A density result on Orlicz-Sobolev spaces in the plane

We show the density of smooth Sobolev functions $W^{k,\infty}(\Omega)\cap C^\infty(\Omega)$ in the Orlicz-Sobolev spaces $L^{k,\Psi}(\Omega)$ for bounded simply connected planar domains $\Omega$ and doubling Young functions $\Psi$.

An important basic question in the theory of function spaces is the relation between different spaces. Answers to this question can be given for instance in terms of embeddings and density results. In this paper, we show that in Orlicz-Sobolev spaces on bounded simply connected planar domains we can approximate functions with bounded derivatives if we consider only the highest order derivatives in the norm. Theorem 1.1. Let k ∈ N, Ψ be a doubling Young function, and Ω ⊂ R 2 be a bounded simply connected domain. Then the subspace W k,∞ (Ω) ∩ C ∞ (Ω) is dense in the space L k,Ψ (Ω).
Recall that for a domain Ω ⊂ R 2 and a Young function Ψ, the version of the Orlicz-Sobolev space L k,Ψ (Ω) used in Theorem 1.1 is defined as The space L k,Ψ (Ω) is equipped with the semi-norm |α|=k ∇ α f L Ψ (Ω) , where · L Ψ (Ω) is the Luxemburg norm. See Section 2 for more basic information on Orlicz and Orlicz-Sobolev spaces.
We will use a Whitney decomposition of the domain Ω ⊂ R 2 and make a polynomial approximation near the boundary ∂Ω. The validity of the approximation is proven using a Ψ − Ψ Poincaré inequality. The form of the polynomial approximation we use here was introduced in [28], where a density result was shown for homogeneous Sobolev spaces on simply connected planar domains. This was then extended to Gromov hyperbolic domains in higher dimensions in [27]. In turn, both of these results were generalizations of density results for first order Sobolev spaces [23,22] which were partly motivated by the recent progress on planar Sobolev extension domains [21,31].
Although for every domain smooth functions are dense in W k,p (Ω) [26] and, more generally, in W k,Ψ (Ω) for doubling Ψ [10], the derivatives of the approximating smooth functions might blow up near the boundary. Therefore, the density of other function spaces, such as W k,q (Ω) in W k,p (Ω) might be false, see [20,22] for this, and for instance [3,19,29] for earlier counter examples on other function spaces. The density of global smooth functions in W 1,p (Ω) is known for instance for Jordan domains [25,23], but the case for higher order Sobolev spaces is still open. Similarly, in the case k ≥ 2, the density result presented in Theorem 1.1 remains still open for the full Orlicz-Sobolev space W k,Ψ (Ω) as well as for the usual Sobolev space W k,p (Ω).
In the same way as for the first order Sobolev spaces [23], for W 1,Ψ (Ω) we get a better density result as a corollary of Theorem 1.1. This is simply because we may first cut a function in W 1,Ψ (Ω) from above and below introducing a small error in the norm, so that the function becomes an L ∞ (Ω) function. The remaining approximations do not change the fact that the function is in L ∞ (Ω).
The paper is organized as follows. In Section 2 we recall the Whitney decomposition and list the required prerequisites from the Orlicz theory. In Section 3 we give a partition of unity for the domain using a Whitney type decomposition. Finally, in Section 4 we show the proof of Theorem 1.1.

Preliminaries
In this paper, we will usually denote constants by C. The value of the constant might change between appearances, even in a chain of inequalities, but the dependence of the constant on a set of fixed parameters is always stated. Sometimes, to clarify the dependence, the parameters are written inside parentheses C(·).

Whitney decomposition.
In this section we recall the Whitney decomposition of a domain in R d . Such decomposition is standard in analysis, see for instance Whitney [34] or the book of Stein [32, Chapter VI]. We will use a version of the decomposition that was used in [28].
We denote the sidelength of a square Q by ℓ(Q). For notational convenience we start the Whitney decomposition below from squares with sidelength 2 −1 . Formally, since we are working with doubling Young functions, by rescaling, we may consider all bounded domains Ω ⊂ R 2 to have diam(Ω) ≤ 1 in which case no Whitney decomposition would have squares larger than the ones used below regardless of the starting scale. A Whitney decomposition in the plane consists of dyadic squares. Let us first recall those.

Definition 2.1 (Dyadic squares). A dyadic interval in R is an interval of the form
A dyadic square in R 2 is a product of dyadic intervals of the same length. That is, a dyadic square is a set of the form for some integers m 1 and , m 2 .
Let us now define a Whitney decomposition following Lemma 2.3 in [28].

Definition 2.2 (Whitney decomposition).
Let Ω be a bounded open subset of R 2 . A Whitney decomposition is a collectionF of dyadic squares inside Ω satisfying the following properties.
Suppose Q 1 , . . . , Q m are Whitney squares such that Q j and Q j+1 touch and 1 4 ≤ We say then {Q 1 , . . . , Q m } is a chain connecting Q 1 to Q m and define the length of that chain to be the integer m.
It follows easily from the convexity and Ψ(0) = 0, that the function t → Ψ(t)/t is increasing. This implies that if Ψ is strictly increasing, then the function Ψ −1 (t)/t is decreasing. A Young function Ψ is doubling if there is a constant C > 0 such that for each t ≥ 0. The smallest constant C satisfying (2.1) is called the doubling constant of Ψ.
Definition 2.4. Given a doubling Young function Ψ and an open set Ω ⊂ R 2 , we denote by L Ψ (Ω), the Orlicz space associated to Ψ, defined by We will not use the Luxemburg norm in this paper, but work with the integrals. This is justified by the following fact.
Lemma 2.5. Let Ψ be a doubling Young function. Then A direct consequence of Jensen's inequality is the following.
Lemma 2.6. Let Ψ be a Young function, u ∈ L 1 loc (R 2 ) and A ⊂ R 2 of positive and finite measure, then 3. Poincaré inequalities and polynomial approximation. From now on, Ψ always refers to a doubling Young function. We will construct an approximation by replacing the original function by approximating polynomials near the boundary of the domain. For this purpose, we will need a few lemmas regarding the polynomials. Here and later on by |E| we denote the Lebesgue measure of a set E ⊂ R 2 .
Lemma 2.7. Let Q be any square in R 2 and P be a polynomial of degree k defined in R 2 . Let E, F ⊂ Q be such that |E|, |F | > η|Q| where η > 0. Then where the constant C depends only on η, k and the doubling constant of Ψ.
Proof. Since any two norms on a finite dimensional vector space are comparable, there exists a constant δ > 0 depending only on k and η so that the set Therefore, by applying monotonicity and doubling properties on Ψ we obtain which gives the claim.
Given a function u ∈ C ∞ (Ω), degree k ∈ N, and a bounded set E ⊂ Ω with |E| > 0, we define (see [17]) the polynomial approximation of u in E, P k (u, E) to be the polynomial of order k − 1 which satisfiesˆE ∇ α (u − P k (u, E)) = 0 for each α = (α 1 , α 2 ) such that |α| = α 1 + α 2 ≤ k − 1. Once k is fixed, we denote the polynomial approximation of u in a dyadic square Q as P Q .
where we have abbreviated E = m i=1 Q i . Proof. By [4, Lemma 1] the claim is true for k = 1 in the case where E is convex. By a change of variables, the claim extends to the case k = 1 and E = m i=1 Q i , for the chain . What remains to show is the case k > 1. We do this by induction. Suppose the claim is true for the order k − 1. Then, using the Poincaré inequality first for the k − 1 orders and the for the first order, we obtain Proof. Let us abbreviate E = m i=1 Q i . Now, using the triangle inequality, Lemma 2.7, the doubling property of Ψ, then triangle inequality again and Lemma 2.8 we obtain

Decomposition and partition of unity
In this section we recall the decomposition of the domain Ω and the associated partition of unity that was obtained in [28]. In order to make the comparison between this paper and [28] easy, we use here the notation from [28].
3.1. Decomposition of the domain. We fix a square Q 0 , which is one of the largest Whitney squares in Ω. For each n ∈ N, the domain Ω is then divided into a core part D n , and a boundary layer, which is the union of setsH i , see Figure 1. The core part D n is the connected component containing Q 0 of the interior of the union of Whitney squares of side-length at least 2 −n . The construction of the boundary parts {H i } l i=1 is more involved. The setsH i are labeled so thatH i ∩H j = ∅ if and only if |i − j| ≤ 1 in a cyclical manner.
The setsH i are expanded by taking a connected component H i of a 2 −n−3 neighbourhood ofH i . The main property of the decomposition is that these expanded sets still satisfy H i ∩ H j = ∅ if and only if |i − j| ≤ 1 in a cyclical manner (Lemma 3.4 in [28]). Moreover, since the neighbourhoods are taken in the Euclidean distance, we may use an Euclidean partition of unity.
For each i we associate a Whitney square Q i ⊂ D n of side length 2 −n so that H i ∩ Q i = ∅ and, more importantly, if |i − j| ≤ 1, there is a chain of Whitney squares Q i,j with length bounded by a universal constant connecting Q i and Q j .  In this section we prove Theorem 1.1, using the results of Section 2 and the partition of union from [28] that was recalled in Section 3. The polynomial approximation is exactly the same as in [28]. What is different is the way the estimates are carried out using Poincaré inequalities. Since the usual Poincaré inequality is replaced by a Ψ − Ψ Poincaré inequality (Proposition 2.8), we need to be more careful with the chains of inequalities.
For n ∈ N fixed, we let D n and {H i } l i=1 be as in Section 3. With these we define a function u n ∈ W k,∞ (Ω) ∩ C ∞ (Ω) by setting for all x ∈ Ω where we have abbreviated P i := P Q i with the choice of squares Q i done in Section 3.2. Clearly, u n ∈ W k,∞ (Ω) ∩ C ∞ (Ω). Therefore, what remains to show is that First of all, by the definition of u n , we have for all x ∈ {z ∈ Ω : ϕ 0 (z) = 1} ⊂ D n−1 .
Also, for all i ∈ {1 . . . , l}, since P i is a degree n − 1 polynomial, we have Therefore, . . , l}. Since the sets D n increasingly exhaust the domain Ω, we have Thus, it remains to show that or equivalently, via Lemma 2.5, that for each multi-index α with |α| = k we havê In order to show (4.1), we estimate for each i ∈ {1, . . . , l} and multi-index α with |α| = k, by using the triangle inequality and Jensen's inequalitŷ We estimate the above two terms separately.
Let us take β ≤ α and writê where the integral is first split into squares Q for which Q ∩ A i ∩ A 0 = ∅. There are only a uniformly bounded amount of such squares Q, and for each such Q we havê Q Ψ(|∇ β u(x) − ∇ β P i (x)|C2 n(|α|−|β|) ) dx using the doubling property of Ψ, triangle inequality, Proposition 2.8 and Lemma 2.9. Next we estimate for β ≤ α, using the fact that H i ∩ H j = ∅ if and only if |i − j| ≤ 1, Lemma 2.7 and Lemma 2.9.
Combining the above estimates and using the fact that there is only a uniform number of overlaps for the estimates we havê giving (4.1). This proves the theorem.