#  $s$-stability for $W^{s,n/s}$-harmonic maps
in homotopy groups

Katarzyna Mazowiecka  Institute of Mathematics,University of Warsaw, Banacha
2, 02-097 Warszawa, Poland<EMAIL_ADDRESS>and  Armin Schikorra
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall,
Pittsburgh, PA 15260, USA<EMAIL_ADDRESS>
###### Abstract.

We study $s$-dependence for minimizing $W^{s,n/s}$-harmonic maps
$u\colon\mathbb{S}^{n}\to\mathbb{S}^{\ell}$ in homotopy classes.
Sacks–Uhlenbeck theory shows that, for each $s$, minimizers exist in a
generating subset of $\pi_{n}({\mathbb{S}}^{\ell})$. We show that this
generating subset can be chosen locally constant in $s$. We also show that as
$s$ varies the minimal $W^{s,n/s}$-energy in each homotopy class changes
continuously. In particular, we provide progress to a question raised by
Mironescu [16] and Brezis–Mironescu [3].

###### Key words and phrases:

Minimizing fractional harmonic maps, homotopy theory, regularity theory,
existence

###### 2010 Mathematics Subject Classification:

58E20, 35B65, 35J60, 35S05

###### Contents

  1. 1 Introduction
  2. 2 Preliminary results
  3. 3 Existence of minimizers and energy identity
  4. 4 Globally improved regularity in the conformal scaling
  5. 5 Continuous dependence

##  1\. Introduction

We study minimizing $W^{s,p}$-harmonic maps between spheres in homotopy
classes, which are defined as maps $u\in
W^{s,p}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$ with least energy

(1.1)
$\mathcal{E}_{s,p}(u)\coloneqq\int_{{\mathbb{S}}^{n}}\int_{{\mathbb{S}}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,\mathrm{d}x\,\mathrm{d}y=[u]^{p}_{W^{s,p}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}$

among maps of the same homotopy. Here $s\in(0,1)$, $p>1$. In (1.1) we take the
$\mathbb{R}^{n+1}$-Euclidean distance and $\mathbb{R}^{\ell+1}$-Euclidean
distance in the numerator and in the denominator, respectively.

A natural question arises: given $\alpha\in\pi_{n}({\mathbb{S}}^{\ell})$, is
the infimum of $\mathcal{E}_{s,p}$ attained in $\alpha$? In other words, can
we find a map $u\in W^{s,p}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$,
$u\in\alpha$ such that

$\mathcal{E}_{s,p}(u)\leq\mathcal{E}_{s,p}(v)\quad\forall v\in
W^{s,p}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell}),\quad v\in\alpha.$

If $p>\frac{n}{s}$ the answer is yes, by standard methods of calculus of
variation, due to the compact embedding of
$W^{s,p}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$ into
$C^{0}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$. If $p<\frac{n}{s}$ it was shown
by Brezis–Nirenberg in [4] that no homotopy theory can be defined for maps in
$W^{s,p}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$ and the infimum energy of
$\mathcal{E}_{s,p}$ in any homotopy class is zero. Thus, throughout this work
we will focus on the critical, conformally invariant case $p=\frac{n}{s}$. By
[4], it is known that the standard notions of homotopy can be extended to
$W^{s,\frac{n}{s}}$-Sobolev maps, see also [15, Section 2] for an overview.

For $s\in(0,1)$ set

(1.2) $\\#_{s}\alpha\coloneqq\inf_{u\in
W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell}),u\in\alpha}\mathcal{E}_{s,\frac{n}{s}}(u),\quad\alpha\in\pi_{n}({\mathbb{S}}^{\ell}).$

In the case of maps between spheres of same dimension
$u\colon{\mathbb{S}}^{n}\to{\mathbb{S}}^{n}$, i.e., when the homotopy classes
are given by their degree we instead write

(1.3) $\\#_{s}d\coloneqq\inf_{u\in
W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell}),\deg(u)=d}\mathcal{E}_{s,\frac{n}{s}}(u),\quad
d\in\mathbb{Z}.$

In general the question whether $\\#_{s}\alpha$ is attained is rather involved
even in the the local case $s=1$, see, e.g., [21, 8, 19] or, for $n=1$ and
$s\in(0,1)$, see [3, Chapter 12]. In [15] we showed111 The case
$s\leq\frac{1}{2}$, $n=1$ was not treated in [15] but is is covered in [12].

######  Theorem 1.1.

For any $\ell,n\geq 1$, with either $(\ell,n)=(1,1)$ or $\ell\geq 2$,
$s\in(0,1)$. There exists a generating set
$X_{s}\subset\pi_{n}({\mathbb{S}}^{\ell})$ such that for any $\alpha\in X_{s}$
the infimum $\\#_{s}\alpha$ is attained.

In this work we are interested in the stability of such results as $s$
changes.

Our first main result is that one can choose the generating set $X_{s}$ from
Theorem 1.1 _locally stable_ as $s$ varies. More precisely we have,

######  Theorem 1.2.

Fix $n,\ell\geq 1$ with either $(\ell,n)=(1,1)$ or $\ell\geq 2$. Let
$\Lambda>0$ and set for $s\in(0,1)$

$X_{s}\coloneqq\left\\{\alpha\in\pi_{n}({\mathbb{S}}^{\ell})\colon\quad\text{there
exists a $W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$-minimizer
$u$ in $\alpha$ and
$[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}\leq\Lambda$}\right\\}.$

Then for any $t\in(0,1)$ there exists $\delta>0$ such that

$Y\coloneqq\bigcap_{s\in(t-\delta,t+\delta)}X_{s}$

spans the same set as $X_{s}$, i.e., $X_{s}\subset{\rm span}Y$, for each
$s\in(t-\delta,t+\delta)$.

See Theorem 5.2 for a more precise statement with respect to the dependencies
of $\delta$.

Let us stress that Theorem 1.2 does _not_ imply that

$X_{s}\coloneqq\\{\alpha\in\pi_{n}({\mathbb{S}}^{\ell})\setminus\\{0\\}\colon\quad\\#_{s}\alpha\text{
is attained}\\}$

is unchanged as $s$ varies. Rather, it says that we can choose the set of
attained generators of $\pi_{n}({\mathbb{S}}^{\ell})$ locally stable. In
particular, for $n=\ell$ (when the homotopy class is identified with the
degree), in principle, it could be possible that

$X_{s}=\begin{cases}\\{1\\}\quad&s<1/2,\\\
\mathbb{Z}\setminus\\{0\\}\quad&s=1/2,\\\ \\{1,2,-3\\}\quad&s\in(1/2,3/4),\\\
\\{2,-3\\}\quad&s\in(3/4,1).\end{cases}$

An important ingredient in the aforementioned Theorem 1.2, and our second main
result, is the following continuity result for the map
$s\mapsto\\#_{s}\alpha$. By lower semicontinuity of the norm, it is elementary
that for $\alpha\in\pi_{n}({\mathbb{S}}^{\ell})$ we have for any $t\in(0,1)$

$\\#_{t}\alpha\leq\liminf_{s\to t^{+}}\\#_{s}\alpha.$

But actually, we have full continuity.

######  Theorem 1.3.

Assume $\ell,n\geq 1$, with either $(\ell,n)=(1,1)$ or $\ell\geq 2$. Let
$\alpha\in\pi_{n}({\mathbb{S}}^{\ell})$. Then the map

$s\mapsto\\#_{s}\alpha,\quad\text{$s\in(0,1)$}$

is continuous.

Besides being a crucial ingredient for the proof of Theorem 1.2, Theorem 1.3
has also several interesting corollaries that we discuss now.

Firstly, it is natural to expect that minimizers exist in the class of degree
one maps in $W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n})$. Towards
this, Berlyand–Mironescu–Rybalko–Sandier obtain in [1, Lemma 3.1] that for
maps in $W^{\frac{1}{2},2}({\mathbb{S}}^{1},{\mathbb{S}}^{1})$ minimizers are
attained for any degree, see also [3, Theorem 12.9]. Moreover, Mironescu
proved a stability result [17, Theorem 2], which asserts that for
$s\in[\frac{1}{2},\frac{1}{2}+\delta)$ degree one minimizers exist in
$W^{s,\frac{1}{s}}({\mathbb{S}}^{1},{\mathbb{S}}^{1})$. As a corollary of
Theorem 1.3 we can extend the latter in the other direction.

######  Corollary 1.4.

For maps from ${\mathbb{S}}^{1}$ to ${\mathbb{S}}^{1}$, there exists
$\delta>0$ such that

$\\#_{s}1=\inf\\{[u]_{W^{s,\frac{1}{s}}}^{\frac{1}{s}}\colon u\in
W^{s,\frac{1}{s}}({\mathbb{S}}^{1},{\mathbb{S}}^{1}),\,\deg u=1\\}$

is attained for all $s\in(\frac{1}{2}-\delta,\frac{1}{2}+\delta)$.

This provides progress towards [16, Open Problem 1] and [3, Open Problem 24.].

More generally, if $\\#_{s}\alpha$ is attained for an
$\alpha\in\pi_{n}({\mathbb{S}}^{\ell})$ it is unclear whether $\\#_{t}\alpha$
is attained for $t\approx s$. Corollary 1.4 works for degree one maps, because
they have the lowest energy level among nontrivial homotopy classes. This
observation is true in any dimension and we have

######  Corollary 1.5.

Fix $n,\ell\geq 1$ with either $(\ell,n)=(1,1)$ or $\ell\geq 2$. Assume that
for some $t\in(0,1)$ and
$\alpha\in\pi_{n}({\mathbb{S}}^{\ell})\setminus\\{0\\}$ we have

$\\#_{t}\alpha\leq\\#_{t}\beta\quad\forall\beta\in\pi_{n}({\mathbb{S}}^{\ell})\setminus\\{0\\}.$

Then not only $\\#_{t}\alpha$ is attained by [15], but also there exists
$\delta>0$ such that $\\#_{s}\alpha$ is attained for all
$s\in(t-\delta,t+\delta)$.

For our next corollary of Theorem 1.3, we consider the
Bourgain–Brezis–Mironescu degree inequality, [2, Theorem 0.6], which says that
for maps $u\in W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n})$

(1.4) $\deg u\leq
C_{n,s}[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n})}^{\frac{n}{s}}.$

Let $\bar{C}_{n,s}$ be the minimal constant, i.e.,

$\bar{C}_{n,s}\coloneqq\sup_{u\in
W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n})}\frac{\deg
u}{[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n})}^{\frac{n}{s}}}<\infty.$

It is natural to discuss the continuity of the map $s\mapsto\bar{C}_{n,s}$.

######  Corollary 1.6.

The map $s\mapsto\bar{C}_{n,s}$ is lower semicontinuous.

More precisely, for any $\Lambda>0$ the map $s\mapsto\bar{C}_{n,s;\Lambda}$
defined by

$\bar{C}_{n,s;\Lambda}\coloneqq\sup_{u\in
W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n}),0<[u]_{W^{s,\frac{n}{s}}}\leq\Lambda}\frac{\deg
u}{[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n})}^{\frac{n}{s}}}$

is continuous.

Corresponding results hold for the Hopf degree, see [25],

$(1-\frac{1}{4n},1)\ni s\mapsto\tilde{C}_{n,s}\coloneqq\sup_{u\in
W^{s,\frac{n}{s}}({\mathbb{S}}^{4n-1},{\mathbb{S}}^{2n}),0<[u]_{W^{s,\frac{4n-1}{s}}}}\frac{[u]_{\pi_{4n-1}({\mathbb{S}}^{2n})}}{[u]_{W^{s,\frac{4n-1}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n})}^{\frac{4n}{s}}},$

and more generally for maps representing rational homotopy groups of spheres,
see [18].

We turn to the main ideas for Theorem 1.3, and thus Theorem 1.2. We use the
following new ingredient

(1.5) $\text{whenever $\\#_{s}\alpha$ is attained, then
$\\#_{t}\alpha\leq\\#_{s}\alpha+\varepsilon$ for $t\approx s$},$

for the precise formulation see Corollary 5.1. To obtain (1.5) we show that
$W^{s,n/s}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$ minimizers actually belong
_globally_ to
$W^{s_{1},\frac{n}{s_{1}}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$ for an
$s_{1}>s$, see Theorem 4.1. Hölder regularity in this situation has been
established in [22, 14], after pioneering work for $n=1$ and $s=\frac{1}{2}$
in [7], but this is a _local_ result on domains where the $BMO$-norm is small.
Smallness of the BMO-norm is of course not a scaling invariant property,
indeed it depends heavily on the specific minimizer $u$ and one cannot deduce
from it a uniform global property. Our higher regularity result in a
conformally invariant Sobolev space, Theorem 4.1, is, on the other hand,
uniform and independent of the specific minimizer $u$. Hence, using stability
of the Sobolev norm $W^{s,n/s}$, Proposition 2.5, we obtain (1.5). Once we
have (1.5), the main results follow from combinatorial observations coupled
with the Sacks–Uhlenbeck theory developed in [15] and the energy identity from
Theorem 3.2.

We conclude this introduction with a few remarks about possible
generalizations of these results.

######  Remark 1.7.

  1. (1)

The modulus of continuity in Theorem 1.3 and the $\delta$ in Theorem 1.2 are
relatively easily to compute. They depend on the regularity theory gain (which
can be calculated explicitly) and they get worse with large $\\#_{s}\alpha$.

  2. (2)

In this paper, to ensure clarity in our presentation, we focus on the case
when the target manifold is a sphere. Nonetheless, it should be easy to extend
the results to the case of a compact Lie group in the target.

  3. (3)

It seems that an extension of our results to general target manifold is more
challenging — the only obstacle is regularity, but it is unclear even for
minimizing maps how to get scaling-invariant higher regularity of Theorem 4.1.

  4. (4)

It would be interesting to study the limiting cases as $s\to 1^{-}$ and $s\to
0^{+}$.

### Notation

We write $\alpha,\beta$, etc. to denote a homotopy group, ${\mathbb{S}}^{n}$
for the sphere in the domain, and ${\mathbb{S}}^{\ell}$ for the target sphere.
For brevity, we write $\precsim$ whenever there is a constant $C$ (not
depending on any crucial quantity) such that $A\leq C$B. Similarly, $A\approx
B$ means $A\precsim B$ and $B\precsim A$.

### Acknowledgement

The work presented in this paper was conducted as part of the Thematic
Research Programme _Analysis and Geometry in Warsaw_ , which received funding
from the University of Warsaw via IDUB (Excellence Initiative Research
University). Part of this work was carried out while K.M. and A.S. were
visiting University of Bielefeld. We like to express our gratitude to the
University for its hospitality. A.S. is an Alexander-von-Humboldt Fellow. A.S.
is funded by NSF Career DMS-2044898. The project is co-financed by the Polish
National Agency for Academic Exchange within Polish Returns Programme -
BPN/PPO/2021/1/00019/U/00001 (K.M.). The project is co-financed by National
Science Centre grant 2022/01/1/ST1/00021 (KM).

##  2\. Preliminary results

Let us emphasize that some of the results of this section can be easily
extended to general target manifolds. For brevity we restrict everything to
sphere targets.

The first result is well-known and follows from the embedding of the critical
Sobolev space into BMO is the following, see, e.g., [15, Lemma 2.10].

######  Proposition 2.1.

For any $\ell,n\in{\mathbb{N}}$ there exists $\lambda=\lambda(\ell,n)>0$ such
that whenever $s\in(0,1]$ and
$\alpha\in\pi_{n}({\mathbb{S}}^{\ell})\setminus\\{0\\}$

$\\#_{s}\alpha\geq\lambda.$

In [27, Theorem 1.2] or [3, Lemma 12.6.] the following is proven.

######  Proposition 2.2.

Whenever $s\in(0,1]$ and $\alpha$ has a free homotopy group decomposition into
$(\alpha_{i})_{i=1}^{N}$ then

$\\#_{s}\alpha\leq\sum_{i=1}^{N}\\#_{s}\alpha_{i}.$

In the case of $W^{\frac{1}{2},2}({\mathbb{S}}^{1},{\mathbb{S}}^{1})$ maps,
minimizers exists for each degree and their exact energy is known. Precisely,
by [1, Lemma 3.1], see also [3, Theorem 12.9 & Theorem 12.10],

######  Theorem 2.3.

For maps from ${\mathbb{S}}^{1}$ to ${\mathbb{S}}^{1}$ we have

(2.1) $\\#_{\frac{1}{2}}d=4\pi^{2}|d|.$

Moreover, $\\#_{\frac{1}{2}}d$ is attained for all $d\in\mathbb{Z}$.

By results in [27] and [2] we obtain that if we know that the energy of a map
is bounded then the map can belong only to a finite subgroup of
$\pi_{n}({\mathbb{S}}^{l})$.

######  Theorem 2.4.

Fix $\Lambda>0$ and let $0<s_{0}<s_{1}<1$, $n,\ell\geq 1$ with either
$(\ell,n)=(1,1)$ or $\ell\geq 2$. Then there exist a _finite_ subgroup
$\mathcal{Q}\subset\pi_{n}({\mathbb{S}}^{\ell})$ such that the following
holds:

Whenever for some $s\in(s_{0},s_{1})$ the map $u\in
W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$ satisfies

$[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}\leq\Lambda$

then $u\in\mathcal{Q}$.

###### Proof.

In the case when $n=\ell$ the assertion follows from the degree estimate in
[2, Theorem 0.6], since for any $s>0$ we have

$\left|\deg
u\right|\precsim[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n})}^{\frac{n}{s}}.$

If $\pi_{1}({\mathbb{S}}^{\ell})=\\{0\\}$, i.e., $\ell\geq 2$, we have for any
$\varepsilon>0$

$\begin{split}&\int_{{\mathbb{S}}^{n}}\int_{{\mathbb{S}}^{n}}\chi_{\\{|f(x)-f(y)|>\varepsilon\\}}\frac{1}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq\varepsilon^{-\frac{n}{s_{0}}}\int_{{\mathbb{S}}^{n}}\int_{{\mathbb{S}}^{n}}\frac{|f(x)-f(y)|^{\frac{n}{s_{0}}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq\varepsilon^{-\frac{n}{s_{0}}}2^{\frac{n}{s_{0}}-\frac{n}{s_{1}}}\,\int_{{\mathbb{S}}^{n}}\int_{{\mathbb{S}}^{n}}\frac{|f(x)-f(y)|^{\frac{n}{s}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq\varepsilon^{-\frac{n}{s_{0}}}2^{\frac{n}{s_{0}}-\frac{n}{s_{1}}}\,\Lambda^{n}.\end{split}$

Hence the assumption in [27, Theorem 1.4] is satisfied and we may conclude. ∎

###  2.1. Continuity for the $s\mapsto W^{s,\frac{n}{s}}$-norm

We need the following continuity result for the fractional Sobolev norm.

######  Proposition 2.5.

Fix $n,\ell\in{\mathbb{N}}$. Let $\Lambda>0$, $t_{0}>0$, $t\in(t_{0},1)$, and
let $s_{1}>t$. For any $\varepsilon>0$ there exists
$\delta=\delta(\varepsilon,\Lambda,|s_{1}-t|,n)>0$ such that the following
holds.

Assume that $u\colon{\mathbb{S}}^{n}\to{\mathbb{S}}^{\ell}$ satisfies

$[u]_{W^{s_{1},\frac{n}{{s_{1}}}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}\leq\Lambda.$

then

$\sup_{r_{1},r_{2}\in(t-\delta,t+\delta)}\left|[u]_{W^{r_{1},\frac{n}{r_{1}}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}^{\frac{n}{r_{1}}}-[u]_{W^{r_{2},\frac{n}{r_{2}}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}^{\frac{n}{r_{2}}}\right|\leq\varepsilon.$

The proof is based on the following elementary lemma.

######  Lemma 2.6.

For any $\varepsilon>0$, $\Gamma>0$, $0<p_{0}<p_{1}$, there exists
$\tilde{\delta}=\tilde{\delta}(p_{0},p_{1},\Gamma,\varepsilon)>0$ such that if
$|p-q|<\tilde{\delta}$, $p,q\in[p_{0},p_{1}]$ then

(2.2)
$|p-q|<\tilde{\delta}\quad\Rightarrow\quad|a^{p}-a^{q}|<\varepsilon\quad\forall
a\in[0,\Gamma].$

###### Proof.

We may assume that $\varepsilon\in(0,1)$ and $\Gamma\geq 1$, $p_{0}\leq
p<q\leq p_{1}$.

Set $\sigma\coloneqq(\frac{1}{2}\,\varepsilon)^{\frac{1}{p_{0}}}\in(0,1)$,
then

$|a^{p}-a^{q}|<2\,\sigma^{p_{0}}\leq\varepsilon\quad\forall a\in[0,\sigma].$

Moreover with the inequality $|1-e^{t}|\leq|t|e^{|t|}$ we find for
$\Lambda_{\varepsilon}\coloneqq\max_{a\in[\sigma,\Gamma]}|\ln a|$,

$|a^{p}-a^{q}|=a^{p}|1-a^{q-p}|\leq\Gamma^{p_{1}}|1-a^{q-p}|\leq|q-p|\,\Gamma^{p_{1}}\,\Lambda_{\varepsilon}e^{\Lambda_{\varepsilon}2p_{1}}\quad\forall
a\in[\sigma,\Gamma].$

So if we set

$\tilde{\delta}\coloneqq\frac{1}{\Gamma^{p_{1}}\Lambda_{\varepsilon}e^{\Lambda_{\varepsilon}2p_{1}}}$

We have shown that

$|a^{p}-a^{q}|<\varepsilon\quad\forall|p-q|<\tilde{\delta},\quad
a\in[0,\Gamma].$

∎

###### Proof of Proposition 2.5.

Pick some $\bar{s}\in(t,s_{1})$ and take
$\frac{t-t_{0}}{2}<\delta<\frac{\bar{s}-t}{2}$ to be specified later. The
relation between the numbers is now

$0<t_{0}<t-\delta<t<t+\delta<\bar{s}<s_{1}<1.$

Fix $r_{1},r_{2}\in(t-\delta,t+\delta)$ such that $r_{2}>r_{1}$. We have

(2.3)
$\begin{split}&\left|[u]_{W^{r_{1},\frac{n}{r_{1}}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}^{\frac{n}{r_{1}}}-[u]_{W^{r_{2},\frac{n}{r_{2}}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}^{\frac{n}{r_{2}}}\right|\\\
&\leq\int_{{\mathbb{S}}^{n}}\int_{{\mathbb{S}}^{n}}\frac{\left||u(x)-u(y)|^{\frac{n}{r_{1}}}-|u(x)-u(y)|^{\frac{n}{r_{2}}}\right|}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\\\
&=\int_{{\mathbb{S}}^{n}}\int_{{\mathbb{S}}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s_{1}}}\left||u(x)-u(y)|^{\frac{n}{r_{1}}-\frac{n}{s_{1}}}-|u(x)-u(y)|^{\frac{n}{r_{2}}-\frac{n}{s_{1}}}\right|}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y.\end{split}$

Set now

$a\coloneqq|u(x)-u(y)|,\quad p\coloneqq\frac{n}{r_{1}}-\frac{n}{s_{1}},\quad
q\coloneqq\frac{n}{r_{2}}-\frac{n}{s_{1}}.$

Since $|u|\equiv 1$ we have $a\in[0,2]$. Also $p,q\in[p_{0},p_{1}]$ for

$p_{0}=\frac{n}{t+\delta}-\frac{n}{s_{1}}\geq\frac{n}{\bar{s}}-\frac{n}{s_{1}}>0\quad\text{and}\quad
p_{1}=\frac{n}{t-\delta}-\frac{n}{s_{1}}\leq\frac{n}{t_{0}}-\frac{n}{s_{1}}<\infty.$

We observe

$|p-q|=\frac{n}{r_{1}r_{2}}\left|r_{2}-r_{1}\right|\leq\frac{2n}{(t_{0})^{2}}\delta.$

Hence choosing $\delta=\tilde{\delta}\frac{(t_{0})^{2}}{2n}$, where
$\tilde{\delta}$ is from Lemma 2.6, and combining (2.3) with (2.2) we get

$\begin{split}\left|[u]_{W^{r_{1},\frac{n}{r_{1}}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}^{\frac{n}{r_{1}}}-[u]_{W^{r_{2},\frac{n}{r_{2}}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}^{\frac{n}{r_{2}}}\right|&\leq\sup_{a\in[0,2]}\left|a^{p}-a^{q}\right|\,\int_{{\mathbb{S}}^{n}}\int_{{\mathbb{S}}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s_{1}}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq\varepsilon\int_{{\mathbb{S}}^{n}}\int_{{\mathbb{S}}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s_{1}}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y,\end{split}$

as desired.

∎

##  3\. Existence of minimizers and energy identity

In this section we show how to deduce an energy identity using [15] and [27].
We begin with recalling the following Lemma, which we will combine later with
Theorem 2.4.

######  Lemma 3.1 ([15, Lemma 7.7]).

Fix $n,\ell\in{\mathbb{N}}$ with either $(\ell,n)=(1,1)$ or $\ell\geq 2$, and
$s\in(0,1)$. There is a number $\theta=\theta(s,n,\ell)$ such that the
following holds222As mentioned before the case $s\leq\frac{1}{2}$, $n=1$ was
not treated in [15] but is is covered in [12].. Let
$\alpha\in\pi_{n}({\mathbb{S}}^{\ell})\setminus\\{0\\}$. Then either
$\\#_{s}\alpha$ is attained or for any $\delta>0$ there exist
$\alpha_{1},\alpha_{2}\in\pi_{n}({\mathbb{S}}^{\ell})\setminus\\{0\\}$
(possibly depending on $\delta$) such that $\alpha=\alpha_{1}+\alpha_{2}$,

(3.1) $\\#_{s}\alpha_{1}+\\#_{s}\alpha_{2}\leq\\#_{s}\alpha+\delta,$

and

(3.2)
$\quad\theta<\\#_{s}\alpha_{i}<\\#_{s}\alpha-\frac{\theta}{2},\quad\text{ for
}i=1,2.$

From Lemma 3.1 we can conclude the following existence and energy identity.

######  Theorem 3.2 (Energy identity).

Fix $n,\ell\in{\mathbb{N}}$, with either $(\ell,n)=(1,1)$ or $\ell\geq 2$, and
$s\in(0,1)$. For each $\alpha\in\pi_{n}({\mathbb{S}}^{\ell})\setminus\\{0\\}$
there exists a finite sequence
$(\alpha_{i})_{i=1}^{N}\subset\pi_{n}({\mathbb{S}}^{\ell})\setminus\\{0\\}$
such that

  1. (1)

$\alpha=\sum_{i=1}^{N}\alpha_{i}$,

  2. (2)

$\\#_{s}\alpha=\sum_{i=1}^{N}\\#_{s}\alpha_{i}$,

  3. (3)

$\\#_{s}\alpha_{i}$ are attained for each $i\in\\{1,\ldots,N\\}$.

###### Proof.

Fix $\Lambda\coloneqq\\#_{s}\alpha+1$.

Fix $\tilde{\delta}\in(0,1)$. If $\\#_{s}\alpha$ is attained then we are done.
If $\\#_{s}\alpha$ is not attained, we apply Lemma 3.1 and decompose
$\alpha=\alpha_{1}+\alpha_{2}$ with

$\\#_{s}\alpha_{1}+\\#_{s}\alpha_{2}\leq\\#_{s}\alpha+2^{-1}\tilde{\delta}\quad\text{and}\quad\theta<\\#_{s}\alpha_{i}<\\#_{s}\alpha-\frac{\theta}{2}\text{
for }i=1,2.$

If both $i\in{1,2}$, $\\#_{s}\alpha_{i}$ are not attained we finish, if at
least one is not attained, say $\\#_{s}\alpha_{1}$ we apply again Lemma 3.1
with $\delta=2^{-2}\tilde{\delta}$. We decompose
$\alpha_{1}=\alpha_{1,1}+\alpha_{1,2}$ and obtain

(3.3)
$\\#_{s}\alpha_{1,1}+\\#_{s}\alpha_{1,2}+\\#_{s}\alpha_{2}\leq\\#_{s}\alpha_{1}+\\#_{s}\alpha_{2}+2^{-2}\tilde{\delta}+2^{-1}\tilde{\delta}\leq\\#_{s}\alpha+\sum_{i=1}^{2}2^{-i}\tilde{\delta}$

with

(3.4)
$\theta<\\#_{s}\alpha_{1,i}<\\#_{s}\alpha_{1}-\frac{\theta}{2}<\\#_{s}\alpha-\theta,$

where $\alpha=\alpha_{1,1}+\alpha_{1,2}+\alpha_{2}$. With an abuse of notation
we relabel $\alpha=\alpha_{1}+\alpha_{2}+\alpha_{3}$.

We apply Lemma 3.1 iteratively, whenever the minimizer of one of the
decomposed terms is not attained — at $\ell$-th step we take
$\delta=2^{-\ell}\tilde{\delta}$ and obtain a decomposition into $\ell+1$
terms with

(3.5)
$\sum_{i=1}^{\ell+1}\\#_{s}\alpha_{i}\leq\\#_{s}\alpha+\sum_{i=1}^{\ell+1}2^{-i}\tilde{\delta}.$

By (3.2) after $2^{\ell-1}$-iterations there is an
$i\in\\{1,\ldots,2^{\ell-1}+1\\}$ for which

(3.6)
$\theta<\\#_{s}\alpha_{i}<\\#_{s}\alpha-\ell\frac{\theta}{2}=\Lambda-1-\ell\frac{\theta}{2}.$

If however $\ell>\frac{2}{\theta}(\Lambda-1-\theta)$ we get a contradiction in
(3.6). Hence, we may iterate at most $2^{L-1}$ times for an
$L=L(\theta,\Lambda)$ obtaining a decomposition into $N_{\tilde{\delta}}$
terms, where $N_{\tilde{\delta}}\leq 2^{L-1}+1$ for all
$\tilde{\delta}\in(0,1)$ and

$\sum_{i=1}^{N_{\tilde{\delta}}}\alpha_{i}=\alpha$

such that $\\#_{s}\alpha_{i}$ must be attained (otherwise we would have
continued the iteration of Lemma 3.1). Moreover,

$\sum_{i=1}^{N_{\tilde{\delta}}}\\#_{s}\alpha_{i}\leq\\#_{s}\alpha+\tilde{\delta}.$

We want to let $\tilde{\delta}\to 0$. Let us stress that as of now the
decomposition of $\alpha_{i}$ depends on $\tilde{\delta}$.

By Theorem 2.4,

$\\{\beta\in\pi_{n}({\mathbb{S}}^{\ell})\colon\quad\\#_{s}\beta\leq\\#_{s}\alpha+1\\}$

is a finite set. Hence, there is only a finite number of possibilities of
$\alpha_{i}=\alpha_{i}(\tilde{\delta})$. Thus, there exists a sequence
$\tilde{\delta}_{k}\to 0$ such that
$\alpha_{i}(\tilde{\delta}_{k})=\alpha_{i}(\tilde{\delta}_{j})$ for all
$j,k\in{\mathbb{N}}$. Thus we have

$\sum_{i=1}^{N}\\#_{s}\alpha_{i}\leq\\#_{s}\alpha+\tilde{\delta}_{k},$

where the left-hand side does not depend on $\tilde{\delta}_{k}$ anymore.
Letting $k\to\infty$ we obtain

$\sum_{i=1}^{N}\\#_{s}\alpha_{i}\leq\\#_{s}\alpha.$

On the other hand, since $\sum_{i=1}^{N}\alpha_{i}=\alpha$ we have by
Proposition 2.2,

$\\#_{s}\alpha\leq\sum_{i=1}^{N}\\#_{s}\alpha_{i}.$

Combining the two last statements we obtain

$\\#_{s}\alpha=\sum_{i=1}^{N}\\#_{s}\alpha_{i}.$

∎

Let us remark in passing that another strategy to obtain Theorem 3.2 would be
to extend the methods [8], which mostly rely on the conformal invariance of
the norms involved, and thus should be applicable to our case.

##  4\. Globally improved regularity in the conformal scaling

In this section we show that $W^{s,\frac{n}{s}}$-minimizers actually belong
_globally_ to $W^{\Theta s,\frac{n}{\Theta
s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$ for a $\Theta>1$. To do so we adapt
the strategy of [22]. It seems to us that we use a somehow unique feature of
the Gagliardo-seminorm and the fractional $p$-Laplacian (i.e., we do not see
how the argument would work for the classical $p$-Laplace): we use
intrinsically a feature of differential stability that was observed for the
fractional $p$-Laplace but is not known for the classical $p$-Laplace, see
[13, 23].

######  Theorem 4.1.

Fix $n,\ell\in{\mathbb{N}}$. For every $0<s_{0}<s_{1}<1$ there exists
$\Theta>1$ and for any $\Lambda>0$ there is a constant
$C(\Lambda)=C(n,\ell,s_{0},s_{1},\Lambda)>0$ such that the following holds:

If $s_{0}<s<s_{1}$ and $u\in
W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$ is a
$W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$-minimizer in its own
homotopy group with

$[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}\leq\Lambda$

then

$[u]_{W^{\Theta s,\frac{n}{\Theta
s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}\leq C(\Lambda).$

Theorem 4.1 is a consequence of Proposition 4.3 and Proposition 4.6 below.

######  Remark 4.2.

  1. (1)

The study of the regularity of fractional harmonic maps was initiated by Da
Lio–Rivière in their celebrated papers [6, 7].

  2. (2)

The argument below applies generally to critical points, not only minimizers.
We only state the a priori estimate versions, since (local) higher regularity
was discussed in [15].

  3. (3)

In [22] it was proven that minimizers as in Theorem 4.1 are Hölder continuous,
see also [14]. Observe that, however, there is no hope to prove

$[u]_{C^{0,\alpha}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}\precsim[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}.$

Indeed, this can be simply disproved by conformal rescaling, concentrating the
map $u$ into one-point. The right-hand side is conformally invariant, and thus
does not change. However, the continuity on the left-hand side becomes worse
and worse.

  4. (4)

Similarly, there is no hope to obtain

$[u]_{W^{s_{1},p_{1}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}\precsim[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}.$

whenever $s_{1}p_{1}>n$.

  5. (5)

While Theorem 4.1, does not imply continuity, and therefore seems to be a
weaker result than [22, 14], it has the crucial advantage of being a global
result on all of ${\mathbb{S}}^{n}$.

By the conformal invariance of the energy we may replace the domain
${\mathbb{S}}^{n}$ by $\mathbb{R}^{n}$ and assume
$u\in\dot{W}^{s,\frac{n}{s}}(\mathbb{R}^{n},{\mathbb{S}}^{\ell})$.

###  4.1. Improved global estimates for harmonic maps

We begin with the Euler–Lagrange equations for $W^{s,\frac{n}{s}}$-harmonic
maps, see, e.g., [22]. For any $\varphi\in
L^{\infty}\cap\dot{W}^{s,\frac{n}{s}}(\mathbb{R}^{n},\mathbb{R}^{\ell+1})$ a
critical point $u\in W^{s,\frac{n}{s}}(\mathbb{R}^{n},{\mathbb{S}}^{\ell})$ of
the energy $\mathcal{E}_{s,\frac{n}{s}}$ satisfies the equation

(4.1)
$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\wedge(u(x)\varphi(x)-u(y)\varphi(y))}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y=0.$

Here $\wedge\colon\mathbb{R}^{\ell}\times\mathbb{R}^{\ell}$ denotes the wedge
product in $\mathbb{R}^{\ell+1}$,

$(u\wedge
v)_{ij}=u^{i}v^{j}-u^{j}v^{i},\quad\text{$i,j\in\\{1,\ldots,\ell+1\\}$}.$

The crucial result is that the equation for fractional
$W^{s,\frac{n}{s}}$-harmonic maps improves globally.

######  Proposition 4.3.

Let $0<s_{0}<s_{1}<1$. There exists a $\theta\in(0,1)$ such that the following
holds:

For any $s\in(s_{0},s_{1})$ and any
$u\in\dot{W}^{s,\frac{n}{s}}(\mathbb{R}^{n},{\mathbb{S}}^{\ell})$, which is
minimizing $W^{s,\frac{n}{s}}$-harmonic map in its own homotopy group we have

$[(-\Delta)^{s}_{\frac{n}{s}}u]_{W^{-\theta s,\frac{n}{n-\theta
s}}(\mathbb{R}^{n})}\leq
C(s_{0},s_{1},n)[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}},$

that is for any $\psi\in\dot{W}^{\theta s,\frac{n}{\theta
s}}(\mathbb{R}^{n},\mathbb{R}^{\ell+1})$,

(4.2)
$\begin{split}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}&\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\,\left(\psi(x)-\psi(y)\right)}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq
C(s_{0},s_{1},n)[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}}[\psi]_{W^{\theta
s,\frac{n}{\theta s}}(\mathbb{R}^{n})}.\end{split}$

In the proof below we will frequently work with the fractional Laplacian
$(-\Delta)^{\frac{s}{2}}$ which can be defined as

$(-\Delta)^{\frac{s}{2}}f=c_{s}\mathcal{F}^{-1}\left(|\xi|^{s}\mathcal{F}f\right),$

where $\mathcal{F}$ is the Fourier transform. When $s\in(0,1)$, with a
different constant, we also have the representation

$(-\Delta)^{\frac{s}{2}}f(x)=c_{s}\int_{\mathbb{R}^{n}}\frac{f(y)-f(x)}{|x-y|^{n+s}}\,\mathrm{d}y.$

The inverse of the fractional Laplacian is the Riesz potential $I^{s}$,
defined as

$I^{s}f=c_{s}\mathcal{F}^{-1}\left(|\xi|^{-s}\mathcal{F}f\right),$

or, for $s\in(0,n)$,

(4.3) $I^{s}f(x)=c_{s}\int_{\mathbb{R}^{n}}|x-y|^{s-n}f(y)\,\mathrm{d}y.$

For mapping properties of the Riesz potential we refer the reader to standard
literature, e.g., [26, 9, 10, 20].

In order to prove Proposition 4.3 we consider the following potential
introduced in [22]:

(4.4)
$T_{t}u(z)\coloneqq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\,(|x-z|^{t-n}-|y-z|^{t-n})}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y.$

Observe that for $t<s$ we have by the representation of the Riesz potential
$I^{t}$, (4.3),

$\int_{\mathbb{R}^{n}}T_{t}u(z)\varphi(z)\,\mathrm{d}z=c\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\,(I^{t}\varphi(x)-I^{t}\varphi(y))}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y.$

From the definition of $T_{t}$ and Hölder’s inequality we have,

(4.5)
$\left|\int_{\mathbb{R}^{n}}T_{t}u(z)\varphi(z)\,\mathrm{d}z\right|\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}-1}\,[I^{t}\varphi]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}.$

Thus, if $t<s$, $T_{t}u$ is a tempered distribution for $u\in
W^{s,\frac{n}{s}}(\mathbb{R}^{n})$.

Observe that even as a distribution we have

(4.6) $\|T_{t}u\|_{L^{\frac{n}{n-t}}(\mathbb{R}^{n})}\precsim\|u\cdot
T_{t}u\|_{L^{\frac{n}{n-t}}(\mathbb{R}^{n})}+\|u\wedge
T_{t}u\|_{L^{\frac{n}{n-t}}(\mathbb{R}^{n})},$

whenever the right-hand side is finite. In the next two lemmas we will
estimate separately both terms on the right-hand side of (4.6): the orthogonal
projection $u\cdot T_{t}u$ and the tangential projection $u\wedge T_{t}u$
(orthogonal and tangential are meant with respect to the tangent space
$T_{u}{\mathbb{S}}^{\ell}$).

######  Lemma 4.4.

Let $0<s_{0}<s_{1}<1$. There exists a $\theta\in(0,1)$ and a constant
$C=C(s_{0},s_{1},n,\ell)>0$ such that the following holds:

For any $s\in(s_{0},s_{1})$, there exists $t<\theta s$ such that if $u$ is a
$W^{s,\frac{n}{s}}(\mathbb{R}^{n},{\mathbb{S}}^{\ell})$-minimizing harmonic
map in its own homotopy group, then for $T_{t}$ as in (4.4)

(4.7) $\|u\cdot T_{t}u\|_{L^{\frac{n}{n-t}}(\mathbb{R}^{n})}\leq
C\,[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}}.$

###### Proof.

We argue similarly as in the proof of [22, Lemma 6.5]. We note that since
$|u|=1$ we have

$(u(x)-u(y))\cdot u(z)=-\frac{1}{2}(u(x)-u(y))\cdot(u(x)+u(y)-2u(z))$

and hence

$\left|u\cdot
T_{t}u(z)\right|\precsim\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-1}|u(x)+u(y)-2u(z)|\,\left||x-z|^{t-n}-|y-z|^{t-n}\right|}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y.$

We observe that for $r\in(0,1)$, we have by [24, Proposition 6.6],

(4.8)
$|u(x)-u(y)|\precsim|x-y|^{r}\,\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u(x)+\mathcal{M}(-\Delta)^{\frac{r}{2}}u(y)\right),$

where $\mathcal{M}$ denotes the Hardy-Littlewood maximal function.

We follow the proof in [22, Proposition 6.3] but replace the use of [22,
Proposition 6.2] by (4.8) and consider the three regimes (similarly as in [22,
Proposition 6.1]):

$\begin{split}&\\{|x-y|\precsim\min\\{|x-z|,|y-z|\\}\\},\quad\text{ in this
case $|x-z|\approx|y-z|$};\\\
&\\{|x-z|\precsim\min\\{|y-z|,|x-y|\\}\\},\quad\text{ in this case
$|y-z|\approx|x-y|$};\\\ &\\{|y-z|\precsim\min\\{|x-z|,|x-y|\\}\\},\quad\text{
in this case $|x-z|\approx|x-y|$}.\end{split}$

we obtain for $\tilde{t}\in(0,t)$ with $r+\tilde{t}\in(0,1)$

(4.9)
$\begin{split}&|u(x)+u(y)-2u(z)|\,\left||x-z|^{t-n}-|y-z|^{t-n}\right|\\\
&\precsim\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u(x)+\mathcal{M}(-\Delta)^{\frac{r}{2}}u(y)+\mathcal{M}(-\Delta)^{\frac{r}{2}}u(z)\right)|x-y|^{r+\tilde{t}}k_{t-\tilde{t},t}(x,y,z),\end{split}$

where $\kappa_{\alpha,\gamma}(x,y,z)$ is given by

(4.10)
$\begin{split}\kappa_{\alpha,\gamma}(x,y,z)&=\min\\{|x-z|^{\alpha-n},|y-z|^{\alpha-n}\\}\\\
&\quad+\left(\frac{|y-z|}{|x-y|}\right)^{\gamma-\alpha}|y-z|^{\alpha-n}\chi_{\\{|x-z|\precsim\min\\{|y-z|,|x-y|\\}\\}}\\\
&\quad+\left(\frac{|x-z|}{|x-y|}\right)^{\gamma-\alpha}|x-z|^{\alpha-n}\chi_{\\{|y-z|\precsim\min\\{|x-z|,|x-y|\\}\\}}.\end{split}$

Moreover, we have by (4.8)

(4.11)
$|u(x)-u(y)|^{\frac{n}{s}-1}\precsim|x-y|^{r(\frac{n}{s}-1)}\left(\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u(x)\right)^{\frac{n}{s}-1}+\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u(y)\right)^{\frac{n}{s}-1}\right).$

Combining (4.11) with (4.9) we obtain for a $\varphi\in
C_{c}^{\infty}(\mathbb{R}^{n})$,
$\|\varphi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}\leq 1$

(4.12) $\begin{split}&\|u\cdot T_{t}u\|_{L^{\frac{n}{n-t}}(\mathbb{R}^{n})}\\\
&\precsim\int_{\mathbb{R}^{n}}u(z)\cdot T_{t}u(z)\varphi(z)\,\mathrm{d}z\\\
&\precsim\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-1}|u(x)+u(y)-2u(z)|\,\left||x-z|^{t-n}-|y-z|^{t-n}\right|}{|x-y|^{2n}}\left|\varphi(z)\right|\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\\\
&\precsim\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|x-y|^{r\frac{n}{s}+\tilde{t}-n-n}k_{t-\tilde{t},t}(x,y,z)U_{r}(x,y,z)\left|\varphi(z)\right|\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z,\end{split}$

where

$\begin{split}&U_{r}(x,y,z)\\\
&\coloneqq\left(\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u(x)\right)^{\frac{n}{s}-1}+\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u(y)\right)^{\frac{n}{s}-1}\right)\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u(x)+\mathcal{M}(-\Delta)^{\frac{r}{2}}u(y)+\mathcal{M}(-\Delta)^{\frac{r}{2}}u(z)\right).\end{split}$

Assuming $r\frac{n}{s}+\tilde{t}-n>0$ we estimate further (4.12) with the help
of [22, Proposition 6.4] and Hölder’s inequality

(4.13) $\begin{split}&\|u\cdot T_{t}u\|_{L^{\frac{n}{n-t}}(\mathbb{R}^{n})}\\\
&\precsim\max_{t_{1}+t_{2}+t_{3}=r\frac{n}{s}-n+t}\int_{\mathbb{R}^{n}}\,I^{t_{1}}\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u(z)\right)^{\frac{n}{s}-1}\,I^{t_{2}}\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u(z)\right)\,I^{t_{3}}\left|\varphi(z)\right|\,\mathrm{d}z\\\
&\precsim\max_{t_{1}+t_{2}+t_{3}=r\frac{n}{s}-n+t}\|I^{t_{1}}\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u\right)^{\frac{n}{s}-1}\|_{L^{\frac{n}{r\frac{n}{s}-r-t_{1}}}(\mathbb{R}^{n})}\,\|I^{t_{2}}\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u\right)\|_{L^{\frac{n}{r-t_{2}}}(\mathbb{R}^{n})}\|I^{t_{3}}|\varphi|\|_{L^{\frac{n}{t-t_{3}}}(\mathbb{R}^{n})}.\end{split}$

Assuming $r\frac{n}{s}-r-t_{1}\geq-r-t+n>0$, $t_{2}\leq r\frac{n}{s}-n+t<r$,
$t_{3}\leq r\frac{n}{s}-n+t<t$ we apply twice Sobolev’s inequality and the
Maximal Theorem to get

(4.14) $\begin{split}\|u\cdot
T_{t}u\|_{L^{\frac{n}{n-t}}(\mathbb{R}^{n})}&\precsim\|\left(\mathcal{M}(-\Delta)^{\frac{r}{2}}u\right)^{\frac{n}{s}-1}\|_{L^{\frac{n}{r\frac{n}{s}-r}}(\mathbb{R}^{n})}\,\|{\mathcal{M}(-\Delta)^{\frac{r}{2}}u}\|_{L^{\frac{n}{r}}(\mathbb{R}^{n})}\|\varphi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}\\\
&\precsim\|(-\Delta)^{\frac{r}{2}}u\|_{L^{(\frac{n}{s}-1)\frac{n}{r\frac{n}{s}-r}}(\mathbb{R}^{n})}^{\frac{n}{s}-1}\,\|{(-\Delta)^{\frac{r}{2}}u}\|_{L^{\frac{n}{r}}(\mathbb{R}^{n})}\|\varphi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}\\\
&=\|(-\Delta)^{\frac{r}{2}}u\|_{L^{\frac{n}{r}}(\mathbb{R}^{n})}^{\frac{n}{s}}\|\varphi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}\\\
&\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}}.\end{split}$

In the last line we used $r<s$.

The conditions on $r$ and $t$ used in estimates (4.13) and (4.14) are

$r<s,\quad n<\frac{n}{s}r+t<n+r,\quad r+t<n.$

If $n=1$ we choose $r=t$, then any
$t\in(\frac{s}{1+s},\min\\{s,\frac{1}{2}\\})$ is admissible, so we can pick

$t\coloneqq\frac{\frac{s}{1+s}+\min\\{s,\frac{1}{2}\\}}{2}\leq
s\frac{\frac{1}{1+s}+1}{2}\leq s\frac{\frac{1}{1+s_{0}}+1}{2}\coloneqq
s\theta.$

Observe that we do not need to make a distinction between the case
$\frac{n}{s}<2$ and $\frac{n}{s}\geq 2$.

If $n\geq 2$ it is even easier, since $r+t<n$ becomes a trivial condition.
This finishes the proof.

∎

Now we estimate the second part on the right-hand side of (4.6).

######  Lemma 4.5.

Let $0<s_{0}<s_{1}<1$. There exists a $\theta\in(0,1)$ such that the following
holds:

For any $s\in(s_{0},s_{1})$, there exists $t<\theta s$ such that if $u$ is a
$W^{s,\frac{n}{s}}(\mathbb{R}^{n},\mathbb{S}^{\ell})$-minimizing harmonic map
in its own homotopy group, then for $T_{t}$ as in (4.4)

(4.15) $\|u\wedge
T_{t}u\|_{L^{\frac{n}{n-t}}(\mathbb{R}^{n})}\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}}.$

###### Proof.

We argue as in [22, Proof of Lemma 3.5].

By duality, there is some $\psi\in C_{c}^{\infty}(\mathbb{R}^{n})$,
$\|\psi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}\leq 1$,

$\|u\wedge
T_{t}u\|_{L^{\frac{n}{n-t}}(\mathbb{R}^{n})}\precsim\int_{\mathbb{R}^{n}}u\wedge
T_{t}u\,\psi.$

Take $R\gg 1$ so that ${\rm supp\,}\psi\subset B(0,R)$. Let $\eta_{R}\in
C_{c}^{\infty}(B(0,2R))$ be a cut-off function such that $\eta_{R}\equiv 1$ in
$B(0,R)$ and set

$\begin{split}\varphi_{1,R}&\coloneqq\eta_{R}I^{t}\psi,\\\
\varphi_{2,R}&\coloneqq(1-\eta_{R})I^{t}\psi.\end{split}$

Then

$\int_{\mathbb{R}^{n}}u\wedge T_{t}u\,\psi=\int_{\mathbb{R}^{n}}u\wedge
T_{t}u\,(-\Delta)^{\frac{t}{2}}\varphi_{1,R}+\int_{\mathbb{R}^{n}}u\wedge
T_{t}u\,(-\Delta)^{\frac{t}{2}}\varphi_{2,R}.$

We observe that with a constant independent of $R\gg 1$

$\|(-\Delta)^{\frac{t}{2}}\varphi_{1,R}\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}\precsim\|\psi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}\leq
1,$

hence

$\begin{split}\int_{\mathbb{R}^{n}}u\wedge T_{t}u\,\psi&\leq\sup_{\varphi\in
C_{c}^{\infty}(\mathbb{R}^{n}),\,\|(-\Delta)^{\frac{t}{2}}\varphi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}\precsim
1}\int_{\mathbb{R}^{n}}u\wedge T_{t}u\,(-\Delta)^{\frac{t}{2}}\varphi\\\
&\quad+\limsup_{R\to\infty}\left|\int_{\mathbb{R}^{n}}u\wedge
T_{t}u\,(-\Delta)^{\frac{t}{2}}\varphi_{2,R}\right|.\end{split}$

For the second term on the right-hand side we observe that similarly as in
(4.5), for suitably small $\varepsilon>0$

$\begin{split}\left|\int_{\mathbb{R}^{n}}u\wedge
T_{t}u\,(-\Delta)^{\frac{t}{2}}\varphi_{2,R}\right|&\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}-1}\,[I^{t}(u\wedge(-\Delta)^{\frac{t}{2}}\varphi_{2,R})]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}\\\
&\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}-1}\,\|(-\Delta)^{\frac{s-t+\varepsilon}{2}}(u\wedge(-\Delta)^{\frac{t}{2}}\varphi_{2,R})\|_{L^{\frac{n}{s+\varepsilon}}(\mathbb{R}^{n})}\\\
&\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}-1}\,(1+[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})})\,\|(-\Delta)^{\frac{s+\varepsilon}{2}}\varphi_{2,R}\|_{L^{\frac{n}{s+\varepsilon}}(\mathbb{R}^{n})},\end{split}$

where in the second estimate we used an embedding in Triebel–Lizorkin spaces,
see [20, Theorem 2.2.3].

Now we observe that, due to the support of $1-\eta_{R}$ and $\psi$ we have for
some $\sigma>0$

$\begin{split}\|(-\Delta)^{\frac{s+\varepsilon}{2}}\varphi_{2,R}\|_{L^{\frac{n}{s+\varepsilon}}(\mathbb{R}^{n})}&\precsim
R^{-s-\varepsilon}\|I^{t}\psi\|_{L^{\frac{n}{s+\varepsilon}}}+\|(1-\eta_{R})(-\Delta)^{\frac{s+\varepsilon-t}{2}}\psi\|_{L^{\frac{n}{s+\varepsilon}}}\\\
&\precsim R^{-\sigma}C({\rm
supp\,}\psi)\|\psi\|_{L^{\frac{n}{t}}}\xrightarrow{R\to\infty}0.\end{split}$

Thus, for some $\varphi\in C_{c}^{\infty}(\mathbb{R}^{n})$,
$\|(-\Delta)^{\frac{t}{2}}\varphi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}\leq 1$,

(4.16) $\begin{split}\|u\wedge
T_{t}u\|_{L^{\frac{n}{n-t}}(\mathbb{R}^{n})}&\precsim\int_{\mathbb{R}^{n}}u\wedge
T_{t}u\,(-\Delta)^{\frac{t}{2}}\varphi\\\
&=-\int_{\mathbb{R}^{n}}\varphi\,(-\Delta)^{\frac{t}{2}}u\wedge
T_{t}u+\underbrace{\int_{\mathbb{R}^{n}}(-\Delta)^{\frac{t}{2}}(\varphi\,u)\wedge
T_{t}u}_{\overset{\eqref{eq:pdemin}}{=}0}-\int_{\mathbb{R}^{n}}H_{(-\Delta)^{\frac{t}{2}}}(\varphi,u)\wedge
T_{t}u.\end{split}$

Here we use the Leibniz term notation

$H_{(-\Delta)^{\frac{t}{2}}}(f,g)\coloneqq(-\Delta)^{\frac{t}{2}}(fg)-f(-\Delta)^{\frac{t}{2}}g-(-\Delta)^{\frac{t}{2}}f\
g.$

For the last term of (4.16) we observe that similarly as in (4.5) for a
suitably small $\varepsilon>0$

(4.17)
$\begin{split}\int_{\mathbb{R}^{n}}H_{(-\Delta)^{\frac{t}{2}}}(\varphi,u)\wedge
T_{t}u&\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}-1}\,[I^{t}H_{(-\Delta)^{\frac{t}{2}}}(\varphi,u)]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}\\\
&\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}-1}\,\|(-\Delta)^{\frac{s-t+\varepsilon}{2}}H_{(-\Delta)^{\frac{t}{2}}}(\varphi,u)\|_{L^{\frac{n}{s+\varepsilon}}(\mathbb{R}^{n})}\\\
&\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}}\,\|(-\Delta)^{\frac{t}{2}}\varphi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}.\end{split}$

The last line works as long $s-t+\varepsilon<t$.

For the remaining estimates we abbreviate

$\text{diff}_{\frac{n}{s}}u(x,y)=|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y)).$

By the representation of the Riesz potential, (4.3), and (4.1) we have

(4.18)
$\begin{split}&\int_{\mathbb{R}^{n}}\varphi\,(-\Delta)^{\frac{t}{2}}u\wedge
T_{t}u\\\
&=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{\text{diff}_{\frac{n}{s}}u(x,y)\wedge\left(I^{t}\left(\varphi\,(-\Delta)^{\frac{t}{2}}u\right)(x)-I^{t}\left(\varphi\,(-\Delta)^{\frac{t}{2}}u\right)(y)\right)}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\\\
&=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{\text{diff}_{\frac{n}{s}}u(x,y)\wedge\left(I^{t}\left(\varphi\,(-\Delta)^{\frac{t}{2}}u\right)(x)-I^{t}\left(\varphi\,(-\Delta)^{\frac{t}{2}}u\right)(y)-\frac{1}{2}(u(x)-u(y))(\varphi(x)+\varphi(y))\right)}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y.\end{split}$

Exactly as in the first lines of the proof of [22, Lemma 6.6] we have

$\begin{split}&\left|I^{t}\left(\varphi\,(-\Delta)^{\frac{t}{2}}u\right)(x)-I^{t}\left(\varphi\,(-\Delta)^{\frac{t}{2}}u\right)(y)-\frac{1}{2}(u(x)-u(y))(\varphi(x)+\varphi(y))\right|\\\
&\precsim\int_{\mathbb{R}^{n}}\left||x-z|^{t-n}-|y-z|^{t-n}\right|\,\left|(-\Delta)^{\frac{t}{2}}u(z)\right|\left|\varphi(x)+\varphi(y)-2\varphi(z)\right|\,\mathrm{d}z.\end{split}$

Thus,

$\begin{split}&\left|\int_{\mathbb{R}^{n}}\varphi\,(-\Delta)^{\frac{t}{2}}u\wedge
T_{t}u\right|\\\
&\precsim\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-1}\left||x-z|^{t-n}-|y-z|^{t-n}\right|\,\left|(-\Delta)^{\frac{t}{2}}u(z)\right|\left|\varphi(x)+\varphi(y)-2\varphi(z)\right|}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z.\end{split}$

This is the same situation as in (4.12): the role of $\varphi\in
L^{\frac{n}{t}}$ in (4.12) is taken here by $(-\Delta)^{\frac{t}{2}}u\in
L^{\frac{n}{t}}$, and the role of $|u(x)+u(y)-2u(z)|$ is taken by
$|\varphi(x)+\varphi(y)-2\varphi(z)|$ and observe that
$(-\Delta)^{\frac{t}{2}}\varphi\in L^{\frac{n}{t}}$. As was discussed there we
can pick $r\leq t$, and thus we have

$\begin{split}\left|\int_{\mathbb{R}^{n}}\varphi\,(-\Delta)^{\frac{t}{2}}u\wedge
T_{t}u\right|&\precsim\|(-\Delta)^{\frac{r}{2}}u\|_{L^{\frac{n}{r}}(\mathbb{R}^{n})}^{\frac{n}{s}-1}\|(-\Delta)^{\frac{r}{2}}\varphi\|_{L^{\frac{n}{r}}(\mathbb{R}^{n})}^{\frac{n}{s}-1}\|(-\Delta)^{\frac{t}{2}}u\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}\\\
&\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}}\,\|(-\Delta)^{\frac{t}{2}}\varphi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}.\end{split}$

We can conclude. (Again it is worth noting that the proof above does not need
to distinguish between the case $\frac{n}{s}\geq 2$ and $\frac{n}{s}\leq 2$).
∎

We are now ready to proceed with the proof of the main result of this section.

###### Proof of Proposition 4.3.

Combining (4.6) with (4.15) and (4.7) we get for a $t<\theta s$, where $t$ and
$\theta$ are as in Lemma 4.4 and Lemma 4.5

(4.19)
$\|T_{t}u\|_{L^{\frac{n}{n-t}}(\mathbb{R}^{n})}\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}}.$

By duality (4.19) implies for any $\varphi\in C^{\infty}\cap
L^{\frac{n}{t}}(\mathbb{R}^{n},\mathbb{R}^{\ell+1})$,

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\,\left(I^{t}\varphi(x)-I^{t}\varphi(y)\right)}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}}\|\varphi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}.$

Thus for any $\psi\in C_{c}^{\infty}(\mathbb{R}^{n})$

(4.20)
$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\,\left(\psi(x)-\psi(y)\right)}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}}\|(-\Delta)^{\frac{t}{2}}\psi\|_{L^{\frac{n}{t}}(\mathbb{R}^{n})}.$

Using Sobolev embedding this implies for $t<t_{2}<s$

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\,\left(\psi(x)-\psi(y)\right)}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\precsim[u]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}^{\frac{n}{s}}[\psi]_{W^{t_{2},\frac{n}{t_{2}}}(\mathbb{R}^{n})}.$

The constant depends on $|t-t_{2}|$, so by taking $\tilde{\theta}$ slightly
larger than $\theta$ and $t_{2}=\tilde{\theta}s$ we have
$\tilde{\theta}s-t>(\tilde{\theta}-\theta)s_{0}$, so the constant can be
chosen uniform and, by density, [20, 2.6.2. Proposition 1.], we obtain (4.2)
for $\tilde{\theta}$. ∎

###  4.2. A fractional version of Iwaniec’s stability result

A fractional version of Iwaniec’s stability result was proposed in [23].
However, the result of [23] does not apply in our situation since it only
considers stability in the differential direction, without adjusting the
integrability. We need the latter, since we need to stay in the scaling
invariant case. Hence, we employ a different version of the Iwaniec’s
stability result [11, Theorem 13.2.1] to obtain the following regularity
result.

######  Proposition 4.6.

For any $0<s_{0}<s_{1}<1$ there exists an
$\varepsilon_{0}=\varepsilon_{0}(s_{0},s_{1},n)>0$ such that the following
holds:

For any $s\in(s_{0},s_{1})$ and any $\Lambda>0$ there exists a constant
$C(\Lambda)$ such that if $u\in L^{\infty}\cap
W^{s,\frac{n}{s}}(\mathbb{R}^{n},\mathbb{R}^{N})$ satisfies for a
$t\in(s-\varepsilon_{0},s]$ and for any
$\psi\in\dot{W}^{t,\frac{n}{t}}(\mathbb{R}^{n},\mathbb{R}^{N})$

(4.21)
$\left|\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\,\left(\psi(x)-\psi(y)\right)}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\right|\leq\Lambda[\psi]_{W^{t,\frac{n}{t}}(\mathbb{R}^{n})},$

then for $r\coloneqq s\frac{n-t}{n-s}\geq s$

$[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}\leq
C(\Lambda,\varepsilon_{0},s_{0},s_{1}).$

We first observe that for $p\coloneqq\frac{n}{s}$ and $\varepsilon\coloneqq
p-\frac{n}{r}=\frac{n}{s}-\frac{n}{r}>0$ we have $\frac{t}{1-\varepsilon}=r$
and

(4.22)
$[u]_{W^{r,\frac{n}{r}}}^{\frac{n}{r}}=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\,|u(x)-u(y)|^{-\varepsilon}(u(x)-u(y))}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y.$

We perform a de facto Hodge decomposition:

(4.23) $|u(x)-u(y)|^{-\varepsilon}(u(x)-u(y))=A(x)-A(y)+G(x,y),$

where, in the terminology of [14], we choose $A$ such that

(4.24) $(-\Delta)^{t}A(x)\coloneqq{\rm
div}_{t}\left(\frac{|u(x)-u(y)|^{-\varepsilon}(u(x)-u(y))}{|x-y|^{t}}\right),$

that is for any $\varphi\in C_{c}^{\infty}(\mathbb{R}^{n})$

$(-\Delta)^{t}A[\varphi]=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{-\varepsilon}(u(x)-u(y))\,(\varphi(x)-\varphi(y))}{|x-y|^{n+2t}}\,\mathrm{d}x\,\mathrm{d}y.$

From the linear theory of partial differential equation we have:

######  Lemma 4.7.

For any $0<t_{0}<t_{1}<1$ there exists an $\varepsilon_{0}>0$ such that
whenever $t\in(t_{0},t_{1})$ and
$A\in\dot{W}^{t,\frac{n}{t}}(\mathbb{R}^{n},\mathbb{R}^{N})$ as in (4.24)
exists and satisfies the estimate

$[A]_{W^{t,\frac{n}{t}}(\mathbb{R}^{n})}\precsim[u]_{W^{\frac{t}{1-\varepsilon},(1-\varepsilon)\frac{n}{t}}(\mathbb{R}^{n})}^{1-\varepsilon}=[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}^{1-\varepsilon}$

with a constant independent of $\varepsilon$ as long as
$\varepsilon\in[0,\varepsilon_{0}]$. $A$ is unique up to constants.

###### Proof.

Proceeding exactly as in [5, Lemma A.1] we have the a priori estimate

$\begin{split}(-\Delta)^{t}A[\varphi]&=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{-\varepsilon}(u(x)-u(y))\,(\varphi(x)-\varphi(y))}{|x-y|^{n+2t}}\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq[u]_{W^{\frac{t}{1-\varepsilon},(1-\varepsilon)\frac{n}{t}}(\mathbb{R}^{n})}^{1-\varepsilon}\,[\varphi]_{W^{t,\frac{n}{n-t}}(\mathbb{R}^{n})}.\end{split}$

Using the identification via Triebel–Lizorkin spaces, [20, Section 2], we have

$\begin{split}[A]_{W^{t,\frac{n}{t}}(\mathbb{R}^{n})}\approx[A]_{\dot{F}^{t,\frac{n}{t}}_{\frac{n}{t}}}\approx[(-\Delta)^{t}A]_{\dot{F}^{-t,\frac{n}{t}}_{\frac{n}{t}}}\approx[(-\Delta)^{t}A]_{\left(\dot{F}^{t,\frac{n}{n-t}}_{\frac{n}{n-t}}\right)^{\ast}}\precsim[u]_{W^{\frac{t}{1-\varepsilon},(1-\varepsilon)\frac{n}{t}}(\mathbb{R}^{n})}^{1-\varepsilon}.\end{split}$

The constants depend only on $t_{0}$ and $t_{1}$ since $t\in(t_{0},t_{1})$. In
particular, $A$ exists since
$(-\Delta)^{t}A\in\left(\dot{F}^{t,\frac{n}{n-t}}_{\frac{n}{n-t}}\right)^{\ast}$.
$A$ is unique up to constants, since
$[A]_{W^{t,\frac{n}{t}}(\mathbb{R}^{n})}=0$ implies that $A$ is a constant. ∎

From Lemma 4.7 and (4.23) we have in particular

$\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|G(x,y)|^{\frac{n}{t}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\right)^{\frac{t}{n}}\precsim[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}^{1-\varepsilon}.$

The latter estimate can, however, be improved.

######  Proposition 4.8.

For any $0<t_{0}<t_{1}<1$ there exists an $\varepsilon_{0}>0$ and a constant
$C=C(t_{0},t_{1},\varepsilon_{0})$ such that for any
$\varepsilon\in(0,\varepsilon_{0})$, and $t\in(t_{0},t_{1})$ and $G$ as in
(4.23),

$\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|G(x,y)|^{\frac{n}{t}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\right)^{\frac{t}{n}}\leq
C\,\left|\varepsilon\right|[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}^{1-\varepsilon}.$

###### Proof.

We follow the approach in [11, Theorem 13.2.1]. Fix $\varepsilon_{0}$ to be
specified later. By density, cf. [20, 2.6.2. Proposition 1.], we may assume
that $u\in C_{c}^{\infty}(\mathbb{R}^{n})$ (observe that we can change $u$ by
a constant without changing the definitions of $G$ and $A$).

For $z\in{\mathbb{C}}$, $|z|\leq\varepsilon_{0}$ we set

(4.25) $G_{z}(x,y)=\left(A_{z}(x)-A_{z}(y)\right)-|u(x)-u(y)|^{z}(u(x)-u(y)),$

where $A_{z}$ is defined as the solution to

(4.26)
$(-\Delta)^{t}A_{z}[\varphi]=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{z}(u(x)-u(y))\,(\varphi(x)-\varphi(y))}{|x-y|^{n+2t}}\,\mathrm{d}x\,\mathrm{d}y,\quad\forall\varphi\in
C_{c}^{\infty}(\mathbb{R}^{n}).$

This is well-defined since $u\in C_{c}^{\infty}(\mathbb{R}^{n})$ and the
right-hand side is a linear functional on a Triebel–Lizorkin space. Assume
that for a $\lambda\in(0,\frac{1}{2})$ we have a $q\in(1,\infty)$ such that

(4.27) $2t-\frac{n}{q}(1+\Re(z))\in(\lambda,1-\lambda).$

Let us remark already that later we will apply this to
$q=\frac{n}{t}(1+\Re(z))$ for $|z|\leq\varepsilon_{0}$, so that $\lambda$ and
$\varepsilon_{0}$ can be chosen depending only on $t_{0}$ and $t_{1}$.

We get by (4.25)

(4.28)
$\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|G_{z}(x,y)|^{\frac{q}{1+\Re(z)}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\right)^{\frac{1+\Re(z)}{q}}\precsim[u]_{W^{\frac{n}{q},q}(\mathbb{R}^{n})}^{1+\Re(z)}+[A_{z}]_{W^{\frac{n}{q}(1+\Re(z)),\frac{q}{1+\Re(z)}}(\mathbb{R}^{n})}.$

Arguing with the identification of Triebel–Lizorkin spaces as in Lemma 4.7 we
have

(4.29)
$[A_{z}]_{W^{\frac{n}{q}(1+\Re(z)),\frac{q}{1+\Re(z)}}(\mathbb{R}^{n})}\approx[A_{z}]_{\dot{F}^{\frac{n}{q}(1+\Re(z))}_{\frac{q}{1+\Re(z)},\frac{q}{1+\Re(z)}}(\mathbb{R}^{n})}\approx[(-\Delta)^{t}A]_{\dot{F}^{\frac{n}{q}(1+\Re(z))-2t}_{\frac{q}{1+\Re(z)},\frac{q}{1+\Re(z)}}(\mathbb{R}^{n})}.$

Moreover,

(4.30)
$\dot{F}^{\frac{n}{q}(1+\Re(z))-2t}_{\frac{q}{1+\Re(z)},\frac{q}{1+\Re(z)}}(\mathbb{R}^{n})=\left(\dot{F}^{2t-\frac{n}{q}(1+\Re(z))}_{(\frac{q}{1+\Re(z)})^{\prime},(\frac{q}{1+\Re(z)})^{\prime}}(\mathbb{R}^{n})\right)^{\ast}=\left(W^{2t-\frac{n}{q}(1+\Re(z)),(\frac{q}{1+\Re(z)})^{\prime}}(\mathbb{R}^{n})\right)^{\ast}\eqqcolon
X^{\ast}.$

Hence, by the equivalence of the norms (the constant depends on $\lambda$)

(4.31)
$\begin{split}&[A_{z}]_{W^{\frac{n}{q}(1+\Re(z)),\frac{q}{1+\Re(z)}}(\mathbb{R}^{n})}\\\
&\approx\sup_{[\varphi]_{X}\leq
1}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{z}(u(x)-u(y))\,(\varphi(x)-\varphi(y))}{|x-y|^{n+2t}}\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq\sup_{[\varphi]_{X}\leq
1}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{1+\Re(z)}}{|x-y|^{\frac{n}{q}(1+\Re(z))}}\,\frac{\left|\varphi(x)-\varphi(y)\right|}{|x-y|^{2t-\frac{n}{q}(1+\Re(z))}}\frac{\,\mathrm{d}x\,\mathrm{d}y}{|x-y|^{n}}\\\
&\precsim[u]_{W^{\frac{n}{q},q}(\mathbb{R}^{n})}^{1+\Re(z)}.\end{split}$

We stress that for $|z|\leq\varepsilon_{0}$ all of the constants above are
independent of $z$ but depend on $\lambda$. Hence, combining the estimates
(4.28) with (4.31) we obtain

(4.32)
$\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|G_{z}(x,y)|^{\frac{q}{1+\Re(z)}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\right)^{\frac{1+\Re(z)}{q}}\precsim[u]_{W^{\frac{n}{q},q}(\mathbb{R}^{n})}^{1+\Re(z)}.$

Fix now $\psi\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{N}$ such
that

(4.33)
$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|\psi(x,y)|^{q}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\leq
1$

and set

$F_{\psi,u}(z)\coloneqq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{\langle
G_{z}(x,y),|\psi(x,y)|^{q-2-\bar{z}}\psi(x,y)\rangle_{{\mathbb{C}}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y.$

Then

$|F_{\psi,u}(z)|\leq\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|G_{z}(x,y)|^{\frac{q}{1+\Re(z)}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\right)^{\frac{1+\Re(z)}{q}}\precsim[u]_{W^{\frac{n}{q},q}(\mathbb{R}^{n})}^{1+\Re(z)},$

where the constants are independent of $q$ as long as the assumption (4.27) is
satisfied.

We observe that $F_{\psi,u}(0)=0$. Indeed, by the definition (4.26) we
have$(-\Delta)^{t}A_{0}=(-\Delta)^{t}u$, hence $A=u+c$ for a constant $c$.
This, by definition (4.25), implies $G_{0}(x,y)\equiv 0$.

Moreover, just as in [11, Theorem 13.2.1], $z\mapsto F_{\psi,u}(z)$ is
holomorphic: since $u\in C_{c}^{\infty}(\mathbb{R}^{n})$ the map
$\partial_{\bar{z}}F_{\psi,u}$ is well-defined, and we can compute explicitly
that the Cauchy–Riemann equations are satisfied.

From Schwarz lemma for holomorphic functions we have for all
$|z|\leq\varepsilon_{0}$

(4.34)
$\left|F_{\psi,u}(z)\right|\precsim|z|[u]_{W^{\frac{n}{p},p}(\mathbb{R}^{n})}^{1+\Re(z)}$

with constant independent of $\psi$ as long as (4.33) is satisfied, the
constant depends only on $\lambda$ and thus on $t_{0}$, $t_{1}$, and
$\varepsilon_{0}$.

Now take $q=\frac{n}{t}(1-\varepsilon)$, $z=-\varepsilon$,
$\varepsilon=\frac{n}{s}-\frac{n}{r}$, $r=s\frac{n-t}{n-s}$, so that
$\frac{t}{1-\varepsilon}=r$. By (4.34) we get

$\begin{split}\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|G_{-\varepsilon}(x,y)|^{\frac{n}{t}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\right)^{\frac{t}{n}}=F_{\psi,u}(-\varepsilon)\leq\sup_{\psi\text{
as in
\eqref{eq:psitestiw}}}F_{\psi,u}(-\varepsilon)\precsim|\varepsilon|[u]_{W^{\frac{n}{q},q}(\mathbb{R}^{n})}^{1+\Re(z)}.\end{split}$

∎

We are ready to proceed with the proof of the main result of this section.

###### Proof of Proposition 4.6.

We have by (4.22) and (4.23)

(4.35)
$\begin{split}[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}^{\frac{n}{r}}&=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\,\left(A(x)-A(y)\right)}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\\\
&\quad+\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\,\left(G(x,y)\right)}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y.\end{split}$

By (4.21) and Lemma 4.7 we have

(4.36)
$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\,\left(A(x)-A(y)\right)}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\leq\Lambda[A]_{W^{t,\frac{n}{t}}(\mathbb{R}^{n})}\precsim\Lambda\,[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}^{1-\varepsilon}.$

As for the second term on the right-hand side of (4.35) we note that
$\frac{r}{n}(\frac{n}{s}-1)+\frac{t}{n}=1$. Hence, by Hölder’s inequality,
Proposition 4.8, and the observation $\frac{n}{s}-\varepsilon=\frac{n}{r}$

(4.37)
$\begin{split}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}&\frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\,\left(G(x,y)\right)}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\\\
&\precsim[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}^{\frac{n}{s}-1}\,\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|G(x,y)|^{\frac{n}{t}}}{|x-y|^{2n}}\,\mathrm{d}x\,\mathrm{d}y\right)^{\frac{t}{n}}\\\
&\precsim\left|\varepsilon\right|[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}^{\frac{n}{r}}.\end{split}$

Combining (4.35) with (4.36) and (4.37) we obtain

(4.38)
$\begin{split}[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}^{\frac{n}{r}}&\precsim\Lambda\,[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}^{1-\varepsilon}+\left|\varepsilon\right|[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}^{\frac{n}{r}}.\end{split}$

Now $s-t\leq\varepsilon_{0}$ implies
$\varepsilon=\frac{n}{s}-\frac{n}{r}=\frac{n}{s}\left(\frac{s-t}{n-t}\right)\leq
C(s_{0},s_{1})\varepsilon_{0}$, so for $\varepsilon_{0}$ suitably small we can
absorb and conclude

$[u]_{W^{r,\frac{n}{r}}(\mathbb{R}^{n})}^{\frac{n}{r}}\precsim
C(\varepsilon_{0},\Lambda,s_{1},s_{0}).$

∎

##  5\. Continuous dependence

The main observation is that the regularity theory of Theorem 4.1 combined
with the stability Proposition 2.5 implies:

######  Corollary 5.1.

Fix $n,\ell\in{\mathbb{N}}$ with with either $(\ell,n)=(1,1)$ or $\ell\geq 2$.
Fix $0<s_{0}<s_{1}<1$ and $\Lambda>0$, then for any $\varepsilon>0$ there
exists $\delta>0$ such that the following holds.

If $s\in(s_{0},s_{1})$ and $\\#_{s}\alpha$ is attained with
$\\#_{s}\alpha\leq\Lambda$ for a homotopy class
$\alpha\in\pi_{n}({\mathbb{S}}^{\ell})$.

Then, for any $\tilde{s}\in(s-\delta,s+\delta)$,

$\\#_{s}\alpha\geq\\#_{\tilde{s}}\alpha-\varepsilon.$

###### Proof.

Let $u\in\alpha$ be the minimizer of $\mathcal{E}_{s,\frac{n}{s}}$. By Theorem
4.1 there is a $\Theta>0$ such that

$[u]_{W^{\Theta s,\frac{n}{\Theta s}}}\leq C(s_{0},s_{1},\Lambda).$

Now we pick $\delta$ from Proposition 2.5 and conclude for any
$\tilde{s}\in(s-\delta,s+\delta)$

$\\#_{s}\alpha=[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n})}^{\frac{n}{s}}\geq[u]_{W^{\tilde{s},\frac{n}{\tilde{s}}}({\mathbb{S}}^{n})}^{\frac{n}{\tilde{s}}}-\varepsilon\geq\\#_{\tilde{s}}\alpha-\varepsilon.$

∎

###  5.1. Continuous dependence of minimal energy: Proof of Theorem 1.3

Theorem 1.3 is a pretty straightforward consequence of Corollary 5.1.

###### Proof of Theorem 1.3.

Fix $\varepsilon>0$ and $\alpha\in\pi_{n}({\mathbb{S}}^{\ell})$. Take any
smooth map $\bar{u}\in C^{\infty}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$ that
represents $\alpha$ then we have, for any $0<s_{0}<s_{1}<1$ the estimate

$\sup_{t\in(s_{0},s_{1})}\\#_{t}\alpha\leq C(\bar{u}).$

That is, we have a uniform energy bound that is needed later in the
application of Corollary 5.1.

In view of Theorem 3.2 there is an $N\in{\mathbb{N}}$ for which

$\\#_{s}\alpha=\sum_{i=1}^{N}\\#_{s}\alpha_{i}\quad\text{for some
$\sum_{i=1}^{N}\alpha_{i}=\alpha$ such that $\\#_{s}\alpha_{i}$ is attained.}$

By Corollary 5.1 there is a $\delta>0$ such that for all
$t\in(s-\delta,s+\delta)$ we have

$\\#_{s}\alpha=\sum_{i=1}^{N}\\#_{s}\alpha_{i}\geq\sum_{i=1}^{N}\left(\\#_{t}\alpha_{i}-\frac{\varepsilon}{N}\right)\geq\\#_{t}\alpha-\varepsilon,$

where the last inequality is a consequence of Proposition 2.2 The converse
inequality follows by reversing the role of $t$ and $s$. ∎

###  5.2. Proof of Corollaries

###### Proof of Corollary 1.4.

Fix $0<s_{0}<1/2<s_{1}<1$. Assume that $\\#_{s}1$ is not attained for some
$s\in(s_{0},s_{1})$. Then by Theorem 3.2,

$\\#_{s}1\geq\\#_{s}d_{1,s}+\\#_{s}d_{2,s}$

for degrees $d_{1,s},d_{2,s}\in\mathcal{Q}\setminus\\{-1,0,1\\}$ such that
$\\#_{s}d_{i,s}$ is attained for $i=1,2$. $\mathcal{Q}$ is a finite set of
integers, in view of Theorem 2.4.

By Theorem 1.3, the family of maps

$(s_{0},s_{1})\ni s\mapsto\\#_{s}d,\quad d\in\mathcal{Q}$

are equicontinuous. So, for every $\varepsilon>0$ there is a $\delta$ such
that for every $s\in(\frac{1}{2}-\delta,\frac{1}{2}+\delta)$ we have

$\\#_{\frac{1}{2}}1\geq\\#_{s}1-\varepsilon\geq\\#_{s}d_{1,s}+\\#_{s}d_{2,s}-\varepsilon\geq\\#_{\frac{1}{2}}d_{1,s}+\\#_{\frac{1}{2}}d_{2,s}-3\varepsilon.$

Combining this with Theorem 2.3 we obtain

$4\pi^{2}=\\#_{\frac{1}{2}}1\geq\\#_{\frac{1}{2}}d_{1,s}+\\#_{\frac{1}{2}}d_{2,s}-3\varepsilon\geq
8\pi^{2}-3\varepsilon.$

For $\varepsilon<\frac{4}{3}\pi^{2}$ this is a contradiction. ∎

In a very similar way as Corollary 1.4 we obtain

###### Proof of Corollary 1.5.

The fact that $\\#_{t}\alpha$ is attained is an immediate consequence of [15,
Lemma 7.7]. Arguing exactly as in the proof of Corollary 1.4 — assuming that
$\\#_{s}\alpha$ is not attained, we obtain for
$\beta_{1,s},\beta_{2,s}\in\pi_{n}({\mathbb{S}}^{\ell})\setminus\\{0\\}$

$\\#_{t}\alpha\geq\\#_{s}\beta_{1,s}+\\#_{s}\beta_{2,s}-\varepsilon\geq\\#_{t}\beta_{1,s}+\\#_{t}\beta_{2,s}-3\varepsilon.$

However, by assumption $\\#_{t}\alpha\leq\\#_{t}\beta_{i,s}$ for $i=1,2$.
Hence we would get $\\#_{t}\alpha\geq 2\,\\#_{t}\alpha-3\varepsilon$. This
gives a contradiction with Proposition 2.1 for sufficiently small
$\varepsilon>0$. ∎

###### Proof of Corollary 1.6.

In view of Theorem 2.4, for each $\Lambda>0$ there exists $D\in\mathbb{N}$
such that

$\begin{split}\bar{C}_{n,s;\Lambda}&\equiv\sup_{u\in
W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n}),0<[u]_{W^{s,\frac{n}{s}}}\leq\Lambda}\frac{\deg
u}{[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n})}^{\frac{n}{s}}}\\\
&=\max_{d\in\mathbb{Z}\setminus\\{0\\},|d|\leq D}\sup_{u\in
W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n}),[u]_{W^{s,\frac{n}{s}}}\leq\Lambda,\deg
u=d}\frac{d}{[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n})}^{\frac{n}{s}}}\\\
&=\max\left\\{\frac{d}{\\#_{s}d}\colon\quad|d|\leq D,\,d\neq
0\quad\text{$\exists u\in
W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{n})$ with $\deg u=d$ and
$[u]_{W^{s,\frac{n}{s}}}\leq\Lambda$}\right\\}.\end{split}$

For each $d\in\mathbb{Z}\setminus\\{0\\}$ the map $s\mapsto\frac{d}{\\#_{s}d}$
is continuous, by Theorem 1.3.

Since

$\bar{C}_{n,s}=\sup_{\Lambda>0}\bar{C}_{n,s;\Lambda}=\lim_{\Lambda\to\infty}\bar{C}_{n,s;\Lambda}$

we have that $s\mapsto\bar{C}_{n,s}$ is lower semicontinuous. ∎

###  5.3. Stability of generators of minimizing $W^{s,\frac{n}{s}}$-harmonic
maps: Proof of Theorem 1.2

Theorem 1.2 is a consequence of the more precise

######  Theorem 5.2.

Fix $n,\ell\geq 1$ with either $(\ell,n)=(1,1)$ or $\ell\geq 2$. Let
$\Lambda>0$ and $0<t_{0}<t_{1}<1$. There exists a
$\delta=\delta(\ell,n,t_{0},t_{1},\Lambda)\in(0,1)$ such that the following
holds for $t\in(t_{0},t_{1})$.

Set

$X_{s}\coloneqq\left\\{\alpha\in\pi_{n}({\mathbb{S}}^{\ell})\colon\quad\text{there
exists a $W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})$-minimizer
$u$ in $\alpha$ and
$[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n},{\mathbb{S}}^{\ell})}\leq\Lambda$}\right\\}$

and

$Y\coloneqq\bigcap_{s\in(t-\delta,t+\delta)}X_{s}.$

Then $X_{s}$ is generated by $Y$, i.e., $X_{s}\subset{\rm span}Y$, for each
$s\in(t-\delta,t+\delta)$.

###### Proof of Theorem 5.2.

Let $\varepsilon>0$ be fixed and chosen below, for this $\varepsilon$ we take
$\delta>0$ from Corollary 5.1.

By Theorem 2.4 there exists a finite number $M\in{\mathbb{N}}$ such that if
for some $s\in(t-\delta,t+\delta)$
$[u]_{W^{s,\frac{n}{s}}({\mathbb{S}}^{n})}<\Lambda$ then
$u\in\mathcal{Q}\coloneqq\\{\alpha_{1},\ldots,\alpha_{M}\\}\subset\pi_{n}({\mathbb{S}}^{\ell})$.
In particular, $X_{s}\subset\mathcal{Q}$ for all $s\in(t-\delta,t+\delta)$.
Observe that for each $s\in(t-\delta,t+\delta)$ we have from Theorem 1.1

$X_{s}\text{ generates }\mathcal{Q}.$

Let us enumerate the elements of $Y=\bigcap_{s\in(t-\delta,t+\delta)}X_{s}$

$Y=\\{\gamma_{1},\ldots,\gamma_{K}\\}\quad\text{ for $K\leq M$}.$

Of course, for now $K=0$ is a possibility.

We define $Z\subset\mathcal{Q}\setminus Y$ as the collection of homotopy
groups $\beta$ where $\\#_{s}\beta$ is not attained for at least one $s$, more
precisely:

$Z\coloneqq\left\\{\alpha\in\pi_{n}({\mathbb{S}}^{\ell})\colon\
\\#_{s}\alpha\leq\Lambda\quad\text{ for an
}s\in(t-\delta,t+\delta)\right\\}\setminus
Y=\\{\beta_{1},\ldots,\beta_{L}\\}.$

We note that $L+K\leq M$. Now assume that for some $s\in(t-\delta,t+\delta)$
there is $\alpha$ such that $\alpha\in X_{s}\setminus Y$. Then there must be
some $\tilde{s}\in(t-\delta,t+\delta)$ with $\alpha\not\in X_{\tilde{s}}$. By
Theorem 3.2 we find $\beta_{k}\in Z$ and $\gamma_{k}\in Y\setminus\\{0\\}$
such that $\\#_{\tilde{s}}\beta_{k}$ and $\\#_{\tilde{s}}\gamma_{k}$ are
attained, $2\leq A+B\leq M$, and

(5.1)
$\alpha=\sum_{k=1}^{A}\beta_{k}+\sum_{k=1}^{B}\gamma_{k},\quad\\#_{\tilde{s}}\alpha=\sum_{k=1}^{A}\\#_{\tilde{s}}\beta_{k}+\sum_{k=1}^{B}\\#_{\tilde{s}}\gamma_{k}.$

Using (5.1) and applying twice Corollary 5.1 we obtain

(5.2)
$\\#_{s}\alpha\geq\\#_{\tilde{s}}\alpha-\varepsilon=\sum_{k=1}^{A}\\#_{\tilde{s}}\beta_{k}+\sum_{k=1}^{B}\\#_{\tilde{s}}\gamma_{k}-\varepsilon\geq\sup_{r\in(t-\delta,t+\delta)}\left(\sum_{k=1}^{A}\\#_{r}\beta_{k}+\sum_{k=1}^{B}\\#_{r}\gamma_{k}\right)-(M+1)\varepsilon.$

In particular we have

(5.3)
$\\#_{s}\alpha\geq\\#_{\tilde{s}}\alpha-\varepsilon\geq\sum_{k=1}^{A}\\#_{s}\beta_{k}+\sum_{k=1}^{B}\\#_{s}\gamma_{k}-(M+1)\varepsilon.$

Observe that this is a contradiction if any of $\beta_{k}$’s on the right-hand
side is equal to $\alpha$ (assuming $\varepsilon$ is small enough).

If $A=0$ then we are done because then $\alpha=\sum\gamma_{k}$ and each
$\gamma_{k}\in Y$.

If $A>0$, then for each $\beta_{k}$ in (5.2) we have two possibilities: either
$\\#_{s}\beta_{k}$ is attained (but still $\beta_{k}\not\in Y$) or
$\\#_{s}\beta_{k}$ is not attained. Rearranging the terms we can assume that

  * •

for the terms $\beta_{1},\ldots,\beta_{{A_{1}}}$, $\\#_{s}\beta_{k}$ is
attained, i.e., $\beta_{k}\in X_{s}\setminus Y$,

  * •

for the terms $\beta_{{A_{1}+1}},\ldots,\beta_{{A}}$ the infimum
$\\#_{s}\beta_{k}$ is not attained, i.e., $\beta_{k}\not\in X_{s}$.

It is possible that $A_{1}=0$ or $A_{1}=A$.

In case $\beta_{k}\in X_{s}\setminus Y$ we find an
$s_{k}\in(t-\delta,t+\delta)$ such that $\beta_{k}\notin X_{s_{k}}$ and we
apply (5.3).

In case $\beta_{k}\notin X_{s}$ we apply Theorem 3.2.

We obtain first using Corollary 5.1 and then using (5.3) and Theorem 3.2

(5.4)
$\begin{split}\sum_{k=1}^{A}\\#_{s}\beta_{k}&\geq\sum_{k=1}^{A_{1}}\\#_{s_{k}}\beta_{k}+\sum_{k=A_{1}+1}^{A}\\#_{s}\beta_{k}-M\varepsilon\\\
&\geq\sum_{k=1}^{A_{1}}\left(\sum_{j=1}^{A_{1,k}}\\#_{s}\beta_{k_{j}}+\sum_{j=1}^{B_{1,k}}\\#_{s}\gamma_{k_{j}}\right)+\sum_{k=A_{1}+1}^{A}\left(\sum_{j=1}^{A_{\tilde{1},k}}\\#_{s}\beta_{\tilde{k}_{j}}+\sum_{j=1}^{B_{\tilde{1},k}}\\#_{s}\gamma_{\tilde{k}_{j}}\right)-(2M^{2}+M)\varepsilon\\\
&=\sum_{i=1}^{\tilde{A}_{1}}\\#_{s}\beta_{\sigma(i)}+\sum_{i=1}^{\tilde{B}_{2}}\\#_{s}\gamma_{\zeta(i)}-(2M^{2}+M)\varepsilon,\end{split}$

where $\tilde{A}_{1},\tilde{B}_{1}\leq 2M^{2}$,
$\sigma(i)\in\\{1,\ldots,L\\}$, $\zeta(i)\in\\{1,\ldots,K\\}$ for each $i$,
and at each step $2\leq A_{1,k}+B_{1,k}\leq M$. Now again, we can repeat the
procedure. However, since there is at most $M$ elements in $Z$ we obtain that
after a finite number of iterations of this procedure the same element
$\beta_{i}$ would appear on the right-hand side and on the left-hand side of
(5.4). This would give a contradiction for a sufficiently small $\varepsilon$
(depending on $M$ and $\lambda$ from Proposition 2.1). Hence, we must obtain
at some moment a decomposition of $\alpha$ into terms belonging only to $Y$. ∎

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