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Lifting in Besov Spaces Petru Mironescu, Emmanuel Russ, Yannick Sire

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Lifting in Besov Spaces Petru Mironescu, Emmanuel Russ, Yannick Sire Lifting in Besov spaces Petru Mironescu, Emmanuel Russ, Yannick Sire To cite this version: Petru Mironescu, Emmanuel Russ, Yannick Sire. Lifting in Besov spaces. 2017. hal-01517735v1 HAL Id: hal-01517735 https://hal.archives-ouvertes.fr/hal-01517735v1 Preprint submitted on 3 May 2017 (v1), last revised 8 Mar 2019 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Lifting in Besov spaces Petru Mironescu * Emmanuel Russ † Yannick Sire‡ April 30, 2017 1 Introduction Let ­ Rn be a bounded simply connected domain and u : ­ S1 a con- ½ ! tinuous (resp. Ck, k 1) function. It is a well-known fact that there exists a ¸ continuous (resp. Ck) real-valued function ' such that u eı'. In other words, Æ u has a continuous (resp. Ck) lifting. The analogous problem when u belongs to the fractional Sobolev space W s,p, s 0, 1 p , received an complete answer in [4]. Let us briefly recall the È · Ç 1 results: 1. when n 1, u has a lifting in W s,p for all s 0 and all p [1, ), Æ È 2 1 2. when n 2 and 0 s 1, u has a lifting in W s,p if and only if sp 1 or ¸ Ç Ç Ç sp n, ¸ 3. when n 2 and s 1, u has a lifting in W s,p if and only if sp 2. ¸ ¸ ¸ Further developments in the Sobolev context can be found in [1, 27, 23, 25]. In the present paper, we address the corresponding question in the frame- work of Besov spaces. More specifically, given s, p, q in suitable ranges de- fined later, we ask whether a map u Bs (­;S1) can be lifted as u eı', with 2 p,q Æ ' Bs (­;R). We say that Bs has the lifting property if and only if the 2 p,q p,q answer is positive. *Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France. Email address: [email protected] †Université Grenoble Alpes, CNRS UMR 5582, 100 rue des mathématiques, 38610 Gieres, France. Email address: [email protected] ‡Johns Hopkins University, Krieger Hall, Baltimore MD, USA. Email address: [email protected] 1 When dealing with circle-valued functions and their phases, it is natural to consider only maps in L1 . This is why we assume that s 0,1 and we take loc È the exponents p and q in the classical range p [1, ), q [1, ].2 2 1 2 1 Since Besov spaces are microscopic modifications of Sobolev (or Slobodeskii) spaces, one expects a global picture similar to the one described before for Sobolev spaces. The analysis in Besov spaces is indeed partly similar to the one in Sobolev spaces, as far as the results and the techniques are concerned. However, several difficulties occur and some cases still remain open. Thus, the analysis of the lifting problem leads us to prove several new properties for Besov spaces (in connection with restriction or absence of restriction proper- ties, sums of integer valued functions which are constant, products of func- tions in Besov spaces, disintegration properties for the Jacobian), which are interesting in their own right. We also provide detailed arguments for clas- sical properties (some embeddings, Poincaré inequalities) which could not be precisely located in the literature. Let us now describe more precisely our results and methods. When sp s s È n, functions in Bp,q are continuous, which readily implies that Bp,q has the lifting property (Case1). s In the case where sp 1, we rely on a characterization of Bp,q in terms Ç s of the Haar basis [3, Théorème 5], to prove that Bp,q has the lifting property (Case2). Assume now that 0 s 1, sp n and q . Let u Bs (­;S1) and Ç · Æ Ç 1 2 p,q let F(x,"): u ½ , where ½ is a standard mollifier. Since Bs , VMO, Æ ¤ " p,q ! for all " sufficiently small and all x ­ we have 1/2 F(x,") 1. Writing ıà 2 Ç j j · F(x,")/ F(x,") e " , where à is C1, and relying on a slight modification of j j Æ " the trace theory for weighted Sobolev spaces developed in [26], we conclude, ıÃ0 s letting "tend to 0, that u e , where Ã0 lim" 0 Ã" Bp,q, and therefore s Æ Æ ! 2 Bp,q still has the lifting property (Case3). Consider now the case where s 1 and sp 2. Arguing as in [4, Section 3], È s¸ it is easily seen that the lifting property for Bp,q will follow from the following property: given u Bs (­;S1), if F : u u Lp(­;Rn), then ( ) curlF 0. 2 p,q Æ ^ r 2 ¤ Æ The proof of ( ) is much more involved than the corresponding one for W s,p ¤ spaces [4, Section 3]. It relies on a disintegration argument for the Jacobians, more generally applicable in W1/p,p. This argument, in turn, relies on the fact that curlF 0 when u VMO and n 2, and a slicing argument. In particular, Æ 2 Æ we need a restriction property for Besov spaces, namely the fact that, for s 0, È 1 p and 1 q p, for all f Bs , the partial maps of f still belong to · Ç 1 · · 2 p,q Bs (see Lemma 6.7 below). Thus, we obtain that, when s 1 and 1 p , p,p È · Ç 1 1 However, we will discuss an appropriate version of the lifting problem when s 0; see · Remark 3.1 and Case 10 below. 2 s We discard the uninteresting case where p . In that case, maps in B ,q are con- Æ 1 s 1 tinuous. Arguing as in Case1 below, we obtain the existence of a B ,q phase for every s 1 1 u B ,q(­;S ). 2 1 2 Bs does have the lifting property when [1 q , n 2, and sp 2], or p,q · Ç 1 Æ Æ [1 q p, n 3, and sp 2], or [1 q , n 2, and sp 2]. · · ¸ Æ · · 1 ¸ È One can improve the conclusion of Lemma 6.7 as follows. For s 0, 1 È · p and 1 q p, for all f Bs , the partial maps of f belong to Bs Ç 1 · · 2 p,q p,q (Proposition 6.10). We emphasize the fact that this type of property holds only under the crucial assumption q p. More precisely, if q p and s 0, then we · È È exhibit a compactly supported function f Bs (R2) such that, for almost every 2 p,q x (0,1), f (x, ) Bs (R) (Proposition 6.11). This phenomenon, which has not 2 ¢ ∉ p, been noticed before,1 shows a picture strikingly different from the one for W s,p, and even more generally for Triebel-Lizorkin spaces [34, Section 2.5.13]. Let us return to the case when 0 s 1, 1 p and n 2. Assume Ç Ç · Ç 1 ¸ now that [1 q and 1 sp n], or [q and 1 sp n]. In this case, · Çs 1 · Ç Æ 1 Ç Ç we show that Bp,q does not have the lifting property. The argument uses embedding theorems and the following fact, for which we provide a proof: let si 0, 1 pi , and [s j p j 1 and 1 q j ], or [s j p j 1 and 1 q j ], È · Ç 1 si Æ · Ç 1 È · · 1 i 1,2. Then, if fi B and f1 f2 only takes integer values, then the Æ 2 pi,qi Å function f1 f2 is constant. Å Assume finally that 0 s , 1 p , n 2 and [1 q and 1 Ç Ç 1 · Ç 1 ¸ · Ç 1 · sp 2] or [q and 1 sp 2]. In this case, Bs does not have the lifting Ç Æ 1 · · p,q property either. We provide a counterexample of topological nature, inspired (x1, x2) s by [4, Section 4]: namely, the function u(x) belongs to Bp,q but Æ ¡ 2 2¢1/2 x1 x2 s Å has no lifting in Bp,q. Contrary to the case of Sobolev spaces, some cases remain open. A first case occurs when s 1, 1 p , p q , n 3, and sp 2. In this È · Ç 1 Ç Çs 1 ¸ Æ situation, since the restriction property for Bp,q does not hold, the argument s sketched before does not work any longer and we do not know if Bp,q has the lifting property. The case where s 1, 1 p , n 3, and [1 q and 2 p n] or Æ · Ç 1 ¸ · Ç 1 · Ç [q and 2 p n] is also open (except when s 1 and p q 2, since in Æ 1 1 Ç ·1,2 Æ Æ Æ this case, B2,2 W has the lifting property). This is related to the fact that Æ ı' 1 it is not known whether the map ' e maps Bp,q into itself.
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