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On Some Refinements of the Embedding of Critical Sobolev ON SOME REFINEMENTS OF THE EMBEDDING OF CRITICAL SOBOLEV SPACES INTO BMO ALMAZ BUTAEV Abstract. We introduce the non-homogeneous analogs of func- tion spaces studied by Van Schaftingen’s. We show that these classes refine the embedding W 1,n(Rn) ⊂ bmo(Rn). The analogous results established on bounded Lipschitz domains and Riemannian manifolds with bounded geometry. 1. Introduction Let f be a locally integrable function on Rn. Given a cube Q ⊂ Rn (henceforth by a cube we will understand a cube with sides parallel to the axes), we denote the average of f over Q by fQ, i.e. 1 fQ = f(x)dx, |Q| ZQ where |Q| is the Lebesgue measure of Q. In 1961 John and Nirenberg introduced the space of functions of bounded mean oscillation (BMO). Definition 1.1. We say that f ∈ BMO(Rn) if 1 kfkBMO := sup |f(x) − fQ|dx < ∞. Q |Q| ZQ Note that k·kBMO is a norm on the quotient space of functions modulo arXiv:1710.04366v2 [math.AP] 15 Jul 2018 constants. Functions of bounded mean oscillations turned out to be the right substitute for L∞ functions in a number of questions in analysis. In par- ticular, the embedding theorem of Gagliardo-Nirenberg-Sobolev (see e.g. [22], Chapter V) asserts that for any p ∈ [1, n) there exists Cp such that kfkLnp/(n−p) ≤ Cpk∇fkLp , ∀f ∈D. The inequality fails for p = n, so we do not have the embedding W 1,n into L∞. However, it follows from the Poincare inequality that for some constant C > 0, kfkBMO ≤ Ck∇fkLn , ∀f ∈D 1 2 A. BUTAEV and therefore W˚ 1,n is continuously embedded into BMO(Rn). Based on one inequality established by Bourgain and Brezis in [6], Van Schaftingen [28] defined a scale of spaces Dk using the k-differential forms i1 ik Φ(x)= φi1,...ik (x)dx ∧···∧ dx 1≤i1<X···<ik≤n as follows Definition 1.2. For 1 ≤ k ≤ n, Dk is defined as Rn ′ Rn Dk( )= {u ∈D ( ): kukDk < ∞}, where n k n R R 1 kukDk := sup{|u(φi1,...,ik )| : Φ ∈D( ;Λ ( )), dΦ=0, kΦkL ≤ 1}. It was shown in [28] that the Dk classes lie strictly between the critical Sobolev spaces and BMO(Rn), refining the classical embedding W˚ 1,n ⊂ BMO. More precisely, the following proper inclusions are con- tinuous 1,n W˚ ⊂ Dn−1 ⊂···⊂ D1 ⊂ BMO. From the point of view of some applications to PDEs, as function spaces Dk (k < n) lack certain “useful” properties: multiplications by smooth cut-off functions are not necessarily bounded operators on Dk and Dk are not invariant under all smooth changes of variables. In this paper, we introduce the non-homogeneous analogs of Van k n Schaftingen’s classes Dk, which we denote by d (R ). Definition 1.3. Let 1 ≤ k ≤ n. We say that u ∈ D′(Rn) belongs to dk(Rn) if (1.1) sup max |u(φI)| < ∞, I kΦkΥ1 (Rn)≤1 k I where the supremum is taken over all k-differential forms Φ= I φI dx , n φ ∈D(R ) and kΦk 1 = kΦk 1 + kdΦk 1 . We will denote this supre- I Υk L L P mum by kukdk . k n n It is useful to compare the defined classes d (R ) with Dk(R ). First k n n of all, d (R ) ⊂ Dk(R ), k = 1, 2,...n as sets. As Banach spaces n k n Dk(R ) are classes of functions modulo constants, while in d (R ) two functions that differ by a non-zero constant are considered as different elements. In contrast to Dk spaces, the smooth change of variables and multi- plications by cut-off functions are invariant operations on dk. In par- ticular, this allows to define dk on certain Riemannian manifolds. In EMBEDDING OF SOBOLEV SPACES INTO BMO 3 Section 2, we recall some facts from the theory of local Hardy spaces, which will be used later. In Section 3, we prove the following theorem Theorem 1.4. d1(Rn) is continuously embedded into the space bmo(Rn) and ∃C > 0 so that for any u ∈ dk(Rn), 1 ≤ k ≤ n kukbmo ≤ Ckukdk . Combining this theorem with the result of Van Schaftingen [26], it shows that the dk classes refine the embedding W 1,n(Rn) ⊂ bmo(Rn), where bmo is the local BMO space of Goldberg [16] in the sense that W 1,n ⊂ dn−1 ⊂···⊂ d1 ⊂ bmo. We also prove that continuous dn−1 functions can be characterized in terms of line integrals, similarly to the inequality of Bourgain, Brezis and Mironescu [8] Theorem 1.5. Let u ∈D(Rn). Then u ∈ dn−1(Rn) if and only if 1 1 sup u(t)τ(t)dt + sup u(t)τ(t)dt < ∞, |γ| Z |γ| Z ∂γ=∅ γ |γ|≥1 γ where the suprema are taken over smooth curves γ with finite lengths |γ|, boundaries ∂γ and unit tangent vectors τ. As an application of dk classes for PDEs, the following fact is estab- lished Theorem 1.6. Let n ≥ 2, F ∈ L1(Rn; Rn) and divF ∈ L1(Rn). Then the system (I − ∆)U = F admits a unique solution U such that • If n =2, then kUk∞ + k∇Uk2 ≤ C(kF k1 + kdivF k1) • If n ≥ 3, then kUkn/(n−2) + k∇Ukn/n−1 ≤ C(kF k1 + kdivF k1) In Section 4, we introduce the localized versions of dk spaces on bounded Lipschitz domains Ω. The main result of Section 4 is the proof of the following fact, which was conjectured by Van Schaftingen [28] for the bmo spaces on domains (see Definition 2.13 below). 1 Theorem 1.7. Any u ∈ d (Ω) is a bmor(Ω) function as there exists C > 0 such that 1 1 kukbmor(Ω) ≤ Ckukd (Ω) ∀u ∈ d (Ω). 4 A. BUTAEV In Section 5, we define dk classes on Riemannian manifolds with bounded geometry and based on the results of Section 3 we prove the refined embeddings between critical Sobolev space and bmo on such manifolds. Theorem 1.8. Let M be the Riemannian manifold with bounded ge- ometry. Then the following continuous embeddings are true W 1,n(M) ⊂ dn−1(M) ⊂···⊂ d1(M) ⊂ bmo(M). 2. Preliminaries Let Ω ⊂ Rn be open. We will use the Schwartz notations: E(Ω) will denote the class of smooth functions on Ω, D(Ω) and S(Ω) will stand for compactly supported smooth functions and smooth functions rapidly decaying at infinity with all their derivatives. By Dk(Ω) we denote the class of k-differential forms with D(Ω) components. All Lp spaces in this paper are considered relative to the Lebesgue measure. I For the differential form of order k,Φ= |I|=k φI dx , we will use the notation P kΦk 1 = kφI k 1 . Lk L XI However, often when it does not create confusion we will omit the subscript k and simply write kΦkL1 or kΦk1. 2.1. Local Hardy and BMO spaces of Goldberg. We recall the definition and basic properties of the local Hardy space h1(Rn) intro- duced by Goldberg [16]. Let us fix φ ∈ S(Rn) such that φ =6 0. For f ∈ L1(Rn), we define the local maximal function mφf(xR) by mφf(x) = sup |φt ∗ f(x)|, 0<t<1 −n y where φt(y)= t φ( t ). Definition 2.1. We say that f belongs to the local Hardy space h1(Rn) 1 n if mφf ∈ L (R ) and we put kfkh1 := kmφfkL1 . It is useful to compare h1 with the classic real Hardy space H1(Rn), which can be defined using the global maximal function Mφ, 1 n Mφf(x) := sup |φt ∗ f(x)|, f ∈ L (R ). t>0 EMBEDDING OF SOBOLEV SPACES INTO BMO 5 Definition 2.2. We say that f belongs to the Hardy space H1(Rn) if 1 n Mφf ∈ L (R ), and we put kfkH1 := kMφfkL1 . It follows from the definitions of the maximal functions that mφf(x) ≤ 1 n 1 1 Mφf(x) for any f ∈ L and x ∈ R . Therefore H ⊂ h . One of the reasons why it is often more convenient to deal with a larger space h1 instead of H1 is that S(Rn) ⊂ h1(Rn), while any f ∈ H1(Rn) has to satisfy Rn f = 0. Moreover, the following result is true. R Lemma 2.3 ([16]). The space D(Rn) is dense in h1(Rn). It is important to note that f ∈ h1(Rn) and f = 0 do not imply RRn that f ∈ H1(Rn) (see Theorem 3 in [16]). However, the following is true Lemma 2.4. If f ∈ h1(Rn), f(x)dx =0 and supp f ⊂ B, where B Rn n R is a bounded subset of R , then there exists CB > 0 such that kfkH1 ≤ CBkfkh1 . 1 Rn Rn Definition 2.5 ([16]). We say that f ∈ Lloc( ) belongs to bmo( ) if 1 1 kfkbmo := sup |f(x) − fQ|dx + sup |f(x)|dx < ∞, l(Q)≤1 |Q| ZQ l(Q)≥1 |Q| ZQ 1 where fQ = |Q| Q f(y)dy and Q are cubes with sides parallel to the axes, of side-lengthR l(Q). It is clear that bmo(Rn) is a subspace of BMO(Rn). Moreover, if n n kfkbmo = 0, then f = 0 a.e. on R , unlike in BMO(R ), where constant functions are identified with f ≡ 0. The following theorem of Goldberg shows the relation between h1 and bmo and the boundedness of pseudo-differential operators of degree zero on h1.
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