Radial extensions in fractional Sobolev spaces

Abstract

Given $f:\partial (-1,1)^n\rightarrow {{\mathbb {R}}}$, consider its radial extension $Tf(X):=f(X/\Vert X\Vert _{\infty })$, $\forall \, X\in [-1,1]^n{\setminus }\{0\}$. Brezis and Mironescu (RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95:121–143, 2001), stated the following auxiliary result (Lemma D.1). If $0<s<1$, $1< p<\infty$ and $n\ge 2$ are such that $1<sp<n$, then $f\mapsto Tf$ is a bounded linear operator from $W^{s,p}(\partial (-1,1)^n)$ into $W^{s,p}((-1,1)^n)$. The proof of this result contained a flaw detected by Shafrir. We present a correct proof. We also establish a variant of this result involving higher order derivatives and more general radial extension operators. More specifically, let B be the unit ball for the standard Euclidean norm $|\ |$ in ${{\mathbb {R}}}^n$, and set $U_af(X):=|X|^a\, f(X/|X|)$, $\forall \, X\in {\overline{B}}{\setminus }\{0\}$, $\forall \,f:\partial B\rightarrow {{\mathbb {R}}}$. Let $a\in {{\mathbb {R}}}$, $s>0$, $1\le p<\infty$ and $n\ge 2$ be such that $(s-a)p<n$. Then $f\mapsto U_af$ is a bounded linear operator from $W^{s,p}(\partial B)$ into $W^{s,p}(B)$.

In [1], the first two authors stated the following

FormalPara Lemma 1

[1, Lemma D.1] Let $0<s<1$, $1<p<\infty$ and $n\ge 2$ be such that $1<sp<n$. Let

$$\begin{aligned} Q:=(-1,1)^n. \end{aligned}$$

(1)

Set

$$\begin{aligned} T f(X):= f(X/\Vert X\Vert _\infty ), \quad \forall \,X\in {\overline{Q}}{\setminus }\{0\},\quad \forall \,f:\partial Q\rightarrow {{\mathbb {R}}}; \end{aligned}$$

(2)

here, $\Vert \ \Vert _\infty$ is the sup norm in ${{\mathbb {R}}}^n$. Then $f\mapsto T f$ is a bounded linear operator from $W^{s,p} (\partial Q)$ into $W^{s,p} (Q)$.

The argument presented in [1] does not imply the conclusion of Lemma 1. Indeed, it is established in [1] (see estimate (D.3) there) that

$$\begin{aligned} |Tf|_{W^{s,p}(Q)}^p\le C\int _{\partial Q}\int _{\partial Q}\frac{|f(x)-f(y)|^p}{\Vert x-y\Vert _\infty ^{n+sp}}\, d\sigma (x) d\sigma (y). \end{aligned}$$

However, this does not imply the desired conclusion in Lemma 1, for which we need the stronger estimate

$$\begin{aligned} |Tf|_{W^{s,p}(Q)}^p\le C\int _{\partial Q}\int _{\partial Q}\frac{|f(x)-f(y)|^p}{\Vert x-y\Vert _\infty ^{n-1+sp}}\, d\sigma (x) d\sigma (y). \end{aligned}$$

In what follows, we establish the following slight generalization of Lemma 1.

FormalPara Lemma 2

Let $0<s\le 1$, $1\le p<\infty$ and $n\ge 2$ be such that $sp<n$. Let Q, T be as in (1), (2). Then $f\mapsto T f$ is a bounded linear operator from $W^{s,p} (\partial Q)$ into $W^{s,p} (Q)$.

Lemma 2 can be generalized beyond one derivative, but for this purpose it is necessary to work on unit spheres arising from norms smoother that $\Vert \ \Vert _\infty$. We consider for example maps $f:\partial B\rightarrow {{\mathbb {R}}}$, with

$$\begin{aligned} B:=\,\text {the Euclidean unit ball in }{{\mathbb {R}}}^n. \end{aligned}$$

(3)

For $a\in {{\mathbb {R}}}$, set

$$\begin{aligned} U_a f(X):=|X|^a f(X/|X|),\quad \forall \,X\in {\overline{B}}{\setminus }\{0\},\quad \forall \,f:\partial B\rightarrow {{\mathbb {R}}}; \end{aligned}$$

(4)

here, $|\ |$ is the standard Euclidean norm in ${{\mathbb {R}}}^n$.

We will prove the following

FormalPara Lemma 3

Let $a\in {{\mathbb {R}}}$, $s>0$, $1\le p<\infty$ and $n\ge 2$ be such that $(s-a)p<n$. Then $f\mapsto U_a f$ is a bounded linear operator from $W^{s,p} (\partial B)$ into $W^{s,p} (B)$.

It is possible to establish directly Lemma 2 by adapting some arguments presented in Step 3 in the proof of Lemma 4.1 in [2]. However, we will derive it from Lemma 3.

FormalPara Proof of Lemma 2 using Lemma 3

Let

$$\begin{aligned} \varPhi :{{\mathbb {R}}}^n\rightarrow {{\mathbb {R}}}^n,\quad \varPhi (X):={\left\{ \begin{array}{ll} \displaystyle \frac{|X|}{\Vert X\Vert _\infty }\, X,&{}\text {if }X\ne 0\\ 0,&{}\text {if }X=0\end{array}\right. },\quad \varLambda :=\varPhi _{|{\overline{B}}}\;\text { and }\;\varPsi :=\varPhi _{|\partial B}. \end{aligned}$$

Clearly,

$$\begin{aligned} \varLambda :{\overline{B}}\rightarrow {\overline{Q}},\ \varPsi :\partial B\rightarrow \partial Q\text { are bi-Lipschitz homeomorphisms} \end{aligned}$$

(5)

and

$$\begin{aligned} T f =[U_0 (f\circ \varPsi )]\circ \varLambda ^{-1}. \end{aligned}$$

(6)

Using (5) and the fact that $0<s\le 1$, we find that

$$\begin{aligned} f\mapsto f\circ \varPsi \text { is a bounded linear operator from }W^{s,p}(\partial Q)\text { into }W^{s,p}(\partial B) \end{aligned}$$

(7)

and

$$\begin{aligned} g\mapsto g\circ \varLambda ^{-1}\text { is a bounded linear operator from }W^{s,p}(B)\text { into }W^{s,p}(Q). \end{aligned}$$

(8)

We obtain Lemma 2 from (6)–(8) and Lemma 3 (with $a=0$). The same argument shows that the conclusion of Lemma 2 holds for the unit sphere and ball of any norm in ${{\mathbb {R}}}^n$. $\square$

FormalPara Proof of Lemma 3

Consider a, s, p and n such that

$$\begin{aligned} a\in {{\mathbb {R}}},\ s>0,\quad 1\le p<\infty ,\quad n\ge 2 \; \text { and } \; (s-a)p<n. \end{aligned}$$

(9)

Considering spherical coordinates on B, we obtain that

$$\begin{aligned} \Vert U_a f\Vert _{L^p(B)}^p= & {} \int _0^1\int _{\partial B}r^ {n-1}|U_af(r\, x)|^p\, d\sigma (x) dr \nonumber \\= & {} \int _0^1\int _{\partial B} r^{n-1+ap}|f(x)|^p\, d\sigma (x) dr \nonumber \\= & {} \frac{1}{n+ap} \Vert f\Vert _{L^p(\partial B)}^p. \end{aligned}$$

(10)

Here, we have used the fact that, by (9), we have $n+ap>n-(s-a)p>0$.

In view of (10), it suffices to establish the estimate

$$\begin{aligned} |U_a f|_{W^{s,p}(B)}^p\le C\, \Vert f\Vert _{W^{s,p}(\partial B)}^p,\quad \forall \,f\in W^{s,p}(\partial B), \end{aligned}$$

(11)

for some appropriate $C=C_{a,s,p,n}$ and semi-norm $|\ |_{W^{s,p}}$ on $W^{s,p}(B)$.

Step 1. Proof of (11) when$0<s<1$. We consider the standard Gagliardo semi-norm on $W^{s,p}(B)$. We have

$$\begin{aligned} \begin{aligned} |U_af|_{W^{s,p}(B)}^p&=\int _B\int _B \frac{|U_af(X)-U_af(Y)|^p}{|X-Y|^{n+sp}}\, dX dY\\&=\int _0^1\int _0^1\int _{\partial B}\int _{\partial B}r^{n-1}\rho ^{n-1} \frac{|U_a f(r\, x)-U_a f(\rho \, y)|^p}{|r\, x-\rho \, y|^{n+s p}}\, d\sigma (x)d\sigma (y) dr d\rho \\&=\int _0^1\int _0^1\int _{\partial B}\int _{\partial B}r^{n-1}\rho ^{n-1} \frac{|r^a\, f(x)-\rho ^a\, f(y)|^p}{|r\, x-\rho \, y|^{n+s p}}\, d\sigma (x)d\sigma (y) dr d\rho \\&=2\int _{\partial B}\int _{\partial B}\int _0^1\int _0^r r^{n-1}\rho ^{n-1} \frac{|r^a\, f(x)-\rho ^a\, f(y)|^p}{|r\, x-\rho \, y|^{n+s p}}\, d\rho dr d\sigma (x) d\sigma (y).\end{aligned} \end{aligned}$$

With the change of variable $\rho =t\, r$, $t\in [0,1]$, we find that

$$\begin{aligned} \begin{aligned} |U_af|_{W^{s,p}(B)}^p&=2\int _0^1 r^{n-(s-a)p-1}\, dr \int _{\partial B}\int _{\partial B}\int _0^1 t^{n-1} \frac{|f(x)-t^a\, f(y)|^p}{|x -t\, y|^{n+sp}}\, dt d\sigma (x) d\sigma (y)\\&=\frac{2}{n-(s-a)\, p}\int _{\partial B}\int _{\partial B}\int _0^1 k(x,y,t)\, dt d\sigma (x) d\sigma (y), \end{aligned} \end{aligned}$$

with

$$\begin{aligned} k(x, y, t):= t^{n-1}\,\frac{|f(x)-t^a\, f(y)|^p}{|x -t\, y|^{n+sp}},\quad \forall \,x,\, y\in \partial B,\, \forall \,t\in [0,1]. \end{aligned}$$

In order to complete this step, it thus suffices to establish the estimates

$$\begin{aligned} I_1:= & {} \int _{\partial B}\int _{\partial B}\int _0^{1/2} k(x,y, t)\, dt d\sigma (x) d\sigma (y)\le C \Vert f\Vert _{L^p(\partial B)}^p, \end{aligned}$$

(12)

$$\begin{aligned} I_2:= & {} \int _{\partial B}\int _{\partial B}\int _{1/2}^1\frac{|f(x)-f(y)|^p}{|x-t\, y|^{n+sp}}\, dt d\sigma (x) d\sigma (y)\le C |f|_{W^{s,p}(\partial B)}^p, \end{aligned}$$

(13)

$$\begin{aligned} I_3:= & {} \int _{\partial B}\int _{\partial B}\int _{1/2}^1\frac{|(1-t^a)\, f(y)|^p}{|x-t\, y|^{n+sp}}\, dt d\sigma (x) d\sigma (y)\le C \Vert f\Vert _{L^{p}(\partial B)}^p; \end{aligned}$$

(14)

here, $|\ |_{W^{s,p}(\partial B)}$ is the standard Gagliardo semi-norm on $\partial B$.

In the above and in what follows, C denotes a generic finite positive constant independent of f, whose value may change with different occurrences.

Using the obvious inequalities

$$\begin{aligned} |x-t\, y|\ge 1-t\ge 1/2,\quad \forall \,x, y\in \partial B,\ \forall \,t\in [0,1/2],\\ |f(x)-t^a\, f(y)|\le (1+t^a)\, (|f(x)|+|f(y)|), \end{aligned}$$

and the fact that, by (9), we have $n+ap>0$, we find that

$$\begin{aligned} I_1\le C\int _0^{1/2}(t^{n-1}+t^{n-1+ap})\, dt\, \Vert f\Vert _{L^p(\partial B)}^p\le C \Vert f\Vert _{L^p(\partial B)}^p, \end{aligned}$$

so that (12) holds.

In order to obtain (13), it suffices to establish the estimate

$$\begin{aligned} \int _{1/2}^1 \frac{1}{|x-t\, y|^{n+sp}}\, dt\le \frac{C}{|x-y|^{n-1+sp}},\quad \forall \,x, y\in \partial B. \end{aligned}$$

(15)

Set $A:=\langle x, y\rangle \in [-1,1]$. If $A\le 0$, then $|x-t\, y|\ge 1$, $\forall \,t\in [1/2,1]$, and then (15) is clear. Assuming $A\ge 0$ we find, using the change of variable $t=A+(1-A^2)^{1/2}\, \tau$,

$$\begin{aligned} \begin{aligned} \int _{1/2}^1 \frac{1}{|x-t\, y|^{n+sp}}\, dt&\le \int _{{\mathbb {R}}}\frac{1}{|x-t\, y|^{n+sp}}\, dt\\&=\int _{{\mathbb {R}}}\frac{1}{(t^2+1-2 A\, t)^{(n+sp)/2}}\, dt\\&=\frac{1}{(1-A^2)^{(n-1+sp)/2}}\, \int _{{\mathbb {R}}}\frac{1}{(\tau ^2+1)^{(n+sp)/2}}\, d\tau \\&=\frac{C}{(1-A^2)^{(n-1+sp)/2}}\le \frac{C}{(2-2 A)^{(n-1+sp)/2}}\\&=\frac{C}{|x-y|^{n-1+sp}}, \end{aligned} \end{aligned}$$

and thus (15) holds again. This completes the proof of (13).

In order to prove (14), we note that

$$\begin{aligned} |1-t^a|^p\le C(1-t)^p,\quad \forall \,t\in [1/2,1], \end{aligned}$$

and that the integral

$$\begin{aligned} J:=\int _{1/2}^1 \int _{\partial B}\frac{ (1-t)^p}{|x-t\, y|^{n+sp}}\, d\sigma (x) dt \end{aligned}$$

does not depend on $y\in \partial B$.

By the above, we have

$$\begin{aligned} I_3\le & {} C\int _{1/2}^1\int _{\partial B}\int _{\partial B}\frac{(1-t)^p|f(y)|^p}{|x-t\, y|^{n+sp}}\, d\sigma (x) d\sigma (y) dt\\= & {} C\, J\, \Vert f\Vert _{L^p(\partial B)}^p, \end{aligned}$$

and thus (14) amounts to proving that $J<\infty$. Since J does not depend on y, we may assume that $y=(0,\ldots , 0, 1)$. Expressing J in spherical coordinates and using the change of variable $t=1-\tau$, $\tau \in [0,1/2]$, we find that

$$\begin{aligned} J=C\int _{1/2}^1\int _0^\pi \frac{\tau ^p \, \sin ^{n-1}\theta }{(\tau ^2+ 4(1-\tau )\sin ^2\theta /2)^{(n+sp)/2}}\, d\theta d\tau . \end{aligned}$$

When $\tau \in [0,1/2]$ and $\theta \in [0,\pi ]$, we have

$$\begin{aligned} \begin{aligned} \frac{\tau ^p \, \sin ^{n-1}\theta }{(\tau ^2+ 4(1-\tau )\sin ^2\theta /2)^{(n+sp)/2}}&\le C \frac{\tau ^p \, \sin ^{n-1}\theta }{(\tau + \sin \theta /2)^{n+sp}}\\&\le C \frac{\tau ^p \, \sin ^{n-1}\theta /2\, \cos \theta /2}{(\tau + \sin \theta /2)^{n+sp}}\\&\le C (\tau + \sin \theta /2)^{p-sp-1}\, \cos \theta /2. \end{aligned} \end{aligned}$$

Inserting the last inequality into the formula of J, we find that

$$\begin{aligned} \begin{aligned} J&\le C\int _0^{1/2}\int _0^\pi (\tau + \sin \theta /2)^{p-sp-1}\, \cos \theta /2\, d\theta d\tau \\ {}&= C\int _0^{1/2}\int _0^1(\tau + \xi )^{p-sp-1}\, d\xi d\tau <\infty , \end{aligned} \end{aligned}$$

the latter inequality following from $p-sp>0$. This completes the proof of (14) and Step 1.

Step 2. Proof of (11) when$s\ge 1$. We will reduce the case $s\ge 1$ to the case $0\le s<1$. Using the linearity of $f\mapsto U_a f$ and a partition of unity, we may assume with no loss of generality that ${\text {supp}}f$ is contained in a spherical cap of the form $\{ x\in \partial B;\, |x-\mathbf{e}|<\varepsilon \}$ for some $\mathbf{e}\in \partial B$ and sufficiently small $\varepsilon$. We may further assume that $\mathbf{e}=(0,0,\ldots , 0, 1)$, and thus

$$\begin{aligned} f\in W^{s,p}(\partial B ; {{\mathbb {R}}}),\quad {\text {supp}}\;f\subset {{\mathcal {E}}}:=\{ x\in \partial B;\, |x- (0,0,\ldots , 0, 1)|<\varepsilon \}. \end{aligned}$$

(16)

Let

$$\begin{aligned} \mathcal{S}:=\{ x\in \partial B;\, |x-(0,0,\ldots , 0, 1)|\le 2\varepsilon \} \quad \text { and } \quad \mathcal{H}:={{\mathbb {R}}}^{n-1}\times \{1\}. \end{aligned}$$

Consider the projection $\varTheta$ with vertex 0 of

$$\begin{aligned} {{\mathbb {R}}}^n_+:=\{ X=(X', X_n)\in {{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}};\, X_n>0\} \end{aligned}$$

onto $\mathcal{H}$, given by the formula $\varTheta (X', X_n)=(X'/X_n,1)$. The restriction $\Pi$ of $\varTheta$ to $\mathcal{S}$ maps $\mathcal{S}$ onto $\mathcal{N}:=\mathcal{B}\times \{1\}$, with

$$\begin{aligned} \mathcal{B}:=\{ X'\in {{\mathbb {R}}}^{n-1};\ |X'|\le r:=2\varepsilon \sqrt{1-\varepsilon ^2}/(1-2\varepsilon ^2)\}, \end{aligned}$$

and is a smooth diffeomorphism between these two sets. We choose $\varepsilon$ such that $r=1/2$, and thus $\mathcal{B}\subset \{ X'\in {{\mathbb {R}}}^{n-1};\, \Vert X'\Vert _\infty \le 1/2\}$.

Set

$$\begin{aligned} g(X'):={\left\{ \begin{array}{ll}|(X',1)|^a\, f(\Pi ^{-1}(X', 1)),&{} \text {if } X'\in \mathcal{B}\\ 0,&{}\text {otherwise} \end{array}\right. }. \end{aligned}$$

(17)

By the above, there exist $C, C'>0$ such that for every $f\in W^{s,p}(\partial B)$ satisfying (16), the function g defined in (17) satisfies

$$\begin{aligned} C\Vert g\Vert _{W^{s,p}({{\mathbb {R}}}^{n-1})}\le \Vert f\Vert _{W^{s,p}(\partial B)}\le C'\Vert g\Vert _{W^{s,p}({{\mathbb {R}}}^{n-1})}. \end{aligned}$$

(18)

On the other hand, set $\mathcal{C}:=\{ (t\, Y', t);\, Y'\in \mathcal{B},\, t>0\}$ and

$$\begin{aligned} V_a g (X', X_n):={\left\{ \begin{array}{ll} (X_n)^a\, g(X'/X_n),&{}\text {if }(X', X_n)\in \mathcal{C}\\ 0,&{}\text {otherwise} \end{array}\right. }. \end{aligned}$$

Then we have $U_a f(X', X_n)= V_a g (X', X_n)$, $\forall \,(X', X_n)\in {\overline{B}}{\setminus }\{ 0\}$.

Write now $s=m+\sigma$, with $m\in {{\mathbb {N}}}$ and $0\le \sigma <1$. When $s=m$, we consider, on $W^{s,p}(B)$, the semi-norm

$$\begin{aligned} |F|_{W^{s,p}(B)}^p=\sum _{\begin{array}{c} \alpha \in {{\mathbb {N}}}^n{\setminus }\{0\}\\ |\alpha |\le m \end{array}}\Vert \partial ^\alpha F\Vert _{L^p(B)}^p. \end{aligned}$$

(19)

When s is not an integer, we consider the semi-norm

$$\begin{aligned} |F|_{W^{s,p}(B)}^p=\sum _{\begin{array}{c} \alpha \in {{\mathbb {N}}}^n{\setminus }\{0\}\\ |\alpha |\le m \end{array}}\Vert \partial ^\alpha F\Vert _{L^p(B)}^p+\sum _{\begin{array}{c} \alpha \in {{\mathbb {N}}}^n\\ |\alpha |= m \end{array}}|\partial ^\alpha F|_{W^{\sigma , p}(B)}^p \end{aligned}$$

(20)

(the semi-norm on ${W^{\sigma , p}(B)}$ is the standard Gagliardo one.)

By the above discussion, in order to obtain (11) it suffices to establish the estimate

$$\begin{aligned} |V_a g|_{W^{s,p}(B)}^p\le C\, \Vert g\Vert _{W^{s,p}({{\mathbb {R}}}^{n-1})}^p, \quad \forall \,g\in W^{s,p}({{\mathbb {R}}}^{n-1})\text { with }{\text {supp}}\, g\subset \mathcal{B}. \end{aligned}$$

(21)

Let $\alpha \in {{\mathbb {N}}}^n{\setminus }\{0\}$ be such that $|\alpha |\le m$. By a straightforward induction on $|\alpha |$, the distributional derivative $\partial ^\alpha [V_a g]$ satisfies

$$\begin{aligned} \partial ^\alpha [V_a g] (X',X_n)=\sum _{|\beta '|\le |\alpha |}V_{a-|\alpha |}[P_{\alpha , \beta '}\, \partial ^{\beta '}g] (X', X_n)\quad \text {in }{{\mathcal {D}}}'(B{\setminus }\{0\}), \end{aligned}$$

(22)

for some appropriate polynomials $P_{\alpha , \beta '}(Y')$, $Y'\in {{\mathbb {R}}}^{n-1}$, depending only on $a\in {{\mathbb {R}}}$, $\alpha \in {{\mathbb {N}}}^n$ and $\beta '\in {{\mathbb {N}}}^{n-1}$.

Thanks to the fact that $g(X'/X_n)=0$ when $(X',X_n)\not \in \mathcal{C}$, we find that for any such $\alpha$ we have

$$\begin{aligned} \begin{aligned} \int _B |\partial ^\alpha [V_a g] |^p\, dx&\le C\sum _{|\beta '|\le |\alpha |}\int _{\mathcal{C}\cap Q} (X_n)^{(a-|\alpha |) p} |\partial ^{\beta '}g(X'/X_n)|^p\, dX' dX_n\\&=\frac{C}{n+(a-|\alpha |)p} \sum _{|\beta '|\le |\alpha |}\int _{ \mathcal{B}}|\partial ^{\beta '}g(Y')|^p\, dY'. \end{aligned} \end{aligned}$$

(23)

Here, we rely on

$$\begin{aligned} \displaystyle \int _0^1 (X_n)^{n-1+(a-|\alpha |)p}\, dX_n=\frac{1}{n+(a-|\alpha |)p}<\infty , \end{aligned}$$

thanks to the assumption (9), which implies that $(|\alpha |-a)p <n$.

Using (23), the fact that $V_a g\in W^{m,p}_{loc}(B{\setminus } \{0\})$ and the assumption that $n\ge 2$, we find that the equality (22) holds also in $\mathcal{D}'(B)$, that $V_a g\in W^{m, p}(B)$ and that

$$\begin{aligned} \Vert V_a g\Vert _{W^{m,p}(B)}^p\le C \Vert g\Vert _{W^{m,p}({{\mathbb {R}}}^{n-1})}^p, \quad \forall \,g\in W^{m,p}({{\mathbb {R}}}^{n-1}) \; \text { with } {\text {supp}}g\subset \mathcal{B}. \end{aligned}$$

(24)

In particular, Eq. (21) holds when s is an integer.

Assume next that s is not an integer. In view of (18), (22) and (24), estimate (21) will be a consequence of

$$\begin{aligned} \begin{aligned} |V_b [P h]|_{W^{\sigma , p}(B)}^p&\le C\, \Vert h\Vert _{W^{\sigma , p}({{\mathbb {R}}}^{n-1})}^p, \quad \forall \,h\in W^{\sigma , p}({{\mathbb {R}}}^{n-1})\\&\qquad \text { with }{\text {supp}}\;h\subset \mathcal{B}, \end{aligned} \end{aligned}$$

(25)

under the assumptions

$$\begin{aligned} 0<\sigma<1,\quad 1\le p<\infty ,\quad n\ge 2,\quad (\sigma -b)p<n \end{aligned}$$

(26)

and

$$\begin{aligned} P\in C^\infty ({{\mathbb {R}}}^{n-1}). \end{aligned}$$

(27)

(Estimate (25) is applied with $b:=a-m$, $P:=P_{\alpha , \beta '}$ and $h:=\partial ^{\beta '} g$.)

In turn, estimate (25) follows from Step 1. Indeed, consider $k:\partial B\rightarrow {{\mathbb {R}}}$ such that ${\text {supp}}\;k\subset \mathcal{B}$ and $U_b k=V_b [Ph]$. (The explicit formula of k can be obtained by “inverting” the formula (17).) By Step 1 and (18), we have

$$\begin{aligned} \begin{aligned} |V_b [Ph]|_{W^{s,p}(B)}^p&= |U_b k|_{W^{s,p}(B)}^p\le C\Vert k\Vert _{W^{s,p}(\partial B)}^p \le C\Vert Ph\Vert _{W^{s,p}({{\mathbb {R}}}^{n-1})}^p\\&\le C\Vert h\Vert _{W^{s,p}({{\mathbb {R}}}^{n-1})}^p. \end{aligned} \end{aligned}$$

This completes Step 2 and the proof of Lemma 3.

$\square$

Finally, we note that the assumptions of Lemma 3 are optimal in order to obtain that $U_a f\in W^{s,p}(B)$.

FormalPara Lemma 4

Let $a\in {{\mathbb {R}}}$, $s>0$, $1\le p<\infty$ and $n\ge 2$. Assume that for some measurable function $f:\partial B\rightarrow {{\mathbb {R}}}$ we have $U_a f\in W^{s,p}(B)$. Then:

• 1. $f\in W^{s,p}(\partial B)$.

• 2. If, in addition, $U_a f$ is not a polynomial, we deduce that $(s-a)p<n$.

FormalPara Proof

• 1. Let $G: (1/2, 1)\times \partial B\rightarrow {{\mathbb {R}}}$, $G(r, x):=r^{-a}\, U_a f(r\, x)$. If $U_af\in W^{s,p}(B)$, then $G\in W^{s,p}((1/2, 1)\times \partial B)$. In particular, we have $G(r, \cdot )\in W^{s, p}(\partial B)$ for a.e. r. Noting that $G (r,x)=f(x)$, we find that $f\in W^{s,p}(\partial B)$.

• 2. Let

  $$\begin{aligned} \varOmega _j:=\{ X\in {{\mathbb {R}}}^n;\, 2^{-j-1}<|X|<2^{-j}\},\quad j\in {{\mathbb {N}}}. \end{aligned}$$

We consider on each $\varOmega _j$ a semi-norm as in (19), (20). Assuming that $U_a f$ is not a polynomial, we have $|U_a f|_{W^{s,p}(\varOmega _0)}>0$. By scaling and the homogeneity of $U_a f$, we have

$$\begin{aligned} |U_a f|_{W^{s,p}(\varOmega _j)}^p=2^{j[(s-a)p-n]}|U_a f|_{W^{s,p}(\varOmega _0)}^p. \end{aligned}$$

Assuming that $U_a f\in W^{s,p}(B)$, we find that

$$\begin{aligned} \infty>|U_a f|_{W^{s,p}(B)}^p\ge \sum _{j\ge 0} |U_a f|_{W^{s,p}(\varOmega _j)}^p=\sum _{j\ge 0}2^{j[(s-a)p-n]}|U_a f|_{W^{s,p}(\varOmega _0)}^p>0, \end{aligned}$$

so that $(s-a)p<n$. $\square$