Abstract
This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order \(s\in (0,1)\) in arbitrary dimensions. It is shown that such fractional harmonic maps are \(C^\infty \) away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of \(s\in (0,1)\) and the stationarity/minimality assumption.
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Notes
The normalization constant \(\gamma _{n,s}\) is chosen in such a way that \(\displaystyle [u]^2_{H^{s}({\mathbb {R}}^n)}=\int _{{\mathbb {R}}^n}(2\pi |\xi |)^{2s}|{{\widehat{u}}}|^2\,{\mathrm{d}}\xi \,\), where \({{\widehat{u}}}\) denotes the (ordinary frequency) Fourier transform of u.
References
Bethuel, F.: On the singular set of stationary harmonic maps. Manuscr. Math. 78, 417–443, 1993
Caffarelli, L.A.; Roquejoffre, J.M.; Savin, O.: Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63, 1111–1144, 2010
Caffarelli, L.A.; Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260, 2007
Coifman, R.; Lions, P.L.; Meyer, Y.; Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286, 1993
Cui, L.; Yang, Q.: On the generalized Morrey spaces. Sib. Math. J. 46, 133–141, 2005
Da Lio, F.: Compactness and bubble analysis for \(1/2\)-harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 201–224, 2015
Da Lio, F.: Fractional harmonic maps into manifolds in odd dimension n > 1. Calc. Var. Partial Differ. Equ. 48, 421–445, 2013
Da Lio, F.: Fractional Harmonic Maps, Recent Developments in Nonlocal Theory. De Gruyter, New York (2018)
Da Lio, F.; Rivière, T.: Three-term commutator estimates and the regularity of \(1/2\)-harmonic maps into spheres. Anal. PDE 4, 149–190, 2011
Da Lio, F.; Rivière, T.: Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps. Adv. Math. 227, 1300–1348, 2011
Da Lio, F.; Schikorra, A.: \(n/p\)-harmonic maps: regularity for the sphere case. Adv. Calc. Var. 7, 1–26, 2014
Da Lio, F.; Schikorra, A.: On regularity theory for \(n/p\)-harmonic maps into manifolds. Nonlinear Anal. 165, 182–197, 2017
Duzaar, F.; Grotowski, J.F.: Energy minimizing harmonic maps with an obstacle at the free boundary. Manuscr. Math. 83, 291–314, 1994
Duzaar, F.; Grotowski, J.F.: A mixed boundary value problem for energy minimizing harmonic maps. Math. Z. 221, 153–167, 1996
Duzaar, F.; Steffen, K.: A partial regularity theorem for harmonic maps at a free boundary. Asymptot. Anal. 2, 299–343, 1989
Duzaar, F.; Steffen, K.: An optimal estimate for the singular set of a harmonic map in the free boundary. J. Reine Angew. Math. 401, 157–187, 1989
Evans, L.C.: Partial regularity for stationary harmonic maps into spheres. Arch. Ration. Mech. Anal. 116, 101–113, 1991
Fabes, E.B.; Kenig, C.E.; Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7, 77–116, 1982
Giaquinta, M., Martinazzi, L.: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Lectures Notes. Scuola Normale Superiore di Pisa, vol. 11. Edizioni della Normale, Pisa 2012
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA (1985)
Gulliver, R.; Jost, J.: Harmonic maps which solve a free-boundary problem. J. Reine Angew. Math. 381, 61–89, 1987
Hardt, R.; Lin, F.H.: Partially constrained boundary conditions with energy minimizing mappings. Commun. Pure Appl. Math. 42, 309–334, 1989
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York 1993
Hélein, F.: Régularité des applications faiblement harmoniques entre une surface et une sphère. C.R. Acad. Sci. Paris Série I 311, 519–524, 1990
Hélein, F.: Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne. C.R. Acad. Sci. Paris Sér. I Math. 312, 591–596, 1991
Ho, K.P.: Sobolev-Jawerth embedding of Triebel-Lizorkin-Morrey-Lorentz spaces and fractional integral operator on Hardy type spaces. Math. Nachr. 287, 1674–1686, 2014
Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems, An Introduction to Geometric Measure Theory, vol. 135. Cambridge University Press, Cambridge (2012)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)
Mazowiecka, K.; Schikorra, A.: Fractional div-curl quantities and applications to nonlocal geometric equations. J. Funct. Anal. 275, 1–44, 2018
Millot, V.; Pegon, M.: Minimizing \(1/2\)-harmonic maps into spheres. Calc. Var. Partial Differ. Equ. 59, Art. 55, 2020
Millot, V.; Pisante, A.: Relaxed energies for \(H^{1/2}\)-maps with values into the circle and measurable weights. Indiana Univ. Math. J. 58, 49–136, 2009
Millot, V.; Sire, Y.: On a fractional Ginzburg-Landau equation and \(1/2\)-harmonic maps into spheres. Arch. Ration. Mech. Anal. 215, 125–210, 2015
Millot, V.; Sire, Y.; Wang, K.: Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces. Arch. Ration. Mech. Anal. 231, 1129–1216, 2019
Millot, V.; Sire, Y.; Yu, H.: Minimizing fractional harmonic maps on the real line in the supercritical regime. Discrete Contin. Dyn. Syst. Ser. A 38, 6195–6214, 2018
Mironescu, P.; Pisante, A.: A variational problem with lack of compactness for \(H^{1/2}(S^1;S^1)\) maps of prescribed degree. J. Funct. Anal. 217, 249–279, 2004
Moser, R.: Intrinsic semiharmonic maps. J. Geom. Anal. 21, 588–598, 2011
Molchanov, S.A.; Ostrovskii, E.: Symmetric stable processes as traces of degenerate diffusion processes. Theory Probab. Appl. 14, 128–131, 1969
Rivière, T.: Everywhere discontinuous harmonic maps into spheres. Acta Math. 175, 197–226, 1995
Rivière, T.: Conservation laws for conformally invariant variational problems. Invent. Math. 168, 1–22, 2007
Roberts, J.: A regularity theory for intrinsic minimising fractional harmonic maps. Calc. Var. Partial Differ. Equ. 57, Art. 109, 2018
Ros-Oton, X.; Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101, 275–302, 2014
Sawano, Y.; Yang, D.; Yuan, W.: New applications of Besov-type and Triebel-Lizorkin-type spaces. J. Math. Anal. Appl. 363, 73–85, 2010
Scheven, C.: Partial regularity for stationary harmonic maps at a free boundary. Math. Z. 253, 135–157, 2006
Schikorra, A.: Regularity of \(n/2\)-harmonic maps into spheres. J. Differ. Equ. 252, 1862–1911, 2012
Schikorra, A.: Integro-differential harmonic maps into spheres. Commun. Partial Differ. Equ. 40, 506–539, 2015
Schikorra, A.: \(\varepsilon \)-regularity for systems involving non-local, antisymmetric operators. Calc. Var. Partial Differ. Equ. 54, 3531–3570, 2015
Schoen, R.M.: Analytic aspects of the harmonic map problem, Seminar on nonlinear partial differential equations (Berkeley, 1983), Math. Sci. Res. Inst. Publ., vol. 2, pp. 321–358. Springer, New York 1984
Schoen, R.M.; Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differ. Geom. 17, 307–335, 1982
Schoen, R.M.; Uhlenbeck, K.: Regularity of minimizing harmonic maps into the sphere. Invent. Math. 78, 89–100, 1984
Shatah, J.: Weak solutions and development of singularities of the \(SU(2)\)-model. Commun. Pure Appl. Math. 41, 459–469, 1998
Sickel, W., Yang, D., Yuan, W.: Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2010
Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112, 2007
Simon, L.: Theorems on Regularity and Singularity of Energy Minimizing Maps. Birkhaüser Verlag, Basel (1996)
Triebel, H.: Theory of Function Spaces. Springer, Basel (2010)
Yang, D.; Yuan, W.: A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces. J. Funct. Anal. 255, 276–2809, 2008
Ziemer, W.P.: Weakly Differentiable Functions. Graduate Texts in Mathematics. Springer, New York (1989)
Acknowledgements
V.M. is supported by the Agence Nationale de la Recherche through the project ANR-14-CE25-0009-01 (MAToS). A.S. is supported by the Simons Fondation through the grant no. 579261.
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Appendices
Appendix A. On the degenerate Laplace equation
In this first appendix, our aim is to recall some of the properties satisfied by weak solutions of the (scalar) degenerate linear elliptic equation
with \(\mathbf{x}_0=(x_0,z_0)\in {\mathbb {R}}^{n+1}\). Those properties are essentially taken from [40], and we reproduce here the statements for convenience of the reader. The notion of weak solution to this equation corresponds to the variational formulation. In other words, we say that \(w\in H^1(B_R(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\) is a weak solution of (A.1) if
for every \(\Phi \in H^1(B_R(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\) such that \(\Phi =0\) on \(\partial B_R(\mathbf{x_0})\).
One may complement (A.1) with a boundary condition of the form \(w=v\) on \(\partial B_R(\mathbf{x}_0)\) for a given \(v\in H^1(B_R(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\). This boundary condition is thus interpreted in the sense of traces. Classically, such a boundary condition uniquely determines the solution of (A.1) which can be characterized by energy minimality.
Lemma A.1
Let \(v\in H^1(B_R(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\). The equation
admits a unique weak solution which is characterized by
for every \(\Phi \in H^1(B_R(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\) satisfying \(\Phi =v\) on \(\partial B_R(\mathbf{x_0})\).
As for the usual Laplace equation, energy minimality can be used to prove that w inherits symmetries from the boundary condition. In our case, we make use of the following lemma.
Lemma A.2
Let \(\mathbf{x}_0\in {\mathbb {R}}^n\times \{0\}\) and \(v\in H^1(B_R,|z|^a{\mathrm{d}}\mathbf{x})\). If v is symmetric with respect to \(\{z=0\}\), then the weak solution w of (A.2) is also symmetric with respect to \(\{z=0\}\).
Concerning interior regularity of weak solutions, the issue is of course near the hyperplane \(\{z=0\}\). Indeed, if the ball \(B_R(\mathbf{x}_0)\) is away from \(\{z=0\}\), then the operator becomes uniformly elliptic with smooth coefficients, and the classical elliptic theory tells us that weak solutions are \(C^\infty \) in the interior. For an arbitrary ball, the general results of [18] about degenerate elliptic equations apply, and they provide at least local Hölder continuity in the interior. Using the invariance of the equation with respect to the x-variables, the regularity can be further improved (see for example [40, Corollary 2.13]).
Some boundary regularity and related maximum principles are also known from the general theory in [23]. We reproduce here the statement in [40, Lemma 2.18].
Lemma A.3
Let \(v\in H^1(B_R(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\cap C^0\big ({{\overline{B}}}_R(\mathbf{x}_0)\big )\). The weak solution w of (A.2) belongs to \(C^0\big ({{\overline{B}}}_R(\mathbf{x}_0)\big )\). Moreover,
A further fundamental property of weak solutions of (A.1) is an energy monotonicity in which one has to distinguish balls centered at a point of \(\{z=0\}\) from balls lying away from \(\{z=0\}\). The two following lemmas are taken from [40, Lemma 2.8] and [40, Lemma 2.17], respectively.
Lemma A.4
Let \(\mathbf{x}_0\in {\mathbb {R}}^n\times \{0\}\) and \(w\in H^1(B_R(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\) a weak solution of (A.1). Assume that either \(s\geqq 1/2\), or that \(s<1/2\) and w is symmetric with respect to the hyperplane \(\{z=0\}\). Then,
for every \(0<\rho \leqq r\leqq R\).
Lemma A.5
Let \(w\in H^1(B_R(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\) be a weak solution of (A.1). If \(\mathbf{x}_0=(x_0,z_0)\in {\mathbb {R}}^{n+1}_+\) and \(R>0\) are such that \(B_R(\mathbf{x}_0)\subseteq {\mathbb {R}}^{n+1}_+\) and \(z_0\geqq \theta R\) for some \(\theta \geqq 2\), then
for a constant \(C=C(n)\).
Appendix B. A Lipschitz estimate for s-harmonic functions
The purpose of this appendix is to provide an interior Lipschitz estimate for weak solutions \(w\in {{\widehat{H}}}^s(D_1)\) of the fractional Laplace equation
The notion of weak solution is understood here according to the weak formulation of the s-Laplacian operator, see (2.3). Interior regularity for weak solutions is known, and it tells us that w is locally \(C^\infty \) in \(D_1\). The following estimate is probably also well known, but we give a proof for convenience of the reader:
Lemma B.1
If \(w\in {{\widehat{H}}}^s(D_1)\) is a weak solution of (B.1), then \(w\in C^\infty (D_{1/2})\), and
for a constant \(C=C(n,s)\).
Proof
As we already mentioned, the regularity theory is already known, and we take advantage of this to only derive estimate (B.2). Let us fix an arbitrary point \(x_0\in D_{1/2}\). We consider the extension \(w^{\mathrm{e}}\) which belongs to \(H^1(B^+_{1/4}(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\) with \(\mathbf{x}_0:=(x_0,0)\) by Lemma 2.9. In view of Lemma 2.12, it satisfies
for every \(\Phi \in H^1(B_{1/4}^+(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\) such that \(\Phi =0\) on \(\partial ^+B_{1/4}^+(\mathbf{x}_0)\). Then we consider the even extension of \(w^{\mathrm{e}}\) to the whole ball \(B_{1/4}(\mathbf{x}_0)\) that we still denote by \(w^{\mathrm{e}}\) (that is \(w^{\mathrm{e}}(x,z)=w^{\mathrm{e}}(x,-z)\)). Then \(w^{\mathrm{e}}\in H^1(B_{1/4}(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\), and arguing as in the proof of Corollary 5.4, we infer that \(w^{\mathrm{e}}\) is a weak solution of (A.1) with \(R=1/4\). According to [40, Corollary 2.13], the weak derivatives \(\partial _i w^{\mathrm{e}}\) belongs to \(H^1(B_{1/8}(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})\) for \(i=1,\ldots ,n\), and they are weak solutions of (A.1) with \(R=1/8\). Now, applying [18, Theorem 2.3.12] to \(w^{\mathrm{e}}\) and \(\partial _i w^{\mathrm{e}}\), we infer that \(w^{\mathrm{e}}\in C^{1,\alpha }(B_{1/16}(\mathbf{x}_0))\) for some exponent \(\alpha =\alpha (n,s)\in (0,1)\),
and
for a constant \(C=C(n,s)\).
On the other hand, for every \(\mathbf{x}\in B_{1/16}(\mathbf{x}_0)\), we have (recall our notation in (5.11))
Combining this estimate with (B.3) and Lemma 2.9 leads to
The same argument applied to \(\nabla _xw^{\mathrm{e}}\) and using (B.4) instead of (B.3) yields
thanks to Lemma 2.9 again. Now the conclusion follows from the fact that \(w^{\mathrm{e}}=w\) and \(\nabla _xw^{\mathrm{e}}=\nabla w\) on \(\partial ^0B^+_{1/16}(\mathbf{x}_0)\). \(\square \)
Appendix C. An embedding theorem between generalized \({\mathcal {Q}}_\alpha \)-spaces
In this appendix, our goal is to prove one of the crucial estimates used in the proof of Theorem 4.1, Corollary C.6 below. It turns out that this estimate does not explicitly appear in the existing literature (to the best of our knowledge), but it can be shortly derived from recent results in harmonic analysis. The purpose of this appendix is thus to explain how to combine those results to reach our goal. First, we need to recall some definitions and notations.
The space \({\mathscr {S}}_\infty ({\mathbb {R}}^n)\) can be defined as the topological subspace of the Schwartz class \({\mathscr {S}}({\mathbb {R}}^n)\) made of all functions \(\varphi \) such that the semi-norm
is finite for every \(M\in {\mathbb {N}}\), where \(\gamma =(\gamma _1,\ldots ,\gamma _n)\in {\mathbb {N}}^n\), \(|\gamma |:=\gamma _1+\ldots +\gamma _n\), and \(\partial ^\gamma :=\partial _1^{\gamma _1}\ldots \partial _n^{\gamma _n}\). Its topological dual is denoted by \({\mathscr {S}}^\prime _\infty ({\mathbb {R}}^n)\), and it is endowed with the weak \(*\)-topology, see for example [54, 55].
The following \({\mathcal {Q}}^{\alpha ,q}_p\)-spaces were introduced in [5, 55], generalizing the notion of \({\mathcal {Q}}_\alpha \)-space (see [51, Section 1.2.4] and references therein), in the sense that \({\mathcal {Q}}_\alpha ({\mathbb {R}}^n)={\mathcal {Q}}^{\alpha ,2}_{n/\alpha }({\mathbb {R}}^n)\).
Definition C.1
([5, 55]) Given \(\alpha \in (0,1)\), \(p\in (0,\infty ]\) and \(q\in [1,\infty )\), define \({\mathcal {Q}}^{\alpha ,q}_p({\mathbb {R}}^n)\) as the space made of elements \(f\in {\mathscr {S}}^\prime _\infty ({\mathbb {R}}^n)\) such that \(f(x)-f(y)\) is a measurable function on \({\mathbb {R}}^n\times {\mathbb {R}}^n\) and
where Q ranges over all cubes of dyadic edge lengths in \({\mathbb {R}}^n\).
Remark C.2
Endowed with \(\Vert \cdot \Vert _{{\mathcal {Q}}^{\alpha ,q}_p({\mathbb {R}}^n)}\), the space \({\mathcal {Q}}^{\alpha ,q}_p({\mathbb {R}}^n)\) is a semi-normed vector space, and
provides an equivalent semi-norm.
The following embeddings between \({\mathcal {Q}}^{\alpha ,q}_p\)-spaces hold:
Theorem C.3
If \(0<\alpha _1<\alpha _2<1\), \(1\leqq q_2<q_1<\infty \), and \(0<\lambda \leqq n\) are such that
then \({\mathcal {Q}}^{\alpha _2,q_2}_{\frac{nq_2}{\lambda }}({\mathbb {R}}^n)\hookrightarrow {\mathcal {Q}}^{\alpha _1,q_1}_{\frac{nq_1}{\lambda }}({\mathbb {R}}^n)\) continuously.
As we briefly mentioned at the beginning of this appendix, this theorem actually follows quite directly from a more general embedding result between some homogeneous Triebel-Lizorkin-Morrey-Lorentz spaces [26] together with an identification result between various definitions of homogeneous Triebel-Lizorkin-Morrey type spaces [42], and a characterization of the \({\mathcal {Q}}^{\alpha ,q}_p\)-spaces within this scale of spaces [55]. We refer to the monograph [51] for what concerns the spaces involved here, and we limit ourselves to their basic definition. To this purpose, we consider a reference bump function \(\psi \in {\mathscr {S}}({\mathbb {R}}^n)\) such that
(In particular, \(\psi \in {\mathscr {S}}_\infty ({\mathbb {R}}^n)\).) For \(j\in {\mathbb {Z}}\), we denote by \(\psi _j\) the function defined by
Definition C.4
Given \(p,q\in (0,\infty )\), \(s\in {\mathbb {R}}\), and \(\tau \in [0,\infty )\), the homogeneous Triebel-Lizorkin space \(\dot{F}^{s,\tau }_{p,q}({\mathbb {R}}^n)\) is defined to be the set of all \(f\in {\mathscr {S}}^\prime _\infty ({\mathbb {R}}^n)\) such that
where Q ranges over all cubes of dyadic edge lengths in \({\mathbb {R}}^n\), and \(j_Q:=-\log _2\ell (Q)\) with \(\ell (Q)\) the edge length of Q.
Definition C.5
Given \(0<p\leqq u<\infty \), \(0<q<\infty \), and \(s\in {\mathbb {R}}\), the homogeneous Triebel-Lizorkin-Morrey space \(\dot{{\mathcal {E}}}^s_{p,q,u}({\mathbb {R}}^n)\) is defined to be the set of all \(f\in {\mathscr {S}}^\prime _\infty ({\mathbb {R}}^n)\) such that
where Q ranges over all cubes of dyadic edge lengths in \({\mathbb {R}}^n\).
Proof of Theorem C.3
In [26], the author introduced a more refined scale of homogeneous Triebel-Lizorkin spaces of Morrey-Lorentz type, denoted by \({\dot{F}}^{s,u}_{M_{p,q,\lambda }}({\mathbb {R}}^n)\). In the case \(u=p=q\), those spaces coincide with the homogeneous Triebel-Lizorkin-Morrey spaces above, namely
for every \(p\in (0,\infty )\), \(\lambda \in (0,n]\), and \(s\in {\mathbb {R}}\). More precisely, their defining semi-norms are equivalent (in one case the supremum is taken over all dyadic cubes, while in the other it is taken over balls). By [26, Theorem 4.1], under condition (C.1) the space \({\dot{F}}^{\alpha _2,q_2}_{M_{q_2,q_2,\lambda }}({\mathbb {R}}^n)\) embeds continuously into \({\dot{F}}^{\alpha _1,q_1}_{M_{q_1,q_1,\lambda }}({\mathbb {R}}^n)\). In other words,
continuously. On the other hand, [42, Theorem 1.1] tells us that
with equivalent semi-norms. Finally, by [55, Theorem 3.1] we have
with equivalent semi-norms. Hence, the conclusion follows from (C.2). \(\square \)
We are now ready to state the important corollary of Theorem C.3 used in the proof of Theorem 4.1. Given \(s\in (0,1)\), \(p\in [1,\infty )\), and an open set \(\Omega \subseteq {\mathbb {R}}^n\), we recall that the Sobolev-Slobodeckij \(W^{s,p}(\Omega )\)-semi-norm of a measurable function f is given by
Corollary C.6
Let \(s\in (0,1)\) and \(f\in L^1({\mathbb {R}}^n)\). If
then,
for a constant \(C=C(n,s)\).
Proof
Since \(f\in L^1({\mathbb {R}}^n)\), it belongs to \({\mathscr {S}}^\prime ({\mathbb {R}}^n)\), and thus to \({\mathscr {S}}^\prime _\infty ({\mathbb {R}}^n)\) (see [54, Sec 5.1.2, p. 237]). Then, condition (C.4) implies that \(f\in {\mathcal {Q}}^{s,2}_{n/s}({\mathbb {R}}^n)\). On the other hand, \({\mathcal {Q}}^{s,2}_{n/s}({\mathbb {R}}^n)\hookrightarrow {\mathcal {Q}}^{s/3,6}_{3n/s}({\mathbb {R}}^n)\) continuously by Theorem C.3. Then the conclusion follows from the definition of \({\mathcal {Q}}^{s/3,6}_{3n/s}({\mathbb {R}}^n)\) together with Remark C.2. \(\square \)
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Millot, V., Pegon, M. & Schikorra, A. Partial Regularity for Fractional Harmonic Maps into Spheres. Arch Rational Mech Anal 242, 747–825 (2021). https://doi.org/10.1007/s00205-021-01693-w
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DOI: https://doi.org/10.1007/s00205-021-01693-w