A fractional version of Rivière’s GL(n)-gauge

We prove that for antisymmetric vector field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}Ω with small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}L2-norm there exists a gauge \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \in L^\infty \cap {\dot{W}}^{1/2,2}({\mathbb {R}}^1,GL(N))$$\end{document}A∈L∞∩W˙1/2,2(R1,GL(N)) such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\text {div}}_{\frac{1}{2}} (A\Omega - d_{\frac{1}{2}} A) = 0. \end{aligned}$$\end{document}div12(AΩ-d12A)=0.This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.


3
As Rivière showed in [27], the GL(N)-gauges have the advantage that they can transform equations of the form into a conservation law This is important since (1.1) is the structure of the equation for harmonic maps, H-surfaces, and more generally the Euler-Lagrange equations of a large class of conformally invariant variational functionals. The GL(N)-gauge transform allows for regularity theory and the study of weak convergence [27]; it also is an important tool for energy quantization, see [16].
In recent years a theory of fractional harmonic maps has developed, beginning with the work by Rivière and the first named author, [9,10]. bubbling analysis was initiated in [6]. Fractional harmonic maps have a variety of applications: they appear as free boundary of minimal surfaces or harmonic maps [8,21,24,31]; they are also related to nonlocal minimal surfaces [22] and to knot energies [2,3].
We recall that in [10] the first named author and Rivière considered nonlocal Schödingertype systems of the form where Ω is an antisymmetric potential in L 2 (ℝ, so(N)) , v ∈ L 2 (ℝ, ℝ N ) . The main technique to establish the sub-criticality of systems (1.2) is to perform a change of gauge by rewriting them after having multiplied v by a well-chosen rotation-valued map P ∈Ẇ 1∕2,2 (ℝ, SO(N)) which is "integrating" Ω in an optimal way. The key point in [9,10] was the discovery of particular algebraic structures (three-term commutators) that play the role of the Jacobians in the case of local systems in 2-D with an antisymmetric potential and that enjoy suitable integrability by compensations properties. In [17] the second and the third named authors introduced a new approach to fractional harmonic maps by considering nonlocal systems with an antisymmetric potential which is seen itself as a nonlocal operator. As we will explain later, such an approach is similar in the spirit to that introduced by Hélein in [15] in the context of harmonic maps.
It begins with the definition of "nonlocal one forms". F ∈ L p ( ⋀ 1 The s-differential, which takes function u ∶ ℝ n → ℝ into 1-forms, is then given by The scalar product for two 1-forms, F ∈ L p ( ⋀ 1 od ℝ n ) and G ∈ L p � ( ⋀ 1 od ℝ n ) , is then given by div(PΩP t − P t ∇P) = 0.
The fractional divergence div s , which takes 1-forms into functions, is then the formal adjoint to d s , namely For more details we refer to Sect. 2. With this notation in mind we now consider equations of the form or in index form where u ∈ (L 2 + L ∞ ) ∩Ẇ 1 2 ,2 (ℝ, ℝ N ) and Ω ij = −Ω ji ∈ L 2 ( ⋀ 1 od ℝ). The main observation in [17] is that the above notation and the above equation are not merely some random definitions of only analytical interest. Rather it was shown that the role of (1.3) for fractional harmonic maps is similar to the role of (1.1) for harmonic maps. In [17] it was shown that there exists a div − curl lemma in the spirit of [5], that fractional harmonic maps into spheres satisfy a conservation law in the spirit of [15], and that fractional harmonic maps into spheres essentially satisfy equations of the form (1.3), in the spirit of [27], and that an analogue of Uhlenbeck's gauge exist. In [20] this argument was further pushed to equations of stationary harmonic map in higher dimensional domains.
We mention that in [7] the authors found quasi conservation laws for nonlocal Schrödinger-type systems of the form where v ∈ L 2 (ℝ) , Ω ∈ L 2 (ℝ, so(N)) , and g is a tempered distribution. As we have already pointed out above, systems (1.4) represent a particular case of systems (1.3) studied in the present paper in the sense that the antisymmetric potential Ω in (1.4) is a pointwise function. The conservation laws found in [7] are a consequence of a stability property of some three-term commutators by the multiplication of P ∈ SO(N) and also of the regularity results obtained previously for such commutators. The reformulation of (1.4) in terms of conservation laws has permitted to get the quantization in the neck regions of the L 2 norms of the negative part of sequences of solutions to systems of the type (1.4).
The conservation laws that we obtain in the current paper are more similar in the spirit to those found in the paper [27] for harmonic maps and concern nonlocal systems (1.3) where the antisymmetric potential acts in general as a nonlocal operator. We hope this technique to be as useful for the question of concentration compactness and energy quantization for systems as it was in the local case in [16]; a question we will study in a future work.
Applying a gauge A ∈ L ∞ ∩Ẇ 1 2 ,2 to the Eq. (1.3), we find (see Lemma 4.1), and Ω satisfies the condition of Theorem 1.1. Then there exists a matrix A such that for converges strongly to f in Ẇ − 1 2 ,2 , and assume that u ∈ (L 2 + L ∞ (ℝ)) ∩Ẇ Here, as usual, we denote Theorem 1.3 will be proven in Sect. 4.

Preliminaries and useful tools
We follow the notation of [17] for the nonlocal operators. For readers convenience we recall it here. We write M(ℝ n ) for the space of all functions f ∶ ℝ n → ℝ measurable with respect to the Lebesgue measure dx and M( ⋀ 1 od ℝ n ) for the space of vector fields F ∶ ℝ n × ℝ n → ℝ measurable with respect to the dx dy |x−y| n measure, where "od" stands for "off diagonal".
For two vector fields F, G ∈ M( ⋀ 1 od ℝ n ) , the scalar product is defined as For any p > 1 the natural L p -space on vector fields F ∶ ℝ n × ℝ n → ℝ is induced by the norm and for D ⊂ ℝ n we define Observe that with this notation we have where is the Gagliardo-Slobodeckij seminorm. Let s ∈ (0.1) and F ∈ M( ⋀ 1 od ℝ n ) . We define the fractional s-divergence in the distributional way whenever the integrals converge.
With this notation we have div s d s = (−Δ) s , i.e., where the fractional Laplacian is defined as A simple observation is the following Moreover, for any F ∈ M( ⋀ 1 od ℝ n ) and u ∈ M(ℝ n ) we have and whenever each term is well-defined.

Proof We have
Thus, As for the latter term, we have Combining (2.3) with (2.4), we obtain (2.1). The proof of (2.2) is similar. ◻

We also denote
We will be using the following "Sobolev embedding" theorem. For the proof see Appendix C. We will also need the following Wente's inequality from [17].

Lemma 2.3 ([17, Corollary 2.3])
Let s ∈ (0, 1) , p > 1 , and let p ′ be the Hölder conjugate of p. Assume moreover that F � ∈ L p ( ⋀ 1 od ℝ) and g ∈ W s,p � (ℝ) with div s F = 0 . Let R be a linear operator such that for some Λ > 0 satisfies where L (2,∞) (ℝ) denote the weak L 2 space. Then any distributional solution u ∈Ẇ   In this section we prove Theorem 1.1. We will be looking for an A in the form A = (I + )P , where P is chosen to be the good gauge from Theorem 2.4. The idea to take perturbation of rotations of the form (I + )P has been taken from [28] in the context of local Schrödinger equations with antisymmetric potentials. This has been also exploited in [7]. .
Next we observe that a(x, y)P T (y)) = 0. since on the right-hand side we have a div-curl term we can apply fractional Wente's inequality, Lemma 2.3, and obtain from (2.7) Combining this with (3.9) and (3.6), we get which implies for sufficiently small that and thus a ≡ const . ◻ Now we will focus on showing that there exists a solution to the Eqs. (3.4) and (3.5). We will do this by using the Banach fixed point theorem.
There is a number 0 < ≪ 1 such that the following holds: (3.12)

Moreover, satisfies the estimate
We will need the following remainder terms estimates.

Lemma 3.4 We have the following estimates and
Proof We observe that for any ∈ C ∞ c (ℝ) we have Let M be the Hardy-Littlewood maximal function and let ∈ (0, 1) . We will use the following fractional counterpart (for the proof see [31, Proposition 6.6]) of the well-known inequality, see [4,14] We begin with the estimate of the first term on the right-hand side of (3.16). We observe that by (3.17) and by the symmetry of the integrals we obtain (3.14) Applying Hölder's inequality (for Lorentz spaces), we obtain where we used the notation from Sect. 2: for s ∈ (0, 1) and q > 1 we write (3.23) where for the estimate of the last term we used again Theorem 2.2, (2.6), with t = 1 2 . Combining This finishes the proof of (3.14).
In order to prove (3.15) we observe Thus, in order to conclude it suffices to apply the estimates (3.21) and (3.25). ◻

Proof of Proposition 3.3 Let
od ℝ) and thus, from Lemma 3.1, we have We apply for this term the nonlocal Hodge decomposition, Lemma A.1: Similarly, if for any two 1 , 2 ∈ X we consider the difference of the corresponding Eq.
(3.26) we get (3.26) (3.31) (Ω P ) * = 0,      [A] By assumption and Lemma 4.1, we have for any ∈ C ∞ c (D 1 ) and for Here with a slight abuse of notation we write for the matrix product u (x) . Therefore, We will pass with → ∞ in (4.5). Roughly speaking, the convergence of most of the terms will be a result of a combination of weak-strong convergence. We first observe that by Theorem 1.
u ⇀ u weakly inẆ 1 2 ,2 (ℝ), u → u locally strongly in L 2 , Let us choose a large R ≫ 1 , such that in particular D 1 ⊂ B(R) . We begin with the first term of (4.5).
Step 1. We claim that (up to a subsequence) Indeed, we observe As for the first term on the right-hand side of (4.8), we observe that since supp ⊂ D 1 ⊂ B(R), By strong convergence in L 2 of A on compact domains, we have and (noting once again that supp ⊂ D 1 ) In the last inequality we used the fact that if x ∈ D 1 and y ∈ ℝ ⧵ B(R) then |x − y| ≿ 1 + |y|. For the last term of (4.10), we similarly use that if y ∈ supp and x ∈ ℝ ⧵ B(R) , then we have |x − y| ≿ 1 + |x| with a constant independent of R.
Step 2. We claim that (up to a subsequence) where A. Indeed, we write Now, in order to obtain (4.17) we split the integral in two The first term on the right-hand side of (4.18) converges to zero as → ∞ . This follows from the weak convergence of od ℝ) (the easy verification of the latter is left to the reader).
As for the second term on the right-hand side of (4.18), we begin with the observation that To estimate the first term of the right-hand side of (4.19), we first note that the support of Ω D, is D × D and then we use Hölder's inequality Now we verify the convergence of the second term of the right-hand side of (4.19). Again we use that the support of Ω D, is D × D and thus by the strong convergence in L 2 of u on compact domains we have We also claim that (4.18) (4.20) To verify this statement, we divide the integral in two The second term on the right-hand side of (4.23) converges to zero as → ∞ , because . We verify the convergence of the first term on the right-hand side of (4.23). First we note that by the strong convergence of u in L 2 on compact domains we have and Finally, we have since supp ⊂ D 1 ⊂ B(R) This gives (4.23) Thus, the convergence of the first term of (4.23) follows from (4.24), (4.25), and (4.27). We proved (4.22). Now (4.15) follows from (4.16) combined with (4.17) and (4.22).
Step 3. We claim that That is, we claim that for any ∈ C ∞ c (ℝ) we have We write As for the second term of (4.29), we observe that by weak convergence of d 1 As for the first term of (4.29), we proceed exactly as in Step 1 and obtain This finishes the proof of (4.28).
Step 4. We claim that (up to a subsequence) where Ω D c = Ω − Ω D and Ω ∈ L 2 ( ⋀ 1 od ℝ) is the one given in the assumptions of the theorem.
Indeed, since Ω D c , (x, y) = 0 whenever both x, y ∈ D we have by the support of , (4.29) Thus, by the strong convergence on compact sets of u in L 2 we obtain Now we estimate the first term of the right-hand side of (4.34). We observe that for all large R, whenever x ∈ supp and y ∉ B(R) , we have |x − y| ≿ 1 + |y| . Therefore, Thus,

By (4.15) we know that Ω
. (4.43) In order to conclude we will need a removability of singularities lemma, compare with [18,Proposition 4.7]. Secondly, by the absolute continuity of the integral. Thus, passing with → ∞ in (4.45) we get for any ∈ C ∞ c (D)
[a]̇Fs p, div s B = div s G − div s (d s a) = div s G − (−Δ) s a = 0, 1 The decomposition is unique if we normalize a which finishes the proof. ◻

Appendix B: Localization
The next proposition follows from a relatively straightforward localization results, see, e.g., [19].
That is, assume y) . Then where G is a bilinear form with the following estimates for any s ∈ (0, 1 2 ) and > 0 In particular we have it is an admissible test function and we have from the Eq. (B.1) Here, � .
This argument works for any s ∈ (0, 1 2  The constants depend on s, p, q, n and are otherwise uniform. While characterizations such as Theorem C.1 are well known for Besov spaces, for Triebel spaces this seems to have been known only for q = p (where it follows from the Besov-space characterization), q = 2 where it is a result due to Stein and Fefferman, [f ] W s p,q (ℝ n ) ≾ [f ]̇Fs p,q (ℝ n ) .