Hodge decompositions for Lie algebroids on manifolds with boundary

Abstract

We investigate when the Chevalley-Eilenberg differential of a complex Lie algebroid on a manifold with boundary admits a Hodge decomposition. We introduce the concepts of Cauchy-Riemann structures, elliptic and non-elliptic boundary points and Levi-forms, which we use to define the notion of q-convexity. We show that the Chevalley-Eilenberg complex of an elliptic, q-convex Lie algebroid admits a Hodge decomposition in degree q. This generalizes the well-known Hodge decompositions for the exterior derivative on real manifolds and the delbar-operator on q-convex complex manifolds. We establish the results in a more general setting, where the differential does not necessarily square to zero and moreover varies in a family, including an analysis of the behaviour on the deformation parameter. As application we give a proof of a classical holomorphic tubular neighbourhood theorem (which implies the Newlander-Nirenberg theorem) based on the Moser trick, and we provide a finite-dimensionality result for certain holomorphic Poisson cohomology groups.

A Appendix

In this appendix we collect some “Leibniz” rules involving Sobolev norms. We will use the notation of Sect. 3.3, and start with some numerical estimates.

Lemma A.1

Let $k\in {\mathbb {R}}$ be a real number.

• i) For all $\xi ,\eta \in {\mathbb {R}}^m$ we have $\Big (\frac{1+|\xi |^2}{1+|\eta |^2}\Big )^k \le 2^{|k|} (1+|\xi -\eta |^2)^{|k|}$.

• ii) Define $K_1(\xi ,\eta ):= (1+|\xi |^2)^{k/2}-(1+|\eta |^2)^{k/2}$, where $\xi ,\eta \in {\mathbb {R}}^m$. Then

  $$\begin{aligned} |K_1(\xi ,\eta )|&\le |k|\cdot |\xi -\eta | \cdot \big ((1+|\xi |^2)^{\frac{k-1}{2}}+(1+|\eta |^2)^{\frac{k-1}{2}}\big ). \end{aligned}$$

• iii) Define $K_3(\xi ,\eta _1,\eta _2):= (1+|\xi |^2)^{\frac{k}{2}}+(1+|\eta _1|^2)^{\frac{k}{2}}-(1+|\eta _2|^2)^{\frac{k}{2}}-(1+|\xi +\eta _1-\eta _2|^2)^{\frac{k}{2}}$, where $\xi ,\eta _1,\eta _2\in {\mathbb {R}}^m$. Setting $C:=k(k-1)$, we have

  $$\begin{aligned}&|K_3(\xi ,\eta _1,\eta _2)|\le C |\xi -\eta _2| \cdot |\eta _1-\eta _2|\cdot \int _0^1 \int _0^1 (1+\big |\xi +t(\eta _1-\eta _2)\nonumber \\&\quad +t'(\eta _2-\xi )\big |^2)^{\frac{k-2}{2}} dtdt'. \end{aligned}$$

Proof

i) See [5, Lemma (A.1.3)].

ii) Consider the smooth function $f(x):=(1+x^2)^{k/2}$ on ${\mathbb {R}}_{\ge 0}$. For all $x,y\in {\mathbb {R}}_{\ge 0}$ we have

$$\begin{aligned} |f(x)-f(y)|\le |x-y|\sup _{x\le z \le y} |f'(z)| \le |k| |x-y| \big ((1+x^2)^{\frac{k-1}{2}}+(1+y^2)^{\frac{k-1}{2}}\big ). \end{aligned}$$

Setting $x=|\xi |$, $y=|\eta |$ and using $| |\xi |- |\eta | |\le |\xi -\eta |$, we obtain (ii).

iii): Define $g(x):=(1+|x|^2)^{k/2}$ so that $K_3(\xi ,\eta _1,\eta _2)=g(\xi )+g(\eta _1)-g(\eta _2)-g(\xi +\eta _1-\eta _2)$. A couple of applications of the fundamental theorem of calculus gives

$$\begin{aligned} K_3(\xi ,\eta _1,\eta _2)=\sum _{i,j} (\eta _2-\xi )^i(\eta _1-\eta _2)^j \int _0^1\int _0^1 \partial _i\partial _jg(\xi +t'(\eta _2-\xi )+t(\eta _1-\eta _2))dtdt', \end{aligned}$$

from which the desired estimate readily follows. $\square$

Below we will write $\alpha \lesssim \beta$ if there is a constant C such that $\alpha \le C \beta$. The only constraint on C is that it is independent of the functions f, g and $\varphi$ appearing in the inequalities below.

Proposition A.2

Set $a:=1+\frac{m}{2}$, where m is the dimension of Euclidean space.

i) For $s\ge 0$ and $f,\varphi \in {\mathcal {S}}$ we have

$$\begin{aligned} || f\varphi ||_s \lesssim ||f||_{s+a} ||\varphi ||+||f||_{a} ||\varphi ||_{s}. \end{aligned}$$

(A.1)

For $s\in {\mathbb {R}}$ and $f,\varphi \in {\mathcal {S}}$ we have

$$\begin{aligned} || f\varphi ||_s \lesssim ||f||_{|s|+a} ||\varphi ||_s. \end{aligned}$$

(A.2)

ii) For $s\in {\mathbb {R}}_{\ge 0}$, $k\in {\mathbb {R}}_{\ge 1}$ and $f,\varphi \in {\mathcal {S}}$ we have

$$\begin{aligned} || [\Lambda ^k,f]\varphi ||_s \lesssim&||f||_{s+k+a} ||\varphi ||+||f||_{k+a} ||\varphi ||_{s}+||f||_{s+1+a} ||\varphi ||_{k-1}+||f||_{1+a} ||\varphi ||_{s+k-1}. \end{aligned}$$

(A.3)

For $s,k\in {\mathbb {R}}$ and $f,\varphi \in {\mathcal {S}}$ we have

$$\begin{aligned} || [\Lambda ^k,f]\varphi ||_s&\lesssim (||f||_{|s+k-1|+1+a} +||f||_{|s|+1+a}) ||\varphi ||_{s+k-1}. \end{aligned}$$

(A.4)

iii) For $s\in {\mathbb {R}}_{\ge 0}$, $k\in {\mathbb {R}}_{\ge 1}$ and $f,\varphi \in {\mathcal {S}}$ we have

$$\begin{aligned} || [\Lambda ^k,[\Lambda ^k,f]] \varphi ||_s \lesssim \;&||f||_{2k+s+a} ||\varphi ||+||f||_{2k+a}||\varphi ||_{s}\nonumber \\&+||f||_{2+a}||\varphi ||_{2k+s-2} +||f||_{s+2+a}||\varphi ||_{2k-2}. \end{aligned}$$

(A.5)

For $s,k\in {\mathbb {R}}$ and $f,\varphi \in {\mathcal {S}}$ we have

$$\begin{aligned} || [\Lambda ^k,[\Lambda ^k,f]] \varphi ||_s&\lesssim (||f||_{|s+2k-2|+2+a}+||f||_{|s|+2+a} )|| \varphi ||_{s+2k-2}. \end{aligned}$$

(A.6)

iv) For $s\in {\mathbb {R}}_{\ge 0}$, $k\in {\mathbb {R}}_{\ge 2}$ and $f,g,\varphi \in {\mathcal {S}}$ we have

$$\begin{aligned} || [[\Lambda ^k,f],&g]\varphi ||_s \lesssim \big ( ||f||_{k-1+s+a} ||g||_{1+a}+||f||_{k-1+a} ||g||_{s+1+a}+||f||_{s+1+a} ||g||_{k-1+a}\nonumber \\&+||f||_{1+a} ||g||_{k-1+s+a}\big )||\varphi ||_{}+(||f||_{k-1+a} ||g||_{1+a}+||f||_{1+a} ||g||_{k-1+a})||\varphi ||_{s} \nonumber \\&+ (||f||_{s+1+a} ||g||_{1+a}+||f||_{1+a} ||g||_{s+1+a})||\varphi ||_{k-2}+||f||_{1+a} ||g||_{1+a}||\varphi ||_{k-2+s}. \end{aligned}$$

(A.7)

For $s,k\in {\mathbb {R}}$ and $f,g,\varphi \in {\mathcal {S}}$ we have

$$\begin{aligned} || [[\Lambda ^k,f],g]\varphi ||_s \lesssim \big (&||f||_{1+|s|+|k-2|+a} ||g||_{1+a}+||f||_{1+|s|+a} ||g||_{1+|k-2|+a} \nonumber \\&+||f||_{1+|k-2|+a} ||g||_{1+|s|+a} +||f||_{1+a} ||g||_{1+|s|+|k-2|+a} \big ) ||\varphi ||_{s+k-2}. \end{aligned}$$

(A.8)

Proof

i) Using $\widehat{f\varphi }={\widehat{f}}\star {\widehat{\varphi }}$, where $\star$ denotes convolution product, we obtain

$$\begin{aligned} || f\varphi ||^2_s&= \int (1+|\xi |^2)^s |\widehat{f\varphi }(\xi )|^2 d\xi \le \int \Big ( \int (1+|\xi |^2)^{s/2}|{\widehat{f}}(\xi -\eta )| |{\widehat{\varphi }}(\eta )|d\eta \Big )^2 d\xi . \end{aligned}$$

If $s\ge 0$ we can use the triangle inequality (raised to a non-negative power) to estimate

$$\begin{aligned} (1+|\xi |^2)^{s/2}\lesssim (1+|\xi -\eta |^2)^{s/2}+ (1+|\eta |^2)^{s/2}. \end{aligned}$$

Using the Cauchy-Schwarz inequality we get

$$\begin{aligned}&\int (1+|\eta |^2)^{s/2}|{\widehat{f}}(\xi -\eta )| |{\widehat{\varphi }}(\eta )|d\eta \\&\quad \le \Big ( \int |{\widehat{f}}(\xi -\eta )| d\eta \Big )^{\frac{1}{2}} \cdot \Big ( \int (1+|\eta |^2)^{s}|{\widehat{f}}(\xi -\eta )| |{\widehat{\varphi }}(\eta )|^2d\eta \Big )^{\frac{1}{2}}, \end{aligned}$$

which implies that

$$\begin{aligned} \int \Big ( \int (1+|\eta |^2)^{s/2}|{\widehat{f}}(\xi -\eta )| |{\widehat{\varphi }}(\eta )|d\eta \Big )^2 d\xi \le \Big ( \int |{\widehat{f}}(\eta )| d\eta \Big )^2 \cdot ||\varphi ||_s^2\lesssim ||f||^2_a ||\varphi ||_s^2. \end{aligned}$$

Here in the last step we applied the Cauchy-Schwarz inequality again to estimate

$$\begin{aligned} \Big (\int |{\widehat{f}}(\eta )| d\eta \Big )^2 \le \int (1+|\eta |^2)^{-a} d\eta \int (1+|\eta |^2)^{a} |{\widehat{f}}(\eta )|^2 d\eta \le C ||f||^2_a \end{aligned}$$

where C is finite because $a>m/2$. In a similar fashion one proves that

$$\begin{aligned} \int \Big ( \int (1+|\xi -\eta |^2)^{s/2}|{\widehat{f}}(\xi -\eta )| |{\widehat{\varphi }}(\eta )|d\eta \Big )^2 d\xi \lesssim ||f||_{s+a} ||\varphi ||^2, \end{aligned}$$

proving (A.1). For arbitrary $s\in {\mathbb {R}}$ we have to replace the triangle inequality with the bound

$$\begin{aligned} (1+|\xi |^2)^{s/2}\lesssim (1+|\xi -\eta |^2)^{|s|/2}(1+|\eta |^2)^{s/2} \end{aligned}$$

which follows from Lemma A.1i). The rest of the steps are then similar to the ones above.

ii) The strategy is the same as in (i). First we observe that

$$\begin{aligned} {\mathcal {F}}([\Lambda ^k,f]\varphi )(\xi )= \int K_1(\xi ,\eta ) {\widehat{f}}(\xi -\eta ) {\widehat{\varphi }}(\eta ) d\eta , \end{aligned}$$

where $K_1$ was defined in Lemma A.1ii). When $k-1\ge 0$ and $s\ge 0$, the triangle inequality together with Lem.A.1ii) imply that

$$\begin{aligned} (1+|\xi |^2)^{s/2}|K_1(\xi ,\eta )|&\lesssim (1+|\xi -\eta |^2)^{\frac{s+k}{2}} +(1+|\xi -\eta |^2)^{\frac{k}{2}}(1+|\eta |^2)^{\frac{s}{2}} \\&+ (1+|\xi -\eta |^2)^{\frac{s+1}{2}}(1+|\eta |^2)^{\frac{k-1}{2}}+(1+|\xi -\eta |^2)^{\frac{1}{2}}(1+|\eta |^2)^{\frac{s+k-1}{2}}. \end{aligned}$$

The same steps as in i) then yield (A.3). For arbitrary s and k we use Lemma A.1ii) to estimate

$$\begin{aligned} (1+|\xi |^2)^{s/2}|K_1(\xi ,\eta )|&\lesssim \big ( (1+|\xi -\eta |^2)^{\frac{1+|s+k-1|}{2}}+(1+|\xi -\eta |^2)^{\frac{1+|s|}{2}}\big )(1+|\eta |^2)^{\frac{k+s-1}{2}}, \end{aligned}$$

giving (A.4).

iii) We have

$$\begin{aligned} {\mathcal {F}}([\Lambda ^k,[\Lambda ^k,f]]\varphi )(\xi )= \int K_2(\xi ,\eta ) {\widehat{f}}(\xi -\eta ) {\widehat{\varphi }}(\eta ) d\eta , \end{aligned}$$

where $K_2(\xi ,\eta ):= \big ( (1+|\xi |^2)^{k/2}-(1+|\eta |^2)^{k/2} \big )^2=K_1(\xi ,\eta )^2$. Using Lemma A.1ii) we obtain

$$\begin{aligned} (1+|\xi |^2)^{s/2}|K_2(\xi ,\eta )|\lesssim&\ (1+|\xi -\eta |^2)^{\frac{s}{2}+k} +(1+|\xi -\eta |^2)^{\frac{s}{2}+1}(1+|\eta |^2)^{k-1} \\&+ (1+|\xi -\eta |^2)^{k}(1+|\eta |^2)^{\frac{s}{2}}+(1+|\xi -\eta |^2)^{}(1+|\eta |^2)^{\frac{s}{2}+k-1}, \end{aligned}$$

valid for $s\ge 0$ and $k\ge 1$. The same steps as in (i) then give (A.5). For arbitrary s and k we use Lemma A.1i) to estimate

$$\begin{aligned} (1+|\xi |^2)^{s/2}|K_2(\xi ,\eta )|&\lesssim \big ( (1+|\xi -\eta |^2)^{|\frac{s}{2}+k-1|+1}+(1+|\xi -\eta |^2)^{|\frac{s}{2}|+1}\big )(1+|\eta |^2)^{\frac{s}{2}+k-1}, \end{aligned}$$

which gives (A.6).

iv): We have

$$\begin{aligned} {\mathcal {F}}([[\Lambda ^k,f],g]\varphi )(\xi )= \iint K_3(\xi ,\eta _1,\eta _2) {\widehat{f}}(\xi -\eta _2) {\widehat{g}}(\eta _2-\eta _1) {\widehat{\varphi }}(\eta _1) d\eta _1d\eta _2, \end{aligned}$$

where $K_3$ was defined in Lemma A.1iv). Consequently, for $s\ge 0$ and $k\ge 2$ we have

$$\begin{aligned} (1+|\xi |^2)^{\frac{s}{2}}|K_3(\xi ,\eta&)|\lesssim (1+|\xi -\eta _2|^2)^{\frac{1}{2}}(1+|\eta _1-\eta _2|^2)^{\frac{1}{2}} \Big ( (1+|\xi -\eta _2|^2)^{\frac{s}{2}} \\&+ (1+|\eta _1-\eta _2|^2)^{\frac{s}{2}}\\&+(1+|\eta _1|^2)^{\frac{s}{2}}\Big )\Big ((1+|\xi -\eta _2|^2)^{\frac{k-2}{2}} \\&+ (1+|\eta _1-\eta _2|^2)^{\frac{k-2}{2}}+(1+|\eta _1|^2)^{\frac{k-2}{2}}\Big ). \end{aligned}$$

Expanding this out gives nine terms, leading to (A.7) by using the same steps as in i).

For arbitrary s and k we use Lemma A.1 ii) to bound

$$\begin{aligned}&\quad (1+\big |\xi +t(\eta _1-\eta _2)+t'(\eta _2-\xi )\big |^2)^{\frac{k-2}{2}}\\&\quad =\Big (\frac{1+\big |\xi +t(\eta _1-\eta _2) +t'(\eta _2-\xi )\big |^2}{1+|\eta _1|^2}\Big )^{\frac{k-2}{2}} (1+|\eta _1|^2)^{\frac{k-2}{2}}\\&\quad \lesssim (1+\big |\xi -\eta _1+t(\eta _1-\eta _2)+t'(\eta _2-\xi )\big |^2)^{|\frac{k-2}{2}|} \\&\quad \quad (1+|\eta _1|^2)^{\frac{k-2}{2}}\\&\quad \lesssim \big ((1+|\xi -\eta _2|^2)^{|\frac{k-2}{2}|}+(1+|\eta _1-\eta _2|^2)^{|\frac{k-2}{2}|}\big )\\&\quad \quad (1+|\eta _1|^2)^{\frac{k-2}{2}}, \end{aligned}$$

where we used $\xi -\eta _1+t(\eta _1-\eta _2)+t'(\eta _2-\xi )=(1-t')(\xi -\eta _2)+(1-t)(\eta _2-\eta _1)$. Proceeding as before yields (A.8). $\square$

There is also a boundary version of Proposition A.2. We continue with the notation of Sect. 3.3, and denote by $\lceil s \rceil$ the smallest integer greater or equal than s. For $K\subset {\mathbb {R}}^m_-$ we denote by $C^\infty _K({\mathbb {R}}^m_-) \subset {\mathcal {S}}({\mathbb {R}}^m_-)$ the functions with support in K.

Proposition A.3

Let $K\subset {\mathbb {R}}^m_-$ be a compact subset and let $a=1+\frac{m}{2}$.

i) For $s\ge 0$ and $f\in C^\infty _K({\mathbb {R}}^m_-)$, $\varphi \in {\mathcal {S}}({\mathbb {R}}^m_-)$ we have

$$\begin{aligned} || f\varphi ||_{\partial ,s} \lesssim |f|_{\lceil s+a\rceil } ||\varphi ||_\partial +|f|_{\lceil a\rceil } ||\varphi ||_{\partial ,s}. \end{aligned}$$

(A.9)

For $s\in {\mathbb {R}}$ and $f\in C^\infty _K({\mathbb {R}}^m_-)$, $\varphi \in {\mathcal {S}}({\mathbb {R}}^m_-)$ we have

$$\begin{aligned} || f\varphi ||_{\partial ,s} \lesssim |f|_{\lceil |s|+a\rceil } ||\varphi ||_{\partial ,s}. \end{aligned}$$

(A.10)

ii) For $s\in {\mathbb {R}}_{\ge 0}, k\in {\mathbb {R}}_{\ge 1}$ and $f\in C^\infty _K({\mathbb {R}}^m_-) ,\varphi \in {\mathcal {S}}({\mathbb {R}}^m_-)$ we have

$$\begin{aligned} || [\Lambda ^k_\partial ,f]\varphi ||_{\partial ,s} \lesssim&\ |f|_{\lceil s+k+a\rceil } ||\varphi ||_\partial +|f|_{\lceil k+a\rceil } ||\varphi ||_{\partial ,s}\nonumber \\&+|f|_{\lceil s+1+a\rceil } ||\varphi ||_{\partial ,k-1}+|f|_{\lceil 1+a\rceil } ||\varphi ||_{\partial ,s+k-1}. \end{aligned}$$

(A.11)

For $s,k\in {\mathbb {R}}$ and $f\in C^\infty _K({\mathbb {R}}^m_-),\varphi \in {\mathcal {S}}({\mathbb {R}}^m_-)$ we have

$$\begin{aligned} || [\Lambda ^k_\partial ,f]\varphi ||_{\partial ,s}&\lesssim (|f|_{\lceil |s+k-1|+1+a\rceil } +|f|_{\lceil |s|+1+a\rceil }) ||\varphi ||_{\partial ,s+k-1}. \end{aligned}$$

(A.12)

iii) For $s\in {\mathbb {R}}_{\ge 0}$, $k\in {\mathbb {R}}_{\ge 1}$ and $f\in C^\infty _K({\mathbb {R}}^m_-), \varphi \in {\mathcal {S}}({\mathbb {R}}^m_-)$ we have

$$\begin{aligned} || [\Lambda ^k_\partial ,[\Lambda ^k_\partial ,f]] \varphi ||_{\partial ,s} \lesssim&\ |f|^2_{\lceil 2k+s+a\rceil } ||\varphi ||^2_\partial +|f|^2_{\lceil 2+a\rceil }||\varphi ||^2_{\partial ,2k+s-2} \nonumber \\&+|f|_{\lceil 2k+a\rceil }||\varphi ||_{\partial ,s}+|f|^2_{\lceil s+2+a\rceil }||\varphi ||^2_{\partial ,2k-2}. \end{aligned}$$

(A.13)

For $s,k\in {\mathbb {R}}$ and $f\in C^\infty _K({\mathbb {R}}^m_-), \varphi \in {\mathcal {S}}({\mathbb {R}}^m_-)$ we have

$$\begin{aligned} || [\Lambda ^k_\partial ,[\Lambda ^k_\partial ,f]] \varphi ||_{\partial ,s}&\lesssim (|f|_{\lceil |s+2k-2|+2+a\rceil }+|f|_{\lceil |s|+2+a\rceil } )|| \varphi ||_{\partial ,s+2k-2}. \end{aligned}$$

(A.14)

iv) For $s\in {\mathbb {R}}_{\ge 0},k\in {\mathbb {R}}_{\ge 2}$ and $f,g\in C^\infty _K({\mathbb {R}}^m_-), \varphi \in {\mathcal {S}}({\mathbb {R}}^m_-)$ we have

$$\begin{aligned} ||&[[\Lambda ^k_\partial ,f],g]\varphi ||_{\partial ,s} \lesssim \big ( |f|_{\lceil k-1+s+a\rceil } |g|_{\lceil 1+a\rceil }+|f|_{\lceil k-1+a\rceil } |g|_{\lceil s+1+a\rceil }+|f|_{\lceil s+1+a\rceil } |g|_{\lceil k-1+a\rceil }\nonumber \\&+|f|_{\lceil 1+a\rceil } |g|_{\lceil k-1+s+a\rceil }\big )||\varphi ||_{\partial }+(|f|_{\lceil k-1+a\rceil } |g|_{\lceil 1+a\rceil }+|f|_{\lceil 1+a\rceil } |g|_{\lceil k-1+a\rceil })||\varphi ||_{\partial ,s} \nonumber \\&+ (|f|_{\lceil s+1+a\rceil } |g|_{\lceil 1+a\rceil }+|f|_{\lceil 1+a\rceil } |g|_{\lceil s+1+a\rceil })||\varphi ||_{\partial ,k-2}+|f|_{\lceil 1+a\rceil } |g|_{\lceil 1+a\rceil }||\varphi ||_{\partial ,k-2+s}. \end{aligned}$$

(A.15)

For $s,k\in {\mathbb {R}}$ and $f,g\in C^\infty _K({\mathbb {R}}^m_-), \varphi \in {\mathcal {S}}({\mathbb {R}}^m_-)$ we have

$$\begin{aligned} || [[\Lambda ^k_\partial ,f],g]\varphi ||_{\partial ,s} \lesssim \big (&|f|_{\lceil 1+|s|+|k-2|+a\rceil } |g|_{1+a}+|f|_{\lceil 1+|s|+a\rceil } |g|_{\lceil 1+|k-2|+a\rceil } \nonumber \\&+|f|_{\lceil 1+|k-2|+a\rceil } |g|_{\lceil 1+|s|+a\rceil } +|f|_{\lceil 1+a\rceil } |g|_{\lceil 1+|s|+|k-2|+a\rceil } \big ) ||\varphi ||_{\partial ,s+k-2}. \end{aligned}$$

(A.16)

Proof

For each value of r we can consider the inequalities of Proposition A.2 for the functions $f(\cdot ,r)$, $g(\cdot ,r)$ and $\varphi (\cdot ,r)$ on ${\mathbb {R}}^{m-1}$. Integrating these over r and using $||f(\cdot ,r) ||_{s}\le C |f(\cdot ,r)|_{\lceil s \rceil }\le C |f|_{\lceil s \rceil }$ for $s\in {\mathbb {R}}_{\ge 0}$ (where C depends on $K\subset {\mathbb {R}}^m_-$), we obtain (A.11)–(A.16). $\square$

Finally we mention a version of Rellich’s lemma for the norm $||\text {D}(\cdot )||_{\partial ,-1/2}$, whose proof is outlined in the appendix of [5] (see the discussion preceding Proposition A.3.1).

Proposition A.4

If a sequence $\varphi _j\in C^\infty _K({\mathbb {R}}^m_-)$ satisfies a uniform bound $||\text {D}\varphi _j||_{\partial ,-1/2}\le C$, then a subsequence converges in $L^2({\mathbb {R}}^m)$.