Nonlinear commutators for the fractional p-Laplacian and applications

Abstract

We prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p-Laplacian. This implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weakly fractional p-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded $\frac{n}{s}$-harmonic maps converge strongly outside at most finitely many points.

1 Introduction

An important tool for obtaining higher differentiability of solutions to elliptic partial differential equations is the method of “differentiating the equation”: As an example, take $u \in W^{1,2}$ a distributional solution to

$$\begin{aligned} \Delta u = g \in L^2_{loc}(\Omega ). \end{aligned}$$

(1.1)

It is easy to obtain such a solution $u \in W^{1,2}$, e.g., by the direct method of the calculus of variations. Actually, any such solution belongs to $W^{2,2}_{loc}$. To see this, we differentiate the equation:

$$\begin{aligned} \Delta \partial _i u =\partial _i g \in \left( W^{1,2}_{loc}(\Omega )\right) ^*. \end{aligned}$$

(1.2)

Now $\partial _i u$ solves an elliptic equation with right-hand side in $(W^{1,2})^*$. Consequently, $\partial _i u$ belongs to $W^{1,2}_{loc}$ and it is shown that $u \in W^{2,2}_{loc}$. Let us have a closer look at this “differentiating the equation”-argument. The distributional Laplacian $(-\Delta ) u$ is defined on testfunctions $\varphi \in C_c^\infty (\Omega )$,

$$\begin{aligned} (-\Delta ) u[\varphi ]:= \int \nabla u\ \nabla \varphi \overset{(1.1)}{=} -\int g \varphi \quad \forall \varphi \in C_c^\infty . \end{aligned}$$

Since $\Delta$ is a linear operator with constant coefficients,

$$\begin{aligned} (-\Delta ) (\partial _i u)[\varphi ] - (-\Delta ) u[-\partial _i\varphi ] = 0. \end{aligned}$$

(1.3)

“Differentiating the equation” (1.2) in distributional sense becomes

$$\begin{aligned} (-\Delta ) (\partial _i u)[\varphi ] = -\int \nabla u \nabla (\partial _i \varphi ) = -\int g\ \partial _i \varphi . \end{aligned}$$

(1.4)

Higher differentiability $u \in W^{2,2}_{loc}$ then follows by duality: Take the supremum over $\varphi$ with $\Vert \nabla \varphi \Vert _{L^2} \le 1$ on both sides of (1.4), and obtain an estimate for $\partial _i \nabla u$ in terms of $g \in L^2$.

The above reasoning relies crucially on (1.3). Of course, we can replace $(-\Delta )$ and $\partial _i$ with more general differential operators of arbitrary order: The s-Laplacian is defined as

$$\begin{aligned} (-\Delta ) ^{s} f = \mathcal {F}^{-1}(c\ |\xi |^{2s} \mathcal {F} f), \end{aligned}$$

where $\mathcal {F}$ and $\mathcal {F}^{-1}$ denote the Fourier transform and its inverse, respectively. As a distribution

$$\begin{aligned} (-\Delta ) ^{s} f[\varphi ] = \int _{\mathbb {R}^n} (-\Delta ) ^{s} f \varphi . \end{aligned}$$

Similarly to (1.3), just via integration by parts,

$$\begin{aligned} (-\Delta ) ^{s+\varepsilon } u[\varphi ] - c(-\Delta ) ^{s} u[(-\Delta ) ^{\varepsilon }\varphi ] = 0, \end{aligned}$$

(1.5)

where c is a constant coming from the choice of the Fourier transform coefficients and the definition of the s-Laplacian. With (1.5) in mind one can prove a finer scale of higher differentiability results. For example,

$$\begin{aligned} u \in W^{1,2}(\Omega )\quad \text{ and }\quad \Delta u \in (W^{1-\varepsilon ,2}(\Omega ))^*\quad \Rightarrow \quad u \in W^{1+\varepsilon ,2}_{loc}(\Omega ). \end{aligned}$$

However, a statement of the form (1.5) is false for some nonlinear operators, in particular it fails for the p-Laplace

$$\begin{aligned} \Delta _{p}u = {\text {div}} (|\nabla u|^{p-2} \nabla u). \end{aligned}$$

And indeed, even p-harmonic functions, i.e. solutions to $\Delta _{p}u = 0$, may not be smooth.

In this paper, we investigate to what extent the “differentiating the equation”-argument can be saved in the case of a nonlocal, nonlinear differential operator which is related to the p-Laplacian: The fractional p-Laplacian.

The fractional p-Laplacian of order $s \in (0,1)$ on a domain $\Omega \subset \mathbb {R}^n$, $(-\Delta )^{s}_{p,\Omega } u$ is a distribution acting on testfunctions $\varphi \in C_c^\infty (\Omega )$ given by

$$\begin{aligned} (-\Delta )^{s}_{p,\Omega } u[\varphi ] := \int _{\Omega }\int _{\Omega } \frac{|u(x)-u(y)|^{p-2} (u(x)-u(y))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx\ dy. \end{aligned}$$

It appears as the first variation of the $\dot{W}^{s,p}$-Sobolev norm

$$\begin{aligned}{}[u]_{W^{s,p}(\Omega )}^p := \int _{\Omega }\int _{\Omega } \frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\ dx\ dy. \end{aligned}$$

In this sense it is related to the classical p-Laplacian which appears as first variation of the $\dot{W}^{1,p}$-Sobolev norm $\Vert \nabla u\Vert _{p}^p$.

If $p = 2$ the fractional p-Laplacian on $\mathbb {R}^n$ becomes the usual fractional Laplace operator $(-\Delta ) ^{s}$. For an overview on the fractional Laplacian and fractional Sobolev spaces we refer to, e.g., [6, 13].

The fractional p-Laplacian has recently received quite some interest, for example we refer to [2, 11, 12, 15, 18–20, 23, 25]. Higher regularity is one interesting and very challenging question where only very partial results are known, e.g. in [2] they obtain for $s \approx 1$ estimates in $C^{1,\alpha }$. We also refer to [5] where they show higher Sobolev regularity when the right-hand side belongs to a Sobolev space.

Since the fractional p-Laplacian is nonlinear, one cannot expect a direct analogue of (1.5). Our first result is a nonlinear commutator estimate which can play the role of (1.5). It measures how and at what price one can “transfer” derivatives to the testfunction. It implies that while an expression such as in (1.5) may not be zero, it is small—on small differential scales. For simplicity we restrict our attention to the case $p \ge 2$.

Theorem 1.1

Let $s \in (0,1)$, $p \in [2,\infty )$, and $\varepsilon \in [0,1-s)$. Take $B \subset \mathbb {R}^n$ a ball or all of $\mathbb {R}^n$. Let $u \in W^{s,p}(B)$ and $\varphi \in C_c^\infty (B)$. For a certain constant c depending on $s, \varepsilon$, p denote the nonlinear commutator

$$\begin{aligned} R_\varepsilon (u,\varphi ) := (-\Delta )^{s+\varepsilon }_{p,B}u[\varphi ] - c(-\Delta )^{s}_{p,B}u[(-\Delta ) ^{\frac{\varepsilon p}{2}}\varphi ]. \end{aligned}$$

Then we have the estimate

$$\begin{aligned} |R_\varepsilon (u,\varphi )| \le C\ \varepsilon \ [u]_{W^{s+\varepsilon ,p}(B)}^{p-1} [\varphi ]_{W^{s+\varepsilon ,p}(\mathbb {R}^n)}. \end{aligned}$$

The fact that the $\varepsilon$ appears in the estimate of $R_\varepsilon (u,\varphi )$ is the main point in Theorem 1.1. For the proof we Taylor expand $R_\varepsilon (\cdot ,\cdot )$ in $\varepsilon$. When computing $\frac{d}{d\delta } R_\delta$ we find a logarithmic potential operator, which we estimate in the following way:

Lemma 1.2

For $p \in (1,\infty )$ we consider the following semi-norm expression for $\varphi \in C_c^\infty (\mathbb {R}^n)$

$$\begin{aligned} A(\varphi ) := \left( \int _{\mathbb {R}^n} \int _{\mathbb {R}^n}\left| \int _{\mathbb {R}^n} k_\alpha (x,y,z)\ (-\Delta ) ^{\frac{\beta }{2}} \varphi (z)\ dz \right| ^p\ \frac{dx\ dy}{|x-y|^{n+\gamma p}} \right) ^{\frac{1}{p}}. \end{aligned}$$

Here, $\alpha , \beta \in (0,n)$, $\gamma \in (0,1)$ so that $s:= \gamma + \beta - \alpha \in (0,1)$, and

$$\begin{aligned} k_\alpha (x,y,z) =\left( |x-z|^{\alpha -n}\log \frac{|x-z|}{|x-y|}-|y-z|^{\alpha -n}\log \frac{|y-z|}{|x-y|} \right) . \end{aligned}$$

Then

$$\begin{aligned} A(\varphi ) \le C [\varphi ]_{W^{s,p}(\mathbb {R}^n)}. \end{aligned}$$

Having Theorem 1.1 serve as a replacement for (1.5), for small enough $\varepsilon$ we obtain estimates “close to the differential order s” for the fractional p-Laplacian.

Theorem 1.3

Let $s \in (0,1)$, $p \in [2,\infty )$, $\Omega \subset \mathbb {R}^n$ open. Take $u \in W^{s,p}(\Omega )$ a solution to

$$\begin{aligned} (-\Delta )^{s}_{p,\Omega } u = f. \end{aligned}$$

Then there is an $\varepsilon _0 > 0$ only depending on s, p, and $\Omega$, so that for $\varepsilon \in (0,\varepsilon _0)$ the following holds: If $f \in (W^{s-\varepsilon (p-1),p}(\Omega ))^*$ then $u \in W^{s+\varepsilon ,p}_{loc}(\Omega )$.

More precisely, for any $\Omega _1 \Subset \Omega$ there is a constant $C= C(\Omega _1,\Omega ,s,p)$ so that

$$\begin{aligned}{}[u]_{W^{s+\varepsilon ,p}(\Omega _1)} \le C\ \Vert f\Vert _{(W_0^{s-\varepsilon (p-1),p}(\Omega ))^*} + C[u]_{W^{s,p}(\Omega )}. \end{aligned}$$

Also, by Sobolev embedding, the higher differentiability $W^{s+\varepsilon ,p}_{loc}$ implies higher integrability i.e. $W^{s,p+\frac{pn}{n-\varepsilon p}}_{loc}$-estimates.

A higher differentiability result similar to Theorem 1.3 was proven by Kuusi et al. [18, 20]. There it is stated only for the case $p = 2$, but the proof goes through for $p \in (1,\infty )$ with only minor modifications. Their method is a generalization of Gehring’s Lemma and dual pairs. Our argument is quite different and allows for a shorter proof. Both techniques are quite robust and can be easily extended to more general nonlinearities:

Theorem 1.4

Let $s \in (0,1)$, $p \in [2,\infty )$, and a domain $\Omega \subset \mathbb {R}^n$. Let $\phi : \mathbb {R}\rightarrow \mathbb {R}$ and K(x, y) be a measurable kernel so that for some $C > 1$,

$$\begin{aligned} |\phi (t)|\le C |t|^{p-1}, \quad \phi (t)t \ge |t|^{p} \quad \forall t \in \mathbb {R}, \end{aligned}$$

and

$$\begin{aligned} C^{-1} |x-y|^{-n-sp} \le K(x,y) \le C |x-y|^{-n-sp}. \end{aligned}$$

We consider for $u \in W^{s,p}(\Omega )$, the distribution $\mathcal {L}_{\phi ,K,\Omega }(u)$

$$\begin{aligned} \mathcal {L}_{\phi ,K,\Omega }(u)[\varphi ] := \int _{\Omega }\int _{\Omega } K(x,y)\ \phi (u(x)-u(y))\ (\varphi (x)-\varphi (y))\ dx\ dy. \end{aligned}$$

Then the conclusions of Theorem 1.3 still hold if the fractional p-Laplace $(-\Delta )^{s}_{p,\Omega }$ is replaced with $\mathcal {L}_{\phi ,K,\Omega }$.

Remark 1.5

(Limiting case as $s \rightarrow 1$) The classical p-Laplacian can be seen as a (rescaled) limit of the fractional p-Laplacian $(-\Delta )^{s}_{p,\Omega }$ as $s \rightarrow 1$, see [4]. Nevertheless, it seems unlikely that as $s \rightarrow 1$ there is a limit differentiability version of Theorem 1.1, and consequently a replacement for Theorems 1.3 and 1.4 if $p > 2$.

There is, however, a nonlinear commutator estimate due to Iwaniec [16] reminiscent of Theorem 1.1. But it concerns integrability instead of differentiability. For any u with $\mathrm{supp\,}u \subset \Omega$ and any $\varepsilon \in (-1,1)$ there are maps v, R so that we have the Hodge decomposition

$$\begin{aligned} |\nabla u|^{\varepsilon } \nabla u = \nabla v + R. \end{aligned}$$

Moreover, $\Vert \nabla v\Vert _{\frac{q}{1+\varepsilon },\Omega } \precsim \Vert \nabla u\Vert ^{1+\varepsilon }_{q,\Omega }$ for all q and, most importantly, by Iwaniec’ nonlinear commutator estimate if $\varepsilon$ is small then R is small:

$$\begin{aligned} \Vert R\Vert _{\frac{p+\varepsilon }{1+\varepsilon },\Omega } \precsim |\varepsilon | \Vert \nabla u\Vert ^{1+\varepsilon }_{p+\varepsilon ,\Omega }. \end{aligned}$$

The additional $\varepsilon$ in the last estimate allows for estimates “close to the integrability order p”. Indeed

$$\begin{aligned} \Vert \nabla u\Vert ^{p+\varepsilon }_{p+\varepsilon , \Omega } = \int _{\Omega } |\nabla u|^{p-2} \nabla u \nabla v + \int _{\Omega } |\nabla u|^{p-2} \nabla u R, \end{aligned}$$

and thus,

$$\begin{aligned} \Vert \nabla u\Vert _{p+\varepsilon , \Omega }^{p+\varepsilon } \precsim |\Delta _p u[v]| + \varepsilon \Vert \nabla u\Vert _{p+\varepsilon ,\Omega }^{p-1}\ \Vert \nabla u\Vert ^{1+\varepsilon }_{p+\varepsilon ,\Omega }. \end{aligned}$$

In particular, if $\varepsilon$ is small enough and $\Delta _p u$ is in $(W_0^{1,\frac{p+\varepsilon }{1+\varepsilon }}(\Omega ))^*$, then $u \in W^{1,p+\varepsilon }(\Omega )$.

The commutator estimate in Theorem 1.1 also allows to estimate very weak solutions—i.e. solutions whose initial regularity assumptions are below the variationally natural regularity:

In the local regime, the distributional p-Laplacian $\Delta _p u[\varphi ]$ is well defined for $\varphi \in C_c^\infty (\Omega )$ whenever $u \in W^{1,p-1}_{loc}(\Omega )$. The variationally natural regularity assumption is however $W^{1,p}$, since $\Delta _p$ appears as first variation of $\Vert \nabla u \Vert _{p,\Omega }^p$. For the p-Laplacian, Iwaniec and Sbordone [17] showed that some very weak p-harmonic functions are in fact classical variational solutions:

Theorem 1.6

(Iwaniec–Sbordone) For any $p \in (1,\infty )$, $\Omega \subset \mathbb {R}^n$, there are exponents $1 < r_1 < p < r_2 < \infty$ so that every (weakly) p-harmonic function,

$$\begin{aligned} \Delta _p u = 0, \end{aligned}$$

satisfying $u \in W^{1,r_1}_{loc}(\Omega )$ indeed belongs to $W^{1,r_2}_{loc}(\Omega )$.

Again, while the p-Laplace improves its solution’s integrability, the fractional p-Laplace improves its solution’s differentiability. The distributional fractional p-Laplace $(-\Delta )^{s}_{p,\Omega }u[\varphi ]$ is well defined for $\varphi \in C_c^\infty (\Omega )$ whenever $u \in W^{q,p-1}(\Omega )$ for any $q > 0$ with $q \ge (\frac{sp-1}{p-1})_+$. We have

Theorem 1.7

For any $s \in (0,1)$, $p \in (2,\infty )$, $\Omega \subset \mathbb {R}^n$, there are exponents $1 < r_1 < p < r_2 < \infty$ and $t_1 < s < t_2$ so that every (weakly) s-p-harmonic map,

$$\begin{aligned} (-\Delta )^{s}_{p,\Omega } u = 0, \end{aligned}$$

satisfying $u \in W^{t_1,r_1}(\Omega )$ indeed belongs to $W^{t_2,r_2}_{loc}(\Omega )$.

The arguments for Theorem 1.7 are quite similar to the ones in Theorem 1.3, and we shall skip them.

Let us state an important application of Theorem 1.3: It is concerning fractional harmonic maps into spheres ${\mathbb S}^N \subset \mathbb {R}^{N+1}$: In [23] we proved that for $s \in (0,1)$ critical points of the energy

$$\begin{aligned} \mathcal {E}_{s} (u) := \int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{\frac{n}{s}}}{|x-y|^{n+s \frac{n}{s}}}\ dx\ dy, \quad u: \Omega \subset \mathbb {R}^n \rightarrow {\mathbb S}^N \end{aligned}$$

are Hölder continuous. Indeed, together with Theorem 1.3 the estimates in [23] imply a sharper result.

Theorem 1.8

($\varepsilon$-regularity for fractional harmonic maps) For any open set $\Omega \subset \mathbb {R}^n$ there is a $\delta > 0$ so that for any $\Lambda > 0$ there exists $\varepsilon > 0$ and the following holds: Let $u \in W^{s,\frac{n}{s}}(\Omega ,{\mathbb S}^N)$ with

$$\begin{aligned}{}[u]_{W^{s,\frac{n}{s}}(\Omega )} \le \Lambda \end{aligned}$$

(1.6)

be a critical point of $\mathcal {E}_s(u)$, i.e.

$$\begin{aligned} \frac{d}{dt}\Big |_{t = 0} \mathcal {E}_s\left( \frac{u+t\varphi }{|u+t\varphi |} \right) = 0 \quad \forall \varphi \in C_c^\infty (\Omega ,\mathbb {R}^N). \end{aligned}$$

(1.7)

If on a ball $2B \subset \Omega$ we have

$$\begin{aligned}{}[u]_{W^{s,\frac{n}{s}}(2B)} \le \varepsilon , \end{aligned}$$

(1.8)

then on the ball B (the ball concentric to 2B with half the radius),

$$\begin{aligned}{}[u]_{W^{s+\delta ,\frac{n}{s}}(B)} \le C_{\Lambda ,B}. \end{aligned}$$

This kind of $\varepsilon$-regularity estimate is crucial for compactness and bubble analysis for fractional harmonic maps. Da Lio obtained quantization results [8] in the $p=2$ regime for $n=1$ and $s=\frac{1}{2}$. With the help of Theorem 1.8 one can extend her compactness estimates to all $s \in (0,1)$, $n \in {\mathbb N}$. More precisely, we have the following result extending the first part of [8, Theorem 1.1].

Theorem 1.9

Let $u_k \in \dot{W}^{s,\frac{n}{s}}(\mathbb {R}^n,{\mathbb S}^{N-1})$ be a sequence of $(s,\frac{n}{s})$-harmonic maps in the sense of (1.7) such that

$$\begin{aligned}{}[u_k]_{W^{s,\frac{n}{s}}(\mathbb {R}^n,{\mathbb S}^{N-1})} \le C. \end{aligned}$$

Then there is $u_\infty \in \dot{W}^{s,\frac{n}{s}}(\mathbb {R}^n,{\mathbb S}^{N-1})$ and a possibly empty set $\{\alpha _1,\ldots ,\alpha _l\}$ such that up to a subsequence we have strong convergence away from $\{\alpha _1,\ldots ,\alpha _l\}$, that is

$$\begin{aligned} u_k \xrightarrow {k \rightarrow \infty } u_\infty \quad \text{ in } W^{s,\frac{n}{s}}_{loc}(\mathbb {R}^n \backslash \{\alpha _1,\ldots ,\alpha _l\}). \end{aligned}$$

A more precise analysis of compactness and the formation of bubbles will be part of a future work.

2 Outline and notation

In Sect. 3 we will prove the commutator estimate, Theorem 1.1. Roughly speaking, we compute the kernel $\kappa _\varepsilon (x,y,z)$ of the commutator and show that its derivative in $\varepsilon$ (which gives a logarithmic potential) induces a bounded operator. The latter estimate is contained in Lemma 1.2 which we shall prove via Littlewood–Paley theory in Sect. 4.

We try to keep the notation as simple as possible. For a ball B, $\lambda B$ denotes the concentric ball with $\lambda$-times the radius. With

$$\begin{aligned} (u)_B := |B|^{-1} \int _B u \end{aligned}$$

we denote the mean value.

The dual norm of the p-Laplacian is denoted as

$$\begin{aligned} \Vert (-\Delta )^{s}_{p,\Omega } u\Vert _{(W^{t,p}_0(\Omega ))^*} \equiv \sup _{\varphi } |(-\Delta )^{s}_{p,\Omega } u [\varphi ]| \end{aligned}$$

where the supremum is taken over $\varphi \in C_c^\infty (\Omega )$ with $[\varphi ]_{W^{t,p}(\mathbb {R}^n)} \le 1$.

We already defined the fractional Laplacian $(-\Delta ) ^{\frac{s}{2}}$. Its inverse $I^{s}$ is the Riesz potential, which for some constant $c \in \mathbb {R}$ can be written as

$$\begin{aligned} I^{s} g(x) = c\int _{\mathbb {R}^n} |x-z|^{s - n} g(z)\ dz. \end{aligned}$$

(2.1)

In the estimates, the constants can change from line to line. Whenever we deem the constant unimportant to the argument, we will drop it, writing $A \precsim B$ if $A \le C\cdot B$ for some constant $C > 0$. Similarly we will use $A \approx B$ whenever A and B are comparable.

3 The commutator estimate: proof of Theorem 1.1

Proof

Recall that for $t \in (0,n)$ there is a constant $c \in \mathbb {R}$ so that for any $\varphi \in C_c^\infty (\mathbb {R}^n)$,

$$\begin{aligned} c \int _{\mathbb {R}^n} |x-z|^{t-n} (-\Delta ) ^{\frac{t}{2}} \varphi (z)\ dz = I^{t} (-\Delta ) ^{\frac{t}{2}} \varphi (x) = \varphi (x). \end{aligned}$$

(3.1)

We write

with

$$\begin{aligned} \kappa _\varepsilon (x,y,z) := \left( \frac{|x-z|^{t+\varepsilon p-n}-|y-z|^{t+\varepsilon p-n}}{|x-y|^{\varepsilon p}} \right) -(|x-z|^{t-n} - |x-y|^{t-n}). \end{aligned}$$

Using again (3.1), this reads as

$$\begin{aligned} R(u,\varphi ):= & {} (-\Delta )^{s+\varepsilon }_{p,B}u[\varphi ]-c(-\Delta )^{s}_{p,B}u[(-\Delta ) ^{\frac{\varepsilon p}{2}}\varphi ] \\= & {} \int \limits _B \int \limits _B \int \limits _{\mathbb {R}^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y)) \kappa _\varepsilon (x,y,z) }{|x-y|^{n+sp}} (-\Delta ) ^{\frac{t+\varepsilon p}{2}} \varphi (z)dz dx dy. \end{aligned}$$

Since $\kappa _0(x,y,z) = 0$ for almost all $x,y,z \in \mathbb {R}^n$,

$$\begin{aligned} \kappa _\varepsilon (x,y,z) = \int _0^\varepsilon \frac{d}{d\delta }\kappa _\delta (x,y,z)\ d\delta . \end{aligned}$$

We denote

$$\begin{aligned} \begin{aligned} k_\delta (x,y,z)&:= |x-y|^{\delta p} \frac{d}{d\delta } \kappa _\delta (x,y,z) \\&=\left( |x-z|^{t+\delta p-n}\log \frac{|x-z|}{|x-y|}-|y-z|^{t+\delta p-n}\log \frac{|y-z|}{|x-y|} \right) . \end{aligned} \end{aligned}$$

Thus, $R(u,\varphi )$ is equal to

$$\begin{aligned} \int \limits _0^\varepsilon \int \limits _B \int \limits _B \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{(s+\varepsilon )(p-1)}} \left( \,\int \limits _{\mathbb {R}^n} \frac{k_\delta (x,y,z)\ (-\Delta ) ^{\frac{t+\varepsilon p}{2}} \varphi (z)dz}{|x-y|^{s+\varepsilon -(\varepsilon -\delta ) p}} \right) \frac{dx\ dy\ d\delta }{|x-y|^n}. \end{aligned}$$

With Hölder inequality we get the upper bound for $|R(u,\varphi )|$

$$\begin{aligned} \varepsilon [u]_{W^{s+\varepsilon ,p}(B)}^{p-1}\ \sup _{\delta \in (0,\varepsilon )} \left( \int \limits _B \int \limits _B \left| \int \limits _{\mathbb {R}^n} \frac{k_\delta (x,y,z)\ (-\Delta ) ^{\frac{t+\varepsilon p}{2}} \varphi (z)dz}{|x-y|^{s+\varepsilon -(\varepsilon -\delta ) p}} \right| ^p \frac{dx\ dy}{|x-y|^n} \right) ^{\frac{1}{p}}. \end{aligned}$$

This falls into the realm of Lemma 1.2, for

$$\begin{aligned} \alpha := t+\delta p, \quad \beta := t+\varepsilon p, \quad \gamma := s+\varepsilon -(\varepsilon -\delta ) p, \quad \gamma +\beta -\alpha = s+\varepsilon . \end{aligned}$$

This concludes the proof. $\square$

4 Logarithmic potential estimate: proof of Lemma 1.2

For the proof of Lemma 1.2 we will use the Littlewood–Paley decomposition: We refer to the Triebel monographs, e.g. [24], and [14] for a complete picture of this theory. We will only need few properties:

For a tempered distribution f we define $f_j$ to be the Littlewood–Paley projections $f_j := P_j f$, where

$$\begin{aligned} P_j f(x) := \int \limits _{\mathbb {R}^n} 2^{jn} p(2^{j} (x-z)) f(z)\ dz. \end{aligned}$$

Here, p is a Schwartz function, and it can be chosen in a way such that

$$\begin{aligned} \sum _{j \in \mathbb {Z}} f_j = f. \end{aligned}$$

(4.1)

For any $j \in \mathbb {Z}$ we have the estimate for Riesz potentials and derivatives (cf. (2.1))

$$\begin{aligned} \Vert I^{s}|(-\Delta ) ^{\frac{t}{2}} f_j| \Vert _{p} \precsim \sum _{i = j-1}^{j+1} 2^{j(t-s)} \Vert f_i \Vert _{p}. \end{aligned}$$

(4.2)

The homogeneous semi-norm for the Triebel space $\dot{F}^s_{p,p} = \dot{B}^s_{p,p}$ is

$$\begin{aligned} \Vert f\Vert _{\dot{F}^s_{p,p}} := \left( \sum _{j \in \mathbb {Z}} 2^{jsp} \Vert f_j\Vert _{p}^p \right) ^{\frac{1}{p}}. \end{aligned}$$

(4.3)

Crucially to us, the Triebel spaces are equivalent to Sobolev spaces: For $s \in (0,1)$ we have the identification

$$\begin{aligned} \Vert f\Vert _{\dot{F}^s_{p,p}} \approx [f]_{W^{s,p}(\mathbb {R}^n)}. \end{aligned}$$

(4.4)

Proof of Lemma 1.2

We denote

$$\begin{aligned} T\varphi (x,y) := \int \limits _{\mathbb {R}^n} k(x,y,z)\ (-\Delta ) ^{\frac{\beta }{2}} \varphi (z)\ dz. \end{aligned}$$

In order to obtain the claimed estimate, we will use two decompositions simultaneously. Firstly, we decompose into slices where $|x-y| \approx 2^{-k}$. For this denote

$$\begin{aligned} \chi _{|y| \approx 2^{-k}} := \chi _{B_{2^{-k}(0)}\backslash B_{2^{-k-1}(0)}}(y). \end{aligned}$$

Secondly, we use the Littlewood–Paley decomposition (4.1). Then

$$\begin{aligned} A(\varphi )^p \precsim \sum _{k\in \mathbb {Z},j \in \mathbb {Z}} I_{j,k}, \end{aligned}$$

where

$$\begin{aligned} I_{j,k} := \int \limits _{\mathbb {R}^n} \int \limits _{\mathbb {R}^n} \chi _{|x-y| \approx 2^{-k}}\left| T\varphi (x,y) \right| ^{p-1}\ |T\varphi _j(x,y)| \ \frac{dx\ dy}{|x-y|^{n+\gamma p}}. \end{aligned}$$

Set

$$\begin{aligned} a_k := \left( \int \limits _{\mathbb {R}^n} \int \limits _{\mathbb {R}^n} \chi _{|x-y| \approx 2^{-k}}\left| T\varphi (x,y)\right| ^{p} \frac{dx\ dy}{|x-y|^{n+\gamma p}} \right) ^{\frac{1}{p}} \end{aligned}$$

and

$$\begin{aligned} b_j := 2^{j(\gamma +\beta -\alpha )}\ \Vert \varphi _j\Vert _{p}. \end{aligned}$$

Note that with (4.3) and (4.4)

$$\begin{aligned} \left( \sum _{k \in \mathbb {Z}} a_k^p \right) ^{\frac{1}{p}} \approx A(\varphi ) \quad \text{ and }\quad \left( \sum _{j \in \mathbb {Z}} b_j^p \right) ^{\frac{1}{p}} \approx \Vert \varphi \Vert _{\dot{F}^s_{p,p}} \approx [\varphi ]_{W^{s,p}(\mathbb {R}^n)}. \end{aligned}$$

(4.5)

With Hölder inequality,

We have to possibilities of estimating $\tilde{I}_{j,k}$, and we are going to interpolate between them:

Firstly, for any small $\sigma \in (0,\alpha )$ we can employ the estimate $|\log \frac{|x-z|}{|x-y|}| \precsim \frac{|x-y|^\sigma }{|x-z|^\sigma } + \frac{|x-z|^\sigma }{|x-y|^\sigma }$. If we recall the Riesz potentials (2.1), we see that

$$\begin{aligned}&\int \limits _{\mathbb {R}^n} |x-z|^{\alpha -n}\log \frac{|x-z|}{|x-y|} |(-\Delta ) ^{\frac{\beta }{2}}\varphi _j(z)|\ dz\\&\quad \precsim |x-y|^{-\sigma } I^{\alpha +\sigma } |(-\Delta ) ^{\frac{\beta }{2}} \varphi _j|(x) + |x-y|^{\sigma } I^{\alpha -\sigma } |(-\Delta ) ^{\frac{\beta }{2}} \varphi _j|(x). \end{aligned}$$

Having in mind (4.2) we obtain the estimate

$$\begin{aligned} \tilde{I}_{j,k}&\precsim 2^{k(\frac{n+\gamma p}{p})}\ 2^{k\sigma } 2^{-k\frac{n}{p}} \Vert I^{\alpha +\sigma } |(-\Delta ) ^{\frac{\beta }{2}} \varphi _j|\Vert _{p} + 2^{k(\frac{n+\gamma p}{p})}\ 2^{-k\sigma } 2^{-k\frac{n}{p}}\\&\quad \,\times \Vert I^{\alpha -\sigma } |(-\Delta ) ^{\frac{\beta }{2}} \varphi _j|\Vert _{p}\\&\precsim 2^{(k-j)(\gamma +\sigma )} (b_{j-1} + b_j + b_{j+1}) + 2^{(k-j)(\gamma -\sigma )} (b_{j-1} + b_j + b_{j+1}). \end{aligned}$$

This is our first estimate:

$$\begin{aligned} \tilde{I}_{j,k} \precsim 2^{(k-j)(\gamma -\sigma )}\ (2^{2\sigma (k-j)}+ 1)\ (b_{j-1} + b_j + b_{j+1}). \end{aligned}$$

(4.6)

Secondly, by a substitution we can write

$$\begin{aligned} T\varphi _j (x,y) = \int _{\mathbb {R}^n} |z|^{\alpha -n}\log \frac{|z|}{|x-y|} \left( (-\Delta ) ^{\frac{\beta }{2}}\varphi _j(z+x) -(-\Delta ) ^{\frac{\beta }{2}}\varphi _j(z+y) \right) dz. \end{aligned}$$

We use now $|f(x) - f(y)| \precsim |x-y| (\mathcal {M}|\nabla f|(x)+\mathcal {M}|\nabla f|(y))$, where $\mathcal {M}$ is the Hardy–Littlewood maximal function. Then, again for any $\sigma > 0$,

$$\begin{aligned}&|T\varphi _j(x,y)| \\&\quad \precsim |x-y| \int _{\mathbb {R}^n} |z|^{\alpha -n}\left| \log \frac{|z|}{|x-y|} \right| \ \mathcal {M}|(-\Delta ) ^{\frac{\beta }{2}}\nabla \varphi _j|(z+x)\ dz \\&\qquad +|x-y| \int _{\mathbb {R}^n} |z|^{\alpha -n} \left| \log \frac{|z|}{|x-y|} \right| \ |\mathcal {M}(-\Delta ) ^{\frac{\beta }{2}}\nabla \varphi _j|(z+x)\ dz \\&\quad \precsim |x-y|^{1-\sigma } I^{\alpha +\sigma } \mathcal {M}|(-\Delta ) ^{\frac{\beta }{2}}\nabla \varphi _j|(x)\\&\qquad + |x-y|^{1-\sigma } I^{\alpha +\sigma } \mathcal {M}|(-\Delta ) ^{\frac{\beta }{2}}\nabla \varphi _j|(y)\\&\qquad + |x-y|^{1+\sigma } I^{\alpha -\sigma } \mathcal {M}|(-\Delta ) ^{\frac{\beta }{2}}\nabla \varphi _j|(x)\\&\qquad + |x-y|^{1+\sigma } I^{\alpha -\sigma } \mathcal {M}|(-\Delta ) ^{\frac{\beta }{2}}\nabla \varphi _j|(y). \end{aligned}$$

Consequently, our second estimate is

Together with (4.6) we thus have

In particular, since $\gamma \in (0,1)$ pick any $0 < \sigma < \min \{\gamma ,1-\gamma \}$—which, as we shall see in a moment, makes the following sums convergent:

With Hölder inequality and (4.5),

$$\begin{aligned} III \precsim \left( \sum _{j \in \mathbb {Z}} b_j^p\right) ^{\frac{1}{p}} \left( \sum _{j \in \mathbb {Z}} a_j^p\right) ^{\frac{p-1}{p}} = A(\varphi )^{p-1}\ [\varphi ]_{W^{s,p}(\mathbb {R}^n)}. \end{aligned}$$

As for I, for any $\varepsilon > 0$,

The same works for II:

Together,

$$\begin{aligned} I + II \precsim \varepsilon ^p [\varphi ]_{W^{s,p}(\mathbb {R}^n)}^p\ + \varepsilon ^{-p'}C_{1-\gamma -\sigma } A(\varphi )^p, \end{aligned}$$

which holds for any $\varepsilon > 0$. Pick

$$\begin{aligned} \varepsilon := [\varphi ]_{W^{s,p}(\mathbb {R}^n)}^{-\frac{1}{p' }}\ A(\varphi )^{\frac{1}{p' }}. \end{aligned}$$

Then

$$\begin{aligned} A(\varphi )^p \le I+II+III \precsim A(\varphi )^{p-1}\ [\varphi ]_{W^{s,p}(\mathbb {R}^n)}. \end{aligned}$$

Lemma 1.2 is proven if we divide both sides by $A(\varphi )^{p-1}$. $\square$

5 Higher differentiability: proof of Theorem 1.3

In view of Lemma 8.1 we can assume w.l.o.g. that $\Omega$ is a bounded open set, and that the support of u is strictly contained in some open set $\Omega _1 \Subset \Omega$. Then Theorem 1.3 follows from

Lemma 5.1

Let $\Omega _1 \Subset \Omega$ two open, bounded sets, $s \in (0,1)$, $p \in [2,\infty )$. Then there exists an $\varepsilon _0 > 0$ so that for any $\varepsilon \in (0,\varepsilon _0)$,

$$\begin{aligned}{}[u]_{W^{s+\varepsilon ,p}(\Omega )}^{p-1} \precsim [u]_{W^{s,p}(\Omega )}^{p-1} + \Vert (-\Delta )^{s}_{p,\Omega } u \Vert _{(W_0^{s-\varepsilon (p-1),p}(\Omega ))^*}. \end{aligned}$$

Proof

We can find finitely many balls $(B_k)_{k =1}^K \subset \Omega$ so that $\bigcup _{k=1}^K B_k \supset \Omega _1$. We denote with $10B_k$ the concentric balls with ten times the radius, and may assume $\bigcup _{k=1}^N 10B_k \subset \Omega$.

Denote

$$\begin{aligned} \Gamma _{s} := [u]_{W^{s,p}(\Omega )}^{p} ,\quad \Gamma _{s+\varepsilon } := [u]_{W^{s+\varepsilon ,p}(\Omega )}^{p}. \end{aligned}$$

We then have

$$\begin{aligned} \Gamma _{s+\varepsilon } \precsim \sum _{k=1}^K [u]_{W^{s+\varepsilon ,p}(2B_k)}^{p} + \sum _{k=1}^K \int _{\Omega \backslash 2B_k} \int _{B_k} \frac{|u(x)-u(y)|^p}{|x-y|^{n+(s+\varepsilon )p}}\ dx\ dy. \end{aligned}$$

As for the second term, because of the disjoint support of the integrals we find

$$\begin{aligned} \int _{\Omega \backslash 2B_k} \int _{B_k} \frac{|u(x)-u(y)|^p}{|x-y|^{n+(s+\varepsilon )p}}\ dx\ dy \precsim (\mathrm{diam\,}B_k)^{-\varepsilon p}\ \Gamma _s. \end{aligned}$$

That is

$$\begin{aligned} \Gamma _{s+\varepsilon } \precsim \sum _{k=1}^K [u]_{W^{s+\varepsilon ,p}(2B_k)}^{p} + \Gamma _s. \end{aligned}$$

With Lemma 8.2 and Poincaré inequality, Proposition 8.3, for any $\delta > 0$,

$$\begin{aligned} \begin{aligned}&\Gamma _{s+\varepsilon } \precsim \delta ^p \Gamma _{s+\varepsilon } + C_\delta \Gamma _s + \sum _{k=1}^K \delta ^{-p'} \left( \sup _{\varphi } (-\Delta )^{s+\varepsilon }_{p,8B_k}u[\varphi ] \right) ^{\frac{p}{p-1}}\\ \end{aligned} \end{aligned}$$

where the supremum is over all $\varphi \in C_c^\infty (4B_k)$ and $[\varphi ]_{W^{s+\varepsilon ,p}(\mathbb {R}^n)} \le 1$. Here we also used that $\bigcup _{k=1}^K 8B_k$ covers no more than $\Omega$. Choosing $\delta$ sufficiently small, we can estimate $\Gamma _{s+\varepsilon }$ by

$$\begin{aligned} \Gamma _s + \sum _{k=1}^K \left( \sup \left\{ |(-\Delta )^{s+\varepsilon }_{p,8B_k} u[\varphi ]|: \ \varphi \in C_c^\infty (4B_k), [\varphi ]_{W^{s+\varepsilon ,p}(\mathbb {R}^n)} \le 1 \right\} \right) ^{\frac{p}{p-1}}. \end{aligned}$$

With Theorem 1.1 this can be estimated by

$$\begin{aligned}&\Gamma _s + \varepsilon ^{\frac{p}{p-1}} \Gamma _{s+\varepsilon }\\&\quad + \sum _{k=1}^K \left( \sup \left\{ \left| (-\Delta )^{s}_{p,8B_k} u[(-\Delta ) ^{\frac{\varepsilon p}{2}}\varphi ]\right| : \ \varphi \in C_c^\infty (4B_k), [\varphi ]_{W^{s+\varepsilon ,p}(\mathbb {R}^n)} \le 1 \right\} \right) ^{\frac{p}{p-1}}. \end{aligned}$$

If $\varepsilon \in [0,\varepsilon _0)$ for $\varepsilon _0$ small enough, we can again absorb $\Gamma _{s+\varepsilon }$. The estimate for $\Gamma _{s+\varepsilon }$ becomes

$$\begin{aligned} \Gamma _s + \sum _{k=1}^K \left( \sup \left\{ \left| (-\Delta )^{s}_{p,8B_k} u[(-\Delta ) ^{\frac{\varepsilon p}{2}}\varphi ]\right| : \ \varphi \in C_c^\infty (4B_k), [\varphi ]_{W^{s+\varepsilon ,p}(\mathbb {R}^n)} \le 1 \right\} \right) ^{\frac{p}{p-1}}. \end{aligned}$$

Next, we need to transform $(-\Delta ) ^{\frac{\varepsilon p}{2}}\varphi$ into a feasible testfunction, and denoting the usual cutoff function with $\eta _{6B_k} \in C_c^\infty (6B_{k})$, $\eta _{6B_k} \equiv 1$ in $5B_{k}$

$$\begin{aligned} (-\Delta ) ^{\frac{\varepsilon p}{2}}\varphi =: \psi + (1-\eta _{6B_k})(-\Delta ) ^{\frac{\varepsilon p}{2}}\varphi \end{aligned}$$

Then $\psi \in C_c^\infty (6B_{k})$

$$\begin{aligned}{}[\psi ]_{W^{s-\varepsilon (p-1),p}(\Omega )} \precsim C_k [\varphi ]_{W^{s+\varepsilon ,p}(\mathbb {R}^n)} \le C_k. \end{aligned}$$

Moreover, the disjoint support of $(1-\eta _{6B_k})$ and $\varphi$ implies (see, e.g., [3, Lemma A.1])

$$\begin{aligned} \left[ (1-\eta _{6B_k})(-\Delta ) ^{\frac{\varepsilon p}{2}}\varphi \right] _{\mathrm{Lip\,}} \le C_k\ [\varphi ]_{W^{s+\varepsilon ,p}(\mathbb {R}^n)}. \end{aligned}$$

Consequently,

$$\begin{aligned} |(-\Delta )^{s}_{p,8B_k} u[(-\Delta ) ^{\frac{\varepsilon p}{2}}\varphi -\psi ]| \precsim [u]_{W^{s,p}(\Omega )}^{p-1}. \end{aligned}$$

Hence, our estimate for $\Gamma _{s+\varepsilon }$ now looks like

$$\begin{aligned} \Gamma _s + \sum _{k=1}^K \left( \sup \left\{ \left| (-\Delta )^{s}_{p,8B_k} u[\psi ]\right| : \ \psi \in C_c^\infty (6B_k), [\psi ]_{W^{s-\varepsilon (p-1),p}(\mathbb {R}^n)} \le 1 \right\} \right) ^{\frac{p}{p-1}}. \end{aligned}$$

Finally, we need to transform the support of $(-\Delta ) ^{\frac{s}{2}}_p$ from $8B_k$ to $\Omega$. Since $\mathrm{supp\,}\psi \subset 6B_k$, the disjoint support of the integrals gives

$$\begin{aligned}&|(-\Delta )^{s}_{p,8B_k} u[\psi ] - (-\Delta )^{s}_{p,\Omega } u[\psi ]|\\&\quad \precsim \int _{\Omega \backslash 8B_k} \int _{7B_k} \frac{|u(x)-u(y)|^{p-1}\ |\psi (x)-\psi (y)|}{|x-y|^{n+sp}}\ dx\ dy\\&\quad \le C_k [u]_{W^{s,p}(\Omega )}^{p-1} [\psi ]_{W^{s-\varepsilon (p-1),p}(\mathbb {R}^n)}. \end{aligned}$$

This implies the final estimate of $\Gamma _{s+\varepsilon }$ by

$$\begin{aligned} \Gamma _s + \left( \sup \left\{ \left| (-\Delta )^{s}_{p,\Omega } u[\psi ]\right| : \ \psi \in C_c^\infty (\Omega ), [\psi ]_{W^{s-\varepsilon (p-1),p}(\mathbb {R}^n)} \le 1 \right\} \right) ^{\frac{p}{p-1}}. \end{aligned}$$

$\square$

6 Differentiability of p-harmonic maps: proof of Theorem 1.8

For $B \subset \mathbb {R}^n$, $t \in (0,1)$, we set

$$\begin{aligned} T_{t,B}u(z) = \int _{B} \int _B \frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\ (|x-z|^{t-n}-|y-z|^{t-n})}{|x-y|^{n+s\frac{n}{s}}}\ dx\ dy. \end{aligned}$$

$T_{t,B}u$ was introduced in [23] because of the following relation

$$\begin{aligned}&c\int _{\mathbb {R}^n} T_{t,B}u(z)\ \varphi (z)\ dz\nonumber \\&\quad = \int _{B}\int _{B} \frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\ (I^{t}\varphi (x)-I^{t}\varphi (y))}{|x-y|^{n+s\frac{n}{s}}}\ dx\ dy.\qquad \end{aligned}$$

(6.1)

From [23, in particular (3.1), Lemma 3.3, 3.4, 3.5] we have the following

Theorem 6.1

Let u satisfy (1.6) and (1.7) in an open set $\Omega$. Assume that on the ball 2B for a small enough $\varepsilon > 0$ (depending on $\Lambda$) (1.8) holds. Then there is $t_0 < s$, $\sigma > 0$, so that for some $\gamma _2 > \gamma _1 \gg 1$ for any ball $B_{\gamma _2 \rho } \subset B$

$$\begin{aligned}{}[u]_{W^{s,\frac{n}{s}}(B_{\rho })} \precsim C_{\Lambda } \rho ^\sigma , \end{aligned}$$

(6.2)

and

$$\begin{aligned} \Vert T_{t_0,B_{\gamma _1 \rho }} u\Vert _{\frac{n}{n-t_0},B_\rho } \le C_{\Lambda } \rho ^\sigma . \end{aligned}$$

(6.3)

Estimate (6.3) looks almost as if $T_{t_0,B_{\gamma _1\rho }}$ belongs locally to a Morrey space. But the domain dependence on $B_{\gamma _1 \rho }$ prevents us from exploiting this immediately. The following proposition removes the domain dependence.

Proposition 6.2

Under the assumptions of Theorem 6.1 there exists $\gamma > 1$, $\sigma > 0$ so that

$$\begin{aligned} \Vert T_{t_0,B} u\Vert _{\frac{n}{n-t_0},B_\rho } \le C_{B,\Lambda } \rho ^\sigma \end{aligned}$$

for any ball so that $B_{\gamma \rho } \subset B$.

Proof

Set $\kappa _1 \ge \kappa _2 \ge \kappa _3 \ge 1$ to be chosen later. Take $\gamma := 2\gamma _1$ with $\gamma _1$ from (6.3). We will always assume $\rho < 1$.

For some $\varphi \in C_c^\infty (B_{\rho ^{\kappa _1}})$, $\Vert \varphi \Vert _{\frac{n}{t_0}} \le 1$ we have

$$\begin{aligned}&\Vert T_{t_0,B} u\Vert _{\frac{n}{n-t_0},B_{\rho ^{\kappa _1}}}\\&\quad \precsim \int _{\mathbb {R}^n} T_{t_0,B} u\ \varphi \\&\quad \overset{(6.1)}{\approx } \int _{B}\int _{B} \frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\ (I^{t_0}\varphi (x)-I^{t_0}\varphi (y))}{|x-y|^{n+s\frac{n}{s}}}\ dx\ dy. \end{aligned}$$

We will now use several cutoffs to slice $\varphi$ into the right form. This kind of arguments and the consequent (tedious) estimates have been used several times in work related to fractional harmonic maps, cf. e.g. [3, 7, 9, 10, 21–23], and we will not repeat them in detail. We will also assume that $\kappa _1 > \kappa _2 > \kappa _3$. If they are equal, to keep the “disjoint support estimates” working one needs to use cutoff functions on twice, four times etc. of the Balls.

For a cutoff function $\eta _{B_{\rho ^{\kappa _2}}} \in C_c^\infty (B_{2\rho ^{\kappa _2}})$, $\eta _{B_{\rho ^{\kappa _2}}} \equiv 1$ on $B_{\rho ^{\kappa _2}}$, we have

$$\begin{aligned} I^{t_0}\varphi := \psi + (1-\eta _{B_{\rho ^{\kappa _2}}}) I^{t_0} \varphi . \end{aligned}$$

Note that $\psi \in C_c^\infty (B_{2\rho ^{\kappa _2}})$ and¹

$$\begin{aligned} \Vert (-\Delta ) ^{\frac{t_0}{2}} \psi \Vert _{\frac{n}{t_0}} + [\psi ]_{W^{t_0,\frac{n}{t_0}}(\mathbb {R}^n)} \precsim \Vert \varphi \Vert _{\frac{n}{t_0}}. \end{aligned}$$

(6.4)

The disjoint support of $(1-\eta )$ and $\varphi$ ensures (see [3, Lemma A.1])

$$\begin{aligned}{}[I^{t_0} \varphi - \psi ]_{W^{s,\frac{n}{s}}(\mathbb {R}^n)} \precsim \rho ^{(\kappa _1-\kappa _2)(n-t_0)}\ \Vert \varphi \Vert _{\frac{n}{t_0}}. \end{aligned}$$

(6.5)

We furthermore decompose

$$\begin{aligned} (-\Delta ) ^{\frac{t_0}{2}} \psi =: \phi + (1-\eta _{B_{\rho ^{\kappa _3}}}) (-\Delta ) ^{\frac{t_0}{2}} \psi . \end{aligned}$$

Then $\phi \in C_c^\infty (B_{2\rho ^{\kappa _3}})$ and

$$\begin{aligned}&\Vert \phi \Vert _{\frac{n}{t_0}} \precsim \Vert \varphi \Vert _{\frac{n}{t_0}}, \end{aligned}$$

(6.6)

$$\begin{aligned}&\Vert \nabla (\psi -I^{t_0}\phi )\Vert _{\infty } \precsim \rho ^{-\kappa _3+(\kappa _2-\kappa _3)n}\ \Vert \varphi \Vert _{\frac{n}{t_0}}. \end{aligned}$$

(6.7)

Again with (6.1), we then have

$$\begin{aligned} \Vert T_{t_0,B} u\Vert _{\frac{n}{n-t_0},B_\rho } \precsim |I| + |II| + |III| + |IV| \end{aligned}$$

where

$$\begin{aligned} I:= & {} \int T_{t_0,B_{\gamma \rho }}u\ \phi ,\\ II:= & {} \int _{B_{\gamma \rho }}\int _{B_{\gamma \rho }} \frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\ ((\psi -I^{t_0}\phi )(x) - (\psi -I^{t_0}\phi )(y))}{|x-y|^{n+s\frac{n}{s}}}\ dx\ dy,\\ III:= & {} \int _{B \backslash B_{\gamma \rho }}\int _{B_{2\rho ^{\kappa _2}}} \frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\ (\psi (x) - \psi (y))}{|x-y|^{n+s\frac{n}{s}}}\ dx\ dy, \end{aligned}$$

and

$$\begin{aligned} IV := \int _{B }\int _{B } \frac{|u(x)-u(y)|^{\frac{n}{s}-2}(u(x)-u(y))\ ((I^{t_0} \varphi -\psi )(x) - (I^{t_0} \varphi -\psi )(y))}{|x-y|^{n+s\frac{n}{s}}}\ dx\ dy. \end{aligned}$$

With (6.6), $\mathrm{supp\,}\phi \subset B_{2\rho ^{\kappa _3}} \subset B_{2\rho }$, and (6.3),

$$\begin{aligned} |I| \precsim \rho ^{\sigma }. \end{aligned}$$

With (6.2), (6.7) (for $\rho$ small enough),

$$\begin{aligned} |II| \precsim [u]_{W^{s,\frac{n}{s}}(B_{\gamma \rho })}^{\frac{n}{s}-1} [\psi -I^{t_0}\phi ]_{W^{s,\frac{n}{s}}(B_{\gamma \rho })} \precsim \rho ^{\sigma (\frac{n}{s}-1)}\ \rho ^{-(\kappa _3-1)} \rho ^{(\kappa _2-\kappa _3)n}. \end{aligned}$$

With the disjoint support of the integrals, Hölder inequality ($\frac{n}{t_0} > \frac{n}{s}$), and (6.4),

$$\begin{aligned} |III| \precsim [u]_{W^{s,\frac{n}{s}}(B)}^{p-1}\ \rho ^{t_0-s}\ \rho ^{\kappa _2(s-t_0)}\ [\psi ]_{W^{t_0,\frac{n}{t_0}}(B)} \precsim \rho ^{(\kappa _2-1)(s-t_0)}. \end{aligned}$$

Lastly, with (6.5)

$$\begin{aligned} |IV| \precsim [u]_{W^{s,\frac{n}{s}}(B)}^{\frac{n}{s}-1} [I^{t_0}\varphi -\psi ]_{W^{s,\frac{n}{s}}(B)} \precsim \rho ^{(\kappa _1-\kappa _2)(n-t_0)}. \end{aligned}$$

If we choose $\kappa _1 = \kappa _2 = \kappa _3 = 1$, we obtain

$$\begin{aligned} \Vert T_{t_0,B} u\Vert _{\frac{n}{n-t_0},B_{\rho }} \precsim 1, \end{aligned}$$

whenever $B_{2\gamma \rho } \subset B$, In particular

$$\begin{aligned} \Vert T_{t_0,B} u\Vert _{\frac{n}{n-t_0},\frac{1}{2\gamma }B} \precsim 1. \end{aligned}$$

(6.8)

On the other hand, we may take

$$\begin{aligned} \kappa _1 > \kappa _2 > \kappa _3 = 1. \end{aligned}$$

Then we have shown that

$$\begin{aligned} \Vert T_{t_0,B} u\Vert _{\frac{n}{n-t_0},B_{\rho ^{\kappa _1}}} \precsim \rho ^{\tilde{\sigma }}, \end{aligned}$$

which holds whenever $B_{\gamma \rho } \subset B$. Equivalently, for an even smaller $\tilde{\sigma }$,

$$\begin{aligned} \Vert T_{t_0,B} u\Vert _{\frac{n}{n-t_0},B_{\rho }} \precsim \rho ^{\tilde{\sigma }}, \end{aligned}$$

which holds whenever $B_{\gamma \rho ^{\frac{1}{\kappa _1}}} \subset B$. With (6.8) this estimate also holds whenever $B_{2\gamma \rho } \subset B$, with a constant depending on the radius of B. $\square$

In [23] it is shown that for $t_1 > t_0$, $T_{t_1,B} u = I^{t_1-t_0} T_{t_0,B} u$. Since according to Proposition 6.2 $T_{t_0,B} u$ belongs to a Morrey space, we can apply Adams estimates on Riesz potential acting on Morrey spaces [1, Theorem 3.1 and Corollary after Proposition 3.4] and obtain an increased integrability estimate for $T_{t_1,B} u$.

Proposition 6.3

Under the assumptions of Theorem 6.1 there are $\gamma > 1$, $t_0 < t_1 < s$, and $p_1 > \frac{n}{n-t_1}$ so that

$$\begin{aligned} \Vert T_{t_1,B} u\Vert _{p_1,B_\rho } \le C_{\Lambda } \rho ^\sigma \end{aligned}$$

for any ball so that $B_{\gamma \rho } \subset B$.

Now we exploit (6.1): For any $\varphi \in C_c^\infty (\mathbb {R}^n)$

$$\begin{aligned} (-\Delta )^{s}_{\frac{n}{s},B} u[\varphi ] = \int _{\mathbb {R}^n} T_{t_1,B} u\ (-\Delta ) ^{\frac{t_1}{2}} \varphi . \end{aligned}$$

Let $\varphi \in C_c^\infty (B_{\frac{1}{4}\rho })$ for $B_{\gamma \rho } \subset B$. With the usual cutoff-function $\eta \in C_c^\infty (B_{\rho })$, $\eta \equiv 1$ on $B_{\frac{1}{2}\rho }$

By the Sobolev inequality for Gagliardo–Norms [23, Theorem 1.6], and the disjoint support [3, Lemma A.1], this implies

$$\begin{aligned} |(-\Delta )^{s}_{\frac{n}{s},B} u[\varphi ]| \precsim C_\Lambda [\varphi ]_{W^{s+t_1-\frac{n}{p_1'},\frac{n}{s}}(\mathbb {R}^n)}. \end{aligned}$$

Since $p_1 > \frac{n}{n-t_1}$, we have $s+t_1-\frac{n}{p_1'} < s$, and the claim of Theorem 1.8 follows from Theorem 1.3 by a covering argument.$\square$

7 Compactness for $\frac{n}{s}$-harmonic maps: proof of Theorem 1.9

From the arguments in [8, Proof of Lemma 2.3.] one has the following:

Proposition 7.1

For $s \in (0,1)$, $p \in (1,\infty )$ let $(u_k)_{k = 1}^\infty \in W^{s,p}(\mathbb {R}^n,{\mathbb S}^{N-1})$, $\Lambda := \sup _{k \in {\mathbb N}} [u_k]_{W^{s,p}(\mathbb {R}^n)} < \infty$ and $\varepsilon _0 > 0$ given. Then up to a subsequence there is $u_\infty \in \dot{W}^{s,p}(\mathbb {R}^n,{\mathbb S}^{N-1})$ and a finite set of points $J = \{a_1,\ldots ,a_l\}$ such that

$$\begin{aligned} u_k \rightharpoonup u_\infty \quad \text{ in } W^{s,p}(\mathbb {R}^n,{\mathbb S}^{N-1}) \text{ as } k \rightarrow \infty , \end{aligned}$$

and for all $x \not \in J$ there is $r = r_x > 0$ so that

$$\begin{aligned} \limsup _{k \rightarrow \infty } [u_k]_{W^{s,p}(B_r(x))} < \varepsilon _0. \end{aligned}$$

This, Theorem 1.8 and the compactness of the embedding $W^{s+\delta ,\frac{n}{s}}(B_r(x)) \hookrightarrow W^{s,\frac{n}{s}}(B_r(x))$ immediately implies that

$$\begin{aligned} u_k \xrightarrow {k \rightarrow \infty } u_\infty \quad \text{ in } W^{s,\frac{n}{s}}_{loc}(\mathbb {R}^n \backslash J). \end{aligned}$$

Appendix A: Useful tools

The following Lemma is used to restrict the fractional p-Laplacian to smaller sets.

Lemma 8.1

(Localization lemma) Let $\Omega _1 \Subset \Omega _2 \Subset \Omega _3 \Subset \Omega \subset \mathbb {R}^n$ be open sets so that $\mathrm{dist\,}(\Omega _1,\Omega _2^c), \mathrm{dist\,}(\Omega _2,\Omega _3^c), \mathrm{dist\,}(\Omega _3,\Omega ^c) > 0$. Let $s \in (0,1)$, $p \in [2,\infty )$.

For any $u \in W^{s,p}(\Omega )$ there exists $\tilde{u} \in W^{s,p}(\mathbb {R}^n)$ so that

• (1) $\tilde{u} - u \equiv const$ in $\Omega _1$

• (2) $\mathrm{supp\,}\tilde{u} \subset \Omega _2$

• (3) $[\tilde{u}]_{W^{s,p}(\mathbb {R}^n)} \precsim \ [u]_{W^{s,p}(\Omega )}$

• (4) For any $t \in (2s-1,s)$,

  $$\begin{aligned} \Vert (-\Delta )^{s}_{p,\Omega _3} \tilde{u}\Vert _{(W^{t,p}_0(\Omega _3))^*} \precsim \Vert (-\Delta )^{s}_{p,\Omega } u\Vert _{(W^{t,p}_0(\Omega ))^*} + [u]_{W^{s,p}(\Omega )}^{p-1}. \end{aligned}$$

The constants are uniform in u and depend only on s, t, p and the sets $\Omega _1$, $\Omega _2$, $\Omega _3$, and $\Omega$.

Proof

Let $\Omega _1 \Subset \Omega$, let $\eta \equiv \eta _{\Omega _1} \in C_c^\infty (\Omega _2)$, $\eta _{\Omega _1} \equiv 1$ on $\Omega _1$. We set

$$\begin{aligned} \tilde{u} := \eta _{\Omega _1}(u-(u)_{\Omega _1}). \end{aligned}$$

Clearly $\tilde{u}$ satisfies property (1) and (2). We have property (3), too:

$$\begin{aligned}{}[\tilde{u}]_{W^{s,p}(\mathbb {R}^n)} \precsim [u]_{W^{s,p}(\Omega )}. \end{aligned}$$

We write

$$\begin{aligned} \tilde{u}(x)-\tilde{u}(y) = \underbrace{\eta (x) (u(x)-u(y))}_{a(x,y)} + \underbrace{(\eta (x)-\eta (y)) (u(y)-(u)_{\Omega _1})}_{b(x,y)}. \end{aligned}$$

Setting

$$\begin{aligned} T(a) := |a|^{p-2} a, \end{aligned}$$

observe that

$$\begin{aligned} |T(a+b) - T(a)| \precsim |b| \left( |a|^{p-2} + |b|^{p-2} \right) . \end{aligned}$$

Also note that

$$\begin{aligned} T(a(x,y)) = \eta ^{p-1}(x) |u(x)-u(y)|^{p-2}(u(x)-u(y)) \end{aligned}$$

We thus have for any $\varphi \in C_c^\infty (\Omega _3)$,

$$\begin{aligned}&(-\Delta )^{s}_{p,\Omega }\tilde{u}[\varphi ]\\&\quad =\int _{\Omega } \int _{\Omega } \frac{|\tilde{u}(x)-\tilde{u}(y)|^{p-2}(\tilde{u}(x)-\tilde{u}(y))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx\ dy\\&\quad = \int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ \eta ^{p-1}(x)\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx\ dy\\&\qquad +\int _{\Omega } \int _{\Omega } \frac{(T(a+b) - T(a))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx\ dy\\&\quad = \int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ ( \eta ^{p-1}(x) \varphi (x)- \eta ^{p-1}(y) \varphi (y))}{|x-y|^{n+sp}}\ dx\ dy\\&\qquad -\int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ (\eta ^{p-1}(x)-\eta ^{p-1}(y))\varphi (y)}{|x-y|^{n+sp}}\ dx\ dy\\&\qquad +\int _{\Omega } \int _{\Omega } \frac{(T(a+b) - T(a))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx\ dy\\&\quad = (-\Delta )^{s}_{p,\Omega }u[\eta ^{p-1}\ \varphi ]\\&\quad \quad -\int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ (\eta ^{p-1}(x)-\eta ^{p-1}(y))\varphi (y)}{|x-y|^{n+sp}}\ dx\ dy\\&\quad \quad +\int _{\Omega } \int _{\Omega } \frac{(T(a+b) - T(a))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx\ dy. \end{aligned}$$

Consequently,

$$\begin{aligned}&| (-\Delta )^{s}_{p,\Omega } \tilde{u}[\varphi ]|\\&\quad \precsim \Vert (-\Delta )^{s}_{p,\Omega }u\Vert _{(W^{t,p}_0(\Omega ))^*}\ [\eta ^{p-1} \varphi ]_{W^{t,p}(\Omega )}\\&\quad \quad + \int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p-1}\ |\eta ^{p-1}(x)-\eta ^{p-1}(y)|\ |\varphi (y)|}{|x-y|^{n+sp}}\ dx\ dy\\&\quad \quad +\int _{\Omega } \int _{\Omega } \frac{|\eta (x)-\eta (y)|\ |u(y)-(u)_{\Omega _1}|\ \eta (x)^{p-2}\ |u(x)-u(y)|^{p-2}\ |\varphi (x)-\varphi (y)|}{|x-y|^{n+sp}}\ dx\ dy\\&\quad \quad +\int _{\Omega } \int _{\Omega } \frac{|\eta (x)-\eta (y)|^{p-1}\ |u(y)-(u)_{\Omega _1}|^{p-1}\ |\varphi (x)-\varphi (y)|}{|x-y|^{n+sp}}\ dx\ dy. \end{aligned}$$

That is for any $t < s$

$$\begin{aligned}&| (-\Delta )^{s}_{p,\Omega } \tilde{u}[\varphi ]|\\&\quad \precsim \Vert (-\Delta )^{s}_{p,\Omega }u\Vert _{(W^{t,p}_0(\Omega ))^*}\ [\eta ^{p-1} \varphi ]_{W^{t,p}(\Omega )}\\&\quad \quad + [u]_{W^{s,p}(\Omega )}^{p-1}\ \left( \int _{\Omega } \int _{\Omega } \frac{|\eta ^{p-1}(x)-\eta ^{p-1}(y)|^p\ |\varphi (y)|^p}{|x-y|^{n+sp}}\ dx\ dy \right) ^{\frac{1}{p}}\\&\quad \quad +[\varphi ]_{W^{t,p}(\Omega )}\ [u]_{W^{s,p}(\Omega )}^{p-2} \left( \int _{\Omega } \int _{\Omega _2} \frac{|\eta (x)-\eta (y)|^{p}\ |u(y)-(u)_{\Omega _1}|^p}{|x-y|^{n+(2s-t)p}}\ dx\ dy \right) ^{\frac{1}{p}}\\&\quad \quad +[\varphi ]_{W^{t,p}(\Omega )} \left( \int _{\Omega } \int _{\Omega _2} \frac{|\eta (x)-\eta (y)|^{p}\ |u(y)-(u)_{\Omega _1}|^p}{|x-y|^{n+(2s-t)p}}\ dx\ dy \right) ^{\frac{p-1}{p}}. \end{aligned}$$

Since $\eta$ is bounded and Lipschitz, $\mathrm{supp\,}\eta \subset \Omega _2$, and $\varphi \in C_c^\infty (\Omega _3)$ we have that

$$\begin{aligned}{}[\eta ^{p-1} \varphi ]_{W^{t,p}(\Omega )} \precsim [\varphi ]_{W^{t,p}(\mathbb {R}^n)}. \end{aligned}$$

Also, choosing some bounded $\Omega _4 \Subset \Omega$ so that $\Omega _3 \Subset \Omega _4$,

$$\begin{aligned}&\int _{\Omega } \int _{\Omega } \frac{|\eta ^{p-1}(x)-\eta ^{p-1}(y)|^p\ |\varphi (y)|^p}{|x-y|^{n+sp}}\ dx\ dy\\&\quad \precsim \int _{\Omega _3} \int _{\Omega _4} |x-y|^{(1-s)p-n}\ \ dx\ |\varphi (y)|^p dy\\&\quad \quad +\int _{\Omega _3} \int _{\mathbb {R}^n\backslash \Omega _4} |x-y|^{-n-sp}\ \ dx\ |\varphi (y)|^p dy\\&\quad \precsim \Vert \varphi \Vert _{p}^p \precsim [\varphi ]_{W^{t,p}(\mathbb {R}^n)}^p. \end{aligned}$$

Finally, using Lipschitz continuity of $\eta$ and that $2s-1 < t < s$

$$\begin{aligned}&\int _{\Omega } \int _{\Omega _2} \frac{|\eta (x)-\eta (y)|^{p}\ |u(y)-(u)_{\Omega _1}|^p}{|x-y|^{n+(2s-t)p}}\ dx\ dy\\&\quad \precsim \int _{\Omega _3} |u(y)-(u)_{\Omega _1}|^p \int _{\Omega _2} |x-y|^{-n+(t+1-2s)p}\ dx\ dy\\&\quad \quad +\int _{\Omega \backslash \Omega _3} |u(y)-(u)_{\Omega _1}|^p \int _{\Omega _2} \frac{1 }{|x-y|^{n+sp}}\ dx\ dy\\&\quad \precsim \int _{\Omega _1} \int _{\Omega _3} |u(y)-u(z)|^p\ dy\ dz\\&\quad \quad +\int _{\Omega _1} \int _{\Omega \backslash \Omega _3} |u(y)-u(z)|^p \int _{\Omega _2} \frac{1 }{|x-y|^{n+sp}}\ dx\ dy\ dz\\ \end{aligned}$$

Note that for $x,z \in \Omega _2$ and $y \in \Omega _3^c$ we have that $|x-y| \approx |y-z|$, and since $\Omega _1,\Omega _2,\Omega _3$ are bounded we then have

$$\begin{aligned} \int _{\Omega } \int _{\Omega _2} \frac{|\eta (x)-\eta (y)|^{p}\ |u(y)-(u)_{\Omega _1}|^p}{|x-y|^{n+(2s-t)p}}\ dx\ dy \precsim [u]_{W^{s,p}(\Omega )}. \end{aligned}$$

Thus we have shown that for any $\varphi \in C_c^\infty (\Omega _3)$,

$$\begin{aligned} | (-\Delta )^{s}_{p,\Omega } \tilde{u}[\varphi ]| \precsim \left( \Vert (-\Delta )^{s}_{p,\Omega }u\Vert _{(W^{t,p}_0(\Omega ))^*} +[u]_{W^{s,p}(\Omega )}^{p-1} \right) \ [\varphi ]_{W^{t,p}(\mathbb {R}^n)}. \end{aligned}$$

Since moreover, $\mathrm{supp\,}\tilde{u} \subset \Omega _2$, for any $\varphi \in C_c^\infty (\Omega _3)$,

$$\begin{aligned} | (-\Delta )^{s}_{p,\Omega _3} \tilde{u}[\varphi ]| \precsim | (-\Delta )^{s}_{p,\Omega } \tilde{u}[\varphi ]| + [u]_{W^{s,p}(\Omega )}^{p-1}\ [\varphi ]_{W^{t,p}(\mathbb {R}^n)}, \end{aligned}$$

we get the claim. $\square$

The next Lemma estimates the $W^{s,p}$-norm in terms of the fractional p-Laplacian.

Lemma 8.2

Let $B \subset \mathbb {R}^n$ be a ball and 4B the concentric ball with four times the radius. Then for any $\delta > 0$, $[u]_{W^{s,p}(B)}^{p}$ can be estimated by

$$\begin{aligned}&\delta ^p [u]_{W^{s,p}(4B)}^{p}\\&\quad + \frac{C}{\delta ^{p'}} \left( \sup _{\varphi } \int _{4B}\int _{4B} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx \ dy \right) ^{\frac{p}{p-1}}\\&\quad + \frac{C}{\delta ^{p'}}\ \mathrm{diam\,}(B)^{-sp}\ \int \limits _{4B} |u(x)-(u)_B|^p\ dx \end{aligned}$$

where the supremum is over all $\varphi \in C_c^\infty (2B)$ and $[\varphi ]_{W^{s,p}(\mathbb {R}^n)} \le 1$.

Proof

Let $\eta \in C_c^\infty (2B)$, $\eta \equiv 1$ in B be the usual cutoff function in 2B.

$$\begin{aligned} \psi (x) := \eta (x) (u(x)-(u)_{B}), \quad \text{ and } \quad \varphi (x) := \eta ^2(x) (u(x)-(u)_{B}). \end{aligned}$$

Then,

$$\begin{aligned}{}[\psi ]_{W^{s,p}(\mathbb {R}^n)} + [\varphi ]_{W^{s,p}(\mathbb {R}^n)} \precsim [u]_{W^{s,p}(2B)}. \end{aligned}$$

(8.1)

We have

$$\begin{aligned}{}[u]_{W^{s,p}(B)}^{p} \le \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2} (\psi (x)-\psi (y))\ (\psi (x)-\psi (y))}{|x-y|^{n+sp}} dx\ dy. \end{aligned}$$

Now we observe

$$\begin{aligned} (\psi (x) - \psi (y))^2= & {} (\psi (x) - \psi (y)) (\eta (x) - \eta (y))(u(x)-(u)_{B})\\&+ \psi (x)(\eta (y)-\eta (x))\ (u(x)-u(y))\\&+ (\varphi (x) - \varphi (y))(u(x)-u(y)). \end{aligned}$$

That is,

$$\begin{aligned}{}[u]_{W^{s,p}(B)}^{p} \precsim I + II + III, \end{aligned}$$

with

$$\begin{aligned} I:= & {} \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2} (u(x)-u(y))(\varphi (x)-\varphi (y))}{|x-y|^{n+sp}} dx\ dy,\\ II:= & {} \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2} |\eta (x)-\eta (y)|\ |\psi (x)-\psi (y)|}{|x-y|^{n+sp}}\ |u(x)-(u)_B|\ dx\ dy,\\ III:= & {} \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-1} |\eta (x)-\eta (y)|}{|x-y|^{n+sp}}\ |\psi (x)| dx\ dy. \end{aligned}$$

With (8.1),

$$\begin{aligned} I \le [u]_{W^{s,p}(4B)}\ \sup _{[\varphi ]_{W^{s,p}(\mathbb {R}^n)} \le 1} \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx \ dy. \end{aligned}$$

As for II,

$$\begin{aligned} II \precsim \Vert \nabla \eta \Vert _{\infty }\ \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2} |\psi (x)-\psi (y)|\ |u(x)-(u)_B| }{|x-y|^{n+sp-1}}\ \ dx\ dy. \end{aligned}$$

For any $t_2 > 0$ so that $t_2 = 1-s$, we have with Hölder’s inequality

Since $t_2 > 0$,

$$\begin{aligned} \int \limits _{4B}\int \limits _{4B} \frac{ |u(x)-(u)_B|^p }{|x-y|^{n- t_2p}}\ \ dx\ dy \precsim (\mathrm{diam\,}B) ^{t_2p}\ \int \limits _{4B} |u(x)-(u)_B|^p\ dx. \end{aligned}$$

So using again (8.1), we arrive at

$$\begin{aligned} II \precsim \mathrm{diam\,}(B)^{-s}\ [u]_{W^{s,p}(4B)}^{p-1}\ \left( \int \limits _{4B} |u(x)-(u)_B|^p\ dx \right) ^{\frac{1}{p}}. \end{aligned}$$

III can be estimated the same way as II, and we have the following estimate for $[u]_{W^{s,p}(B)}^{p}$

$$\begin{aligned}&[u]_{W^{s,p}(4B)}\ \sup _{\varphi } \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx \ dy\\&\quad +[u]_{W^{s,p}(4B)}^{p-1}\ \mathrm{diam\,}(B)^{-s}\ \left( \int \limits _{4B} |u(x)-(u)_B|^p\ dx \right) ^{\frac{1}{p}}. \end{aligned}$$

We conclude with Young’s inequality. $\square$

The next Proposition follows immediately from Jensen’s inequality and the definition of $[u]_{W^{t,p}(\lambda B)}^p$.

Proposition 8.3

(A Poincaré type inequality) Let B be a ball and for $\lambda \ge 1$ let $\lambda B$ be the concentric ball with $\lambda$ times the radius. Then for any $t \in (0,1)$, $p \in (1,\infty )$,

$$\begin{aligned} \int \limits _{\lambda B} |u(x)-(u)_B|^p\ dx \precsim \lambda ^{n+tp} \mathrm{diam\,}(B)^{t p} \ [u]_{W^{t,p}(\lambda B)}^p. \end{aligned}$$