A BMO-Type Characterization of Higher Order Sobolev Spaces

We obtain a new characterization of the higher Sobolev space Wm,p(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W^{m,p}(\mathbb R^{n})$\end{document}, m∈ℕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m\in \mathbb N$\end{document} and p∈(1,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\in (1, +\infty )$\end{document} and of the space BVm, the space of functions of higher order bounded variation. The characterizations are in term of BMO-type seminorms. The results unify and substantially extend previous results in Fusco et al. (ESAIM Control Optim. Calc Var., 24(2), 835–847 2018) and Farroni et al. (J. Funct. Anal., 278(9), 108451 2020).


Introduction
Let W m,p loc (R n ) (m ∈ N, 1 ≤ p < ∞), denote the Sobolev space of functions belonging to L p loc (R n ) whose distribution derivatives up to order m belong to L p loc (R n ).In [3], the Authors studied a characterization of W m,p based on J. Bourgain, H. Brezis and P. Mironescu's approach introduced in [5] (see also [7]).In particular they prove that if f ∈ W m−1,p (Ω), 1 < p < ∞ and Ω is a smooth bounded domain in R n then f belongs to W m,p (Ω) if and only if, where ρ ε , with ε > 0, are radial mollifiers and R m−1 f is the Taylor (m − 1) remainder of f .For p = 1, the condition (1) describes BV m .
Here we say that a W m−1,1 (Ω) is of m-th order bounded variation BV m if its m-th order partial derivatives in the sense of distributions are finite Radon measures.Spaces of this kind have been studied in [10] as applications in mathematical imaging in the setting of isotropic and anisotropic variants of the TV-model (see also [13]).
Another characterization of W m,p , 1 < p < ∞, (BV m for p = 1) formulated in terms of the m-th differences has been presented in [4].
In this article we are concerned with a characterization of W m,p 1 < p < ∞, (BV m for p = 1) as the limit of certain BMO-type seminorms similar to the one introduced by J. Bourgain, H. Brezis, P. Mironescu in [6].
In [15] the Authors showed that a function f ∈ L p loc (R n ) belongs to the Sobolev space W 1,p loc (R n ), 1 < p < +∞, if and only if lim where and the supremum on the right hand side is taken over all families G ε of disjoint ε-cubes Q = Q (x 0 , ε) of side length ε, centered in x 0 , with arbitrary orientation.Moreover, if f ∈ W 1,p loc (R n ) and p ≥ 1 then where γ (n, p) := max where . Following some ideas in [1], an analogous representation formula is obtained for the total variation of SBV functions in [14] (see also [11]).For related results see also [9,12].
Here, given a function f ∈ W m−1,p loc (R n ), p ≥ 1, for any ε > 0, we consider where the families G ε are as above and is the polynomial of degree m − 1 centered at x 0 , given by In particular, for m = 1 and m = 2 we have: Our main Theorem reads as follows: The constant in Eq. 7 is given by where N = n m and we refer to Section 2 for the notation.
Note that this Theorem is exactly an extension of Theorem 2.2 in [15] to the higher order case; indeed, in the case m = 1, since Q x • ν dx = 0, the constant β(n, 1, p) coincides with the one defined in Eq. 4.
A drawback of the formula Eq. 7 is that one does not recover the function in BV m .However, we are able to show that it is possible to characterize the functions in

Notation and Preliminaries
We denote by For m, n ≥ 1, we denote by N j = n m−j for j = 0, . . ., m.Given ν ∈ R N 0 we denotes its components by ν i 1 ,...,i k ,...,i m with i k = 1 . . .n. Taking x ∈ R n , x = (x i k ) i k ∈{1,...,n} we define the product ν • x as the element of R N 1 given by The product of ν ∈ R N 0 and m times the vector the monomial of degree |α| = n i=1 α i .In the same way, is a weak partial derivative of order |α|.Sometimes, we use the convention that D 0 u = u.Moreover, let ∇ m u be a vector with the components D α u, |α| = m.

The Sobolev Space W m,p
Definition 1 Let ⊂ R n be an open set, let m ∈ N, and let 1 ≤ p < ∞.The Sobolev space W m,p ( ) is the space of all functions u ∈ L p ( ) which admit α−th weak derivative D α u in L p ( ) for every α ∈ N n with 1 ≤ |α| ≤ m.
The space W m,p ( ) is endowed with the norm be an open set, let m ∈ N, and let 1 ≤ p < ∞.The homogeneous Sobolev space Ẇ m,p ( ) is the space of all functions u ∈ L 1 loc ( ) whose α−th weak derivative D α u belongs to L p ( ) for every α ∈ N n with |α| = m.
Note that the inclusion W m,p ( ) ⊆ Ẇ m,p ( ) holds.Moreover, as a consequence of Poincarè's inequality for sufficiently regular domains of finite measure the spaces Ẇ m,p ( ) and W m,p ( ) actually coincide.
The space Ẇ m,p ( ) is equipped with the seminorm Sometimes we will also use the equivalent seminorm u → |α|=m D α u L p ( ) .
The equivalence of the norm permit to have a useful density result as in [17,Remark 11.28].Indeed, if u ∈ Ẇ m,p ( ) then for every σ > 0 there exists Let ⊂ R n be an open bounded set and let E ⊂ be a Lebesgue measurable set with finite positive measure.Let 1 ≤ p ≤ +∞ and let m ∈ N.Then, for every u ∈ W m,p ( ), there exists a polynomial P m−1 E [u] of degree m − 1 such that for every multi-index α ∈ R n with 0 ≤ |α| ≤ m − 1 (see [17,Exercise 13.26]), for every u ∈ W m,p and for every k = 0, . . ., m − 1.
Notice that for m = 1 the previous Theorem is the classical Poincarè inequality and the polynomial P [u] is the mean of u over .In particular, if u ∈ W m,p (Q ) with Q = Q (x 0 , ε), then there exists a unique polynomial P m−1 Q [u] of degree m − 1 such that Eq. 9 holds and there exists a constant Next, we consider the Sobolev-Gagliardo-Nirenberg's embedding in W m,p (see Lemma 2.1 in [18]). Let Then there exists a unique polynomial P m−1 Q [u] of degree m − 1 such that Eq. 9 holds and there exists a where p = np n−mp .Moreover, the following easy properties of P [u] holds: • Linearity: where u ε (x) := u(εx).
We write for the Taylor polynomial of order m and for the Taylor remainder of order m.

Functions of Higher-Order Bounded Variation
is of bounded variation (for short u ∈ BV (Ω)) if u has a distributional gradient in form of a Radon measure of finite total mass and write We define the space of (real valued) functions of m-th order bounded variation, i.e. the set of all functions, whose distributional gradients up to order m − 1 are represented through 1-integrable tensor-valued functions and whose m-th distributional gradient is a tensor-valued Radon measure of finite total variation.Here S k (R n ) denotes the set of all symmetric tensors of order k with real components, which is naturally isomorphic to the set of all k-linear symmetric maps (R n ) k → R (see [10]).
It becomes a Banach space with the norm Here the total variation of ∇ m−1 u is denoted by |∇ m u|( ) and defined by where the supremum is taken over all The definition of BV m generalizes that of the classical space of functions of bounded variation and many results about BV can be obtained in BV m similarly (see [16]).We recall a higher-order variant of the famous Poincaré inequality, which will be useful throughout the sequel: an open and bounded subset with Lipschitz boundary, m ∈ N, 1 ≤ p < ∞.Then there exist a constant C > 0, depending only on , m and n such that for all u ∈ BV m ( ) In particular, the following version of Poincare's inequality holds.Let u ∈ BV m (Q ) with Q = Q (x 0 , ε), then there exists a unique polynomial P m−1 Q [u] of degree m − 1 such that Eq. 9 holds and there exists a constant By the nature of its definition, the space BV m inherits the Poincare-Wirtinger inequality which can be proved exactly as the corresponding first order result. Let Then there exists a unique polynomial P m−1 Q [u] of degree m − 1 such that Eq. 9 holds and there exists a constant We end this subsection with a higher-order variant of the compactness result in BV (Theorem 3.23 in [2]).for some constant M > 0. Then there is a subsequence

Other Useful Inequalities
The following tools will be useful in the sequel.
Given δ ∈ (0, 1), from the convexity of the function t → |t| p we get for every a, b ∈ R Taking into account Eq. 14, we also obtain the following pointwise inequality

The Local Version of the Functional K ε (f , m, p)
We define the following local counterpart of Eq. 2 which will be use in Step 3 of proof of Theorem 1 where the supremum on the right hand side is taken over all families G ε of disjoint open cubes of sidelenght ε and arbitrary orientation contained in .This quantity is strictly related to the L p norm of ∇ m f .Indeed, for p < n m with p = np n−mp , by using Hölder inequality, we have Thus, there exists a constant C depending only on Q, m, p such that for Q = εQ + x 0 , by Eqs.19 and 11, we get Summing over all sets Q in G ε , we obtain and therefore . We conclude this subsection, by observing that if ν ∈ S N−1 is a vector maximizing the integral in Eq. 8, x 0 ∈ R n and Q η (x 0 ) is a cube of side length η with center in x 0 then 3 The Case m = 2 In this section we deal with the case m = 2.In this case it is easier to make some explicit computations.Moreover we give an estimates on the constant β(n, 2, p) in terms of the Laplacian of the function f ∈ W 2,p .We prove the following Proposition 2 Let f ∈ W 2,p and β(n, 2, p) as in Eq. 8. Then the following estimate from below holds true First, by virtue of Eq. 9, it is possible to characterize P [u] for m = 2. Fixed x 0 ∈ , a generic polynomial of degree 1 centered in x 0 is given by By Eq. 9 with |α| = 0, we have Moreover, for every i = 1, . . ., n, again Eq. 9 for |α| = 1 gives and we write a = ∇u(x) dx.
Then the polynomial P 1 (u) is where, with a slight abuse of notation, we mean Remark 1 We observe that if is symmetric with respect to x 0 , the polynomial P 1 [u] has a simpler form, indeed ∇u, y dy = 0, and then Proof of Proposition 5 We observe that when m = 2, p ≥ 1, Eq. 8 reads as In this case ν • x 2 can equivalently be write as

Ax, x
where A ∈ M(n) is a matrix n × n and •, • denote the usual scalar product in R n .It is worth to remark that Firstly we observe that denoting by e i the canonical basis of R n , by O ∈ O(n) an orthogonal matrix and by R ∈ SO(n) a rotation around the origin taking Moreover, given A ∈ S(n) a symmetric matrix there exist O ∈ O(n) and D ∈ D(n) such that A = ODO −1 .Thus we have Then we can estimate from below β(n, 2, p) using Eqs.26 and 27, proving Eq. 22. Indeed, setting ∇ 2 f (0) = A we have Moreover setting y = min y i , we have

A Characterization of W m,p
Proof of Theorem 1 We divide the proof in three steps, proving first the limsup and liminf inequalities in Eq. 7 and then the validy of Eq. 6.
As a starting point we fix a bounded open set ⊂ R n and f ∈ W m,p ( ).Given σ > 0, there exists a function g ∈ C ∞ c ( ) such that f − g W m,p ( ) < σ and we choose ε > 0 such that Let us take now a family G ε of disjoint open cubes Q of side length ε and arbitrary orientation and let us denote by G ε the subfamily of G ε made by all cubes Q ∈ G ε such that Q ⊂ .

Step1 (limsup inequality)
We are going to show that lim sup We may assume, without loss of generality, that |∇ m f | ∈ L p (Ω).Using Eq. 14 and the linearity of for any Q ∈ G ε we have: where M δ = (1 + δ) p /δ p .We recall the notation in Section 2, so denoting by x 0 the center of the cube Q and for all x ∈ Q we write where We now estimate the two terms in Eq. 30.Let us focus on the first addendum: using again Eq. 14 we have Moreover, applying again Eqs. 14 and 29 we have Let us focus now on the second addendum in Eq. 30.By Poincaré inequality in W m,p (see Theorem 2), we have where C p is the Poincaré constant for cubes.

Step2 (liminf inequality)
We fix ⊂ R n , we assume again that f ∈ W m,p loc ( ) and we fix σ > 0 and g ∈ C ∞ c ( ) as in the previous Step.We prove that lim inf So, for η ∈ (0, 1) we consider the set With a clever use of Lemma 2.95 of [2] (as in Proposition 3.6 of [14]) it is possible to find k sufficiently small pairwise disjoint open sets S j ⊂ S N−1 covering S N−1 .Precisely, For all j = 1, . . ., k we denote which are open sets with the property For ε > 0 we consider the family F ε of all open cubes with faces parallel to the coordinate planes, side length ε, centered at all points of the form εv, with v ∈ Z n .Then for all j = 1, . . ., k we choose M j ∈ S j and we denote by R j ∈ SO(n) a rotation that takes e 1 into M j .Note that in this way, denoting by x the center of the cube Q ∈ F ε , we have (see Eq. 21), 1 For all j = 1, . . ., k we denote by R j (Q h,j ), Q h,j ∈ F ε , h = 1, . . ., m j , the elements of G ε contained in A j .By Eq. 36 there exists ε(σ, η) such that if ε < ε(σ, η) then We denote by x h,j the center of the cube R j (Q h,j ) and we argue as in Step 1. Indeed we have Now, adding on j and h the previous inequality, recalling Eq. 36, we have where the constants may change from line to line and depend only on p, n and | |.We conclude choosing η small enough and consequently ε small, where again C may change from line to line and depend on p, n and | |.To conclude we take the supremum over all the families G ε and let first ε → 0, σ → 0, δ → 0 and ↑ R n , proving Eq. 35.
Step3 (proof of Eq. 6) We fix σ > 0, ⊂ R n and observe that there exist r > 0 and a finite family of pairwise disjoint open cubes Q(x i , r) such that Moreover we fix 0 < ε < r and we set f ε (x) = ( ε * f )(x), where is a standard mollifier with compact support in the unit cube Q and ε (x) = ε −n (x/ε).For every Q(x i , r) we consider a family H ε of pairwise disjoint cubes Q j = z j + εQ ⊂ Q(x i , r), for j = 1, . . ., k.
We compute now noindent Moreover, by Eqs. 29 and 14, we have Summing up all the cubes in H ε , we obtain where the last inequality follows since kε n ≤ r n .Taking the supremum with respect to all families H ε and the liminf with respect to ε, we have Summing up with respect to i and using Eq.38 we have Letting σ → 0, δ → 0 and ↑ R n , we conclude.
Remark 2 We observe that Theorem 7 hold also in an open set Ω with the same proof replacing K ε (f, m, p) by the quantity K ε (f, m, p, ) defined in Eq. 18.
Corollary 1 Let p > 1, n > mp, p = np n−mp , ⊂ R n and G ε a pairwise disjoint family of translations Q of εQ contained in Then, following three statements are equivalent: Proof In this proof the constant C may change from line to line.
We prove that iii) ⇒ ii).By Hölder's inequality it holds Summing over all sets Q in G ε and passing to the supremum, we conclude.We prove that i) ⇒ iii).Using the Sobolev-Gagliardo-Nirenberg inequality Eq. 11, we obtain that there exists a constant C = C(n, m, p) such that Summing over Q in G ε and passing to the supremum over all families G ε the proof is completed.The equivalence i) ⇔ ii) is proved in [8].

A Characterization of Higher Order Bounded Variation
In this section we deal with the case p = 1.This case is not included in Theorem 1 since Eq.6 hold only for p > 1.
The case m = 1 was treated in [15].They proved that (see Proposition 2.4 of [15]) if Precisely, they prove that for where the total variation of f in ⊂ R n , possibly equal to +∞, is defined by setting We prove a similar characterization for the case m > 1.Now an equivalence similar to Eq. 43 involve the space BV m (R n ) of functions of m-th order bounded variation (see Section 2).Precisely, we prove the following Moreover, there is a positive constants C, independent of f , such that Proof To prove the first inequality in Eq. 44 we argue as in Step 3 of Theorem 1.In particular, we have Taking the supremum with respect to all families H ε and the liminf with respect to ε, we have 1 1 + δ lim inf Summing up with respect to i and using Eq.38 we obtain We conclude letting σ → 0, δ → 0, Ω ↑ R n .In order to prove the estimate from above in Eq. 44, it is is sufficient to apply the Poincare' inequality in BV m (see Section 2).
Corollary 2 Let n > m, 1 = n n−m , ⊂ R n and G ε is any pairwise disjoint family of translations Q of εQ contained in .Then, the following three statements are equivalent: Proof We prove that iii) ⇒ ii).By Hölder's inequality it holds The conclusion follows by summing over all sets Q in G ε .We prove that i) ⇒ iii).By using Eq. 12 there exists a constant C = C(n.m)such that The conclusion follows again by summing over all sets Q in G ε .The equivalence i) ⇔ ii) is proved in [8].