightarrow 0^+} \varphi (t)/t=\infty \), and this contradicts with condition (4).
Thus we have proved the following remarks.
Remark 5.1
Remark 5.2
Notes
Acknowledgements
Open access funding provided by University of Helsinki including Helsinki University Central Hospital. The authors would like to thank the referee for reading the manuscript carefully and for giving valuable comments.
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