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Partial Regularity for Fractional Harmonic Maps into Spheres

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

  1. 1.

    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.

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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

$$\begin{aligned} {\mathrm{div}}(|z|^a\nabla w)= 0 \quad \text {in }B_R(\mathbf{x_0}), \end{aligned}$$
(A.1)

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

$$\begin{aligned} \int _{B_R(\mathbf{x}_0)}|z|^a\nabla w\cdot \nabla \Phi \,{\mathrm{d}}\mathbf{x}=0 \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathrm{div}}(|z|^a\nabla w)= 0 &{}\text {in }B_R(\mathbf{x_0}),\\ w=v &{} \text {on }\partial B_R(\mathbf{x}_0), \end{array}\right. } \end{aligned}$$
(A.2)

admits a unique weak solution which is characterized by

$$\begin{aligned} \int _{B_R(\mathbf{x}_0)}|z|^a|\nabla w|^2\,{\mathrm{d}}\mathbf{x}\leqq \int _{B_R(\mathbf{x}_0)}|z|^a|\nabla \Phi |^2\,{\mathrm{d}}\mathbf{x} \end{aligned}$$

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,

$$\begin{aligned} \min _{{{\overline{B}}}_R(\mathbf{x}_0)} w=\min _{\partial B_R(\mathbf{x}_0)} v \quad \text {and}\quad \max _{{{\overline{B}}}_R(\mathbf{x}_0)} w=\max _{\partial B_R(\mathbf{x}_0)} v. \end{aligned}$$

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,

$$\begin{aligned} \frac{1}{\rho ^{n+2-2s}}\int _{B_\rho (\mathbf{x}_0)}|z|^a|\nabla w|^2\,{\mathrm{d}}\mathbf{x}\leqq \frac{1}{r^{n+2-2s}}\int _{B_r(\mathbf{x}_0)}|z|^a|\nabla w|^2\,{\mathrm{d}}\mathbf{x} \end{aligned}$$

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

$$\begin{aligned} \Big (\frac{2}{R}\Big )^{n+1}\int _{B_{R/2}(\mathbf{x}_0)} |z|^a|\nabla w|^2\,{\mathrm{d}}\mathbf{x}\leqq \Big (1+\frac{C}{\theta -1}\Big )\frac{1}{R^{n+1}}\int _{ B_{R} ( \mathbf{x}_0 ) } |z|^a|\nabla w|^2\,{\mathrm{d}}\mathbf{x}, \end{aligned}$$

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

$$\begin{aligned} (-\Delta )^sw=0 \quad \text {in }H^{-s}(D_1). \end{aligned}$$
(B.1)

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

$$\begin{aligned} \Vert w\Vert ^2_{L^\infty (D_{1/2})}+\Vert \nabla w\Vert ^2_{L^\infty (D_{1/2})}\leqq C\big ({\mathcal {E}}_s(w,D_1)+\Vert w\Vert ^2_{L^2(D_1)}\big ), \end{aligned}$$
(B.2)

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

$$\begin{aligned} \int _{B_{1/4}^+(\mathbf{x}_0)}z^a\nabla w^{\mathrm{e}}\cdot \nabla \Phi \,{\mathrm{d}}x=0 \end{aligned}$$

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 \(\alpha =\alpha (n,s)\in (0,1)\),

$$\begin{aligned} {[}w^{\mathrm{e}}]_{C^{0,\alpha }(B_{1/16}(\mathbf{x}_0))}\leqq C\Vert w^{\mathrm{e}}\Vert _{L^2(B_{1/8}(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})}, \end{aligned}$$
(B.3)

and

$$\begin{aligned} {[}\nabla _x w^{\mathrm{e}}] _{C^{0,\alpha }(B_{1/16}(\mathbf{x}_0))}\leqq C\Vert \nabla _x w^{\mathrm{e}}\Vert _{L^2(B_{1/8}(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})}, \end{aligned}$$
(B.4)

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))

$$\begin{aligned} |w^{\mathrm{e}}(\mathbf{x})|\leqq & {} \Big | w^{\mathrm{e}}(\mathbf{x})-\frac{1}{|B_{1/16}|_a}\int _{B_{1/16}(x_0)}|z|^a w^{\mathrm{e}}(\mathbf{y}){\mathrm{d}}\mathbf{y} \Big |\\&+ \frac{1}{|B_{1/16}|_a}\int _{B_{1/16}(x_0)}|z|^a|w^{\mathrm{e}}(\mathbf{y})|{\mathrm{d}}\mathbf{y}\\\leqq & {} C\big ([w^{\mathrm{e}}]_{C^{0,\alpha }(B_{1/16}(\mathbf{x}_0))} +\Vert w^{\mathrm{e}}\Vert _{L^2(B_{1/16}(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})} \big ). \end{aligned}$$

Combining this estimate with (B.3) and Lemma 2.9 leads to

$$\begin{aligned} \Vert w^{\mathrm{e}}\Vert ^2_{L^\infty (B_{1/16}(x_0))} \leqq C\big ({\mathcal {E}}_s(w,D_1)+\Vert w\Vert ^2_{L^2(D_1)}\big ). \end{aligned}$$

The same argument applied to \(\nabla _xw^{\mathrm{e}}\) and using (B.4) instead of (B.3) yields

$$\begin{aligned} \Vert \nabla _x w^{\mathrm{e}}\Vert ^2_{L^\infty (B_{1/16}(x_0))} \leqq C \Vert \nabla _ xw^{\mathrm{e}}\Vert ^2_{L^2(B_{1/8}(\mathbf{x}_0),|z|^a{\mathrm{d}}\mathbf{x})} \leqq C{\mathcal {E}}_s(w,D_1), \end{aligned}$$

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

$$\begin{aligned} \Vert \varphi \Vert _M :=\sup _{|\gamma |\leqq M}\sup _{\xi \in {\mathbb {R}}^n}\big |\partial ^\gamma {\widehat{\varphi }} (\xi )\big |(|\xi |^M+|\xi |^{-M}) \end{aligned}$$

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

$$\begin{aligned} \Vert f\Vert _{{\mathcal {Q}}^{\alpha ,q}_p({\mathbb {R}}^n)}:=\sup _Q\, |Q|^{\frac{1}{p}-\frac{1}{q}}\left( \iint _{Q\times Q}\frac{|f(x)-f(y)|^q}{|x-y|^{n+\alpha q}}\,{\mathrm{d}}x{\mathrm{d}}y\right) ^{1/q}<+\infty , \end{aligned}$$

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

$$\begin{aligned} N_{\alpha ,p,q}(f):=\sup _{D_r(x_0)\subseteq {\mathbb {R}}^n} r^{\frac{n}{p}-\frac{n}{q}}\left( \iint _{D_r(x_0)\times D_r(x_0)}\frac{|f(x)-f(y)|^q}{|x-y|^{n+\alpha q}}\,{\mathrm{d}}x{\mathrm{d}}y\right) ^{1/q} \end{aligned}$$

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

$$\begin{aligned} \alpha _1-\frac{\lambda }{q_1}=\alpha _2-\frac{\lambda }{q_2}, \end{aligned}$$
(C.1)

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

$$\begin{aligned} {\mathrm{spt}}\,{\widehat{\psi }} \subseteq \Big \{\xi \in {\mathbb {R}}^n : \frac{1}{2}\leqq |\xi |\leqq 2\Big \}\quad \text {and}\quad |{{\widehat{\psi }}}(\xi )|\geqq C>0 \text { for }\frac{3}{5}\leqq |\xi |\leqq \frac{5}{3}. \end{aligned}$$

(In particular, \(\psi \in {\mathscr {S}}_\infty ({\mathbb {R}}^n)\).) For \(j\in {\mathbb {Z}}\), we denote by \(\psi _j\) the function defined by

$$\begin{aligned} \psi _j(x):=2^{jn}\psi (2^jx). \end{aligned}$$

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

$$\begin{aligned} \Vert f\Vert _{\dot{F}^{s,\tau }_{p,q}({\mathbb {R}}^n)}:=\sup _Q\, \frac{1}{|Q|^{\tau }}\left( \int _Q\bigg (\sum _{j=j_Q}^\infty \big (2^{js}|\psi _j*f(x)|\big )^q\bigg )^{p/q} {\mathrm{d}}x\right) ^{1/p}<+\infty , \end{aligned}$$

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

$$\begin{aligned} \Vert f\Vert _{\dot{{\mathcal {E}}}^s_{p,q,u}({\mathbb {R}}^n)}:=\sup _Q\, |Q|^{\frac{1}{u}-\frac{1}{p}}\left( \int _Q\bigg (\sum _{j\in {\mathbb {Z}}} \big (2^{js}|\psi _j*f(x)|\big )^p\bigg )^{q/p} {\mathrm{d}}x\right) ^{1/q}<+\infty , \end{aligned}$$

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

$$\begin{aligned} {\dot{F}}^{s,p}_{M_{p,p,\lambda }}({\mathbb {R}}^n)=\dot{{\mathcal {E}}}^s_{p,p,\frac{np}{\lambda }}({\mathbb {R}}^n) \end{aligned}$$

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,

$$\begin{aligned} \dot{{\mathcal {E}}}^{\alpha _2}_{q_2,q_2,\frac{nq_2}{\lambda }}({\mathbb {R}}^n)\hookrightarrow \dot{{\mathcal {E}}}^{\alpha _1}_{q_1,q_1,\frac{nq_1}{\lambda }}({\mathbb {R}}^n) \end{aligned}$$
(C.2)

continuously. On the other hand, [42, Theorem 1.1] tells us that

$$\begin{aligned} \dot{{\mathcal {E}}}^{\alpha _1}_{q_1,q_1,\frac{nq_1}{\lambda }}({\mathbb {R}}^n)= \dot{F}^{\alpha _1,\frac{n-\lambda }{nq_1}}_{q_1,q_1}({\mathbb {R}}^n)\quad \text {and}\quad \dot{{\mathcal {E}}}^{\alpha _2}_{q_2,q_2,\frac{nq_2}{\lambda }}({\mathbb {R}}^n)= \dot{F}^{\alpha _2,\frac{n-\lambda }{nq_2}}_{q_2,q_2}({\mathbb {R}}^n), \end{aligned}$$

with equivalent semi-norms. Finally, by [55, Theorem 3.1] we have

$$\begin{aligned} \dot{F}^{\alpha _1,\frac{n-\lambda }{nq_1}}_{q_1,q_1}({\mathbb {R}}^n) = {\mathcal {Q}}^{\alpha _1,q_1}_{\frac{nq_1}{\lambda }}({\mathbb {R}}^n) \quad \text {and}\quad \dot{F}^{\alpha _2,\frac{n-\lambda }{nq_2}}_{q_2,q_2}({\mathbb {R}}^n) = {\mathcal {Q}}^{\alpha _2,q_2}_{\frac{nq_2}{\lambda }}({\mathbb {R}}^n), \end{aligned}$$

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

$$\begin{aligned} {[}f]_{W^{s,p}(\Omega )}:=\left( \iint _{\Omega \times \Omega } \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}}\,{\mathrm{d}}x{\mathrm{d}}y\right) ^{1/p}. \end{aligned}$$
(C.3)

Corollary C.6

Let \(s\in (0,1)\) and \(f\in L^1({\mathbb {R}}^n)\). If

$$\begin{aligned} \sup _{D_r(x)\subseteq {\mathbb {R}}^n} r^{2s-n}[f]^2_{H^s(D_r(x))}<+\infty , \end{aligned}$$
(C.4)

then,

$$\begin{aligned} \sup _{D_r(x)\subseteq {\mathbb {R}}^n} r^{\frac{2s-n}{3}}[f]^2_{W^{s/3,6}(D_r(x))}\leqq C \sup _{D_r(x)\subseteq {\mathbb {R}}^n} r^{2s-n}[f]^2_{H^s(D_r(x))}, \end{aligned}$$

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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