Moduli space and deformations of special Lagrangian submanifolds with edge singularities

Abstract

Special Lagrangian submanifolds are submanifolds of a Calabi–Yau manifold calibrated by the real part of the holomorphic volume form. In this paper we use elliptic theory for edge-degenerate differential operators on singular manifolds to study the moduli space of deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold.

Appendices

Appendix 1: Vector-valued Sobolev embeddings

In Sects. 4 and 6 we used results from this “Appendix”. Most of the results here are based on vector-valued Sobolev embeddings. In the first section of this “Appendix” we recall the basics of vector-valued Sobolev embeddings and derive some consequences related to cone and edge-Sobolev spaces. In the second part we prove the estimate (11.1) following a similar result of Dreher and Witt [8]. This estimate implies the Banach Algebra property of our edge-Sobolev spaces on M, (11.39), and the regularity of the product of elements in ${\mathcal {W}}^{s,\gamma }(M)$, (11.40). In order to simplify the notation we will denote $a\approx b$ and $a\lesssim b$ if $a=\kappa b$ or $a\le \kappa b$ respectively with a positive constant $\kappa$ depending only on s and $\gamma$.

Let’s consider the classical Sobolev spaces $W^{m,p}({\mathbb {R}}^{q})$ (see [5] for a detailed introduction). A classical tool in the analysis of partial differential equations on ${\mathbb {R}}^{q}$ is the set of Sobolev embeddings, see [5, section 9.3].

Theorem 10.1

Let $m\in {\mathbb {Z}}$, $m>1$ and $p\in [1,+\infty ).$

$$\begin{aligned}&\text {If}\quad \frac{q}{p}>m \quad \text {then }\quad W^{m,p}({\mathbb {R}}^{q})\hookrightarrow L^{k}({\mathbb {R}}^{q}) \quad \text {where } \frac{1}{q}=\frac{1}{p}-\frac{m}{q}. \end{aligned}$$

(10.1)

$$\begin{aligned}&\text {If}\quad \frac{q}{p}=m \quad \text {then }\quad W^{m,p}({\mathbb {R}}^{q})\hookrightarrow L^{k}({\mathbb {R}}^{q}) \quad \text {for all }\quad k\in [p,+\infty ). \end{aligned}$$

(10.2)

$$\begin{aligned}&\text {If}\quad \frac{q}{p}<m \quad \text {then }\quad W^{m,p}({\mathbb {R}}^{q})\hookrightarrow L^{\infty }({\mathbb {R}}^{q})\text { and}\quad W^{m,p}({\mathbb {R}}^{q})\hookrightarrow {\mathcal {C}}^{r}({\mathbb {R}}^{q}) \end{aligned}$$

(10.3)

where $r=[s-\frac{q}{2}]$ i.e. r is the integer part of $s-\frac{q}{2}$.

In this “Appendix” we are interested in the vector-valued version of this theorem i.e. given a Banach space ${\mathfrak {B}}$ we want a version for the ${\mathfrak {B}}$-valued Sobolev spaces $W^{m,p}({\mathbb {R}}^{q},{\mathfrak {B}}).$ There are many books and monographs dealing with vector-valued spaces of all kinds like $L^{p}({\mathbb {R}}^{q},{\mathfrak {B}}),{\mathcal {C}}^{k}({\mathbb {R}}^{q},{\mathfrak {B}})$ and ${\mathcal {S}}({\mathbb {R}}^{q},{\mathfrak {B}})$, see [3, 12, 36]. In many cases they work in the more general context where ${\mathfrak {B}}$ is a Fréchet or locally convex Hausdorff space. For our specific purposes we follow closely [23]. Here Kreuter analyzes carefully the validity of Theorem 10.1 for the spaces $W^{m,p}({\mathbb {R}}^{q},{\mathfrak {B}})$ where ${\mathfrak {B}}$ is a Banach space.

Recall that the vector-valued space of distributions is defined as the space of continuous operators from $C^{\infty }_{0}({\mathbb {R}}^{q})$ to ${\mathfrak {B}}$ i.e. we have ${\mathcal {D'}}({\mathbb {R}}^{q},{\mathfrak {B}}):={\mathcal {L}}(C^{\infty }_{0}({\mathbb {R}}^{q}),{\mathfrak {B}}).$ The vector-valued $L^{p}$-spaces, $L^{p}({\mathbb {R}}^{q},{\mathfrak {B}})$, are defined by means of the Bochner integral. The Bochner integral is constructed by means of ${\mathfrak {B}}$-valued step functions in a similar way to the standard Lebesgue integral. See [1, Appendix A.4] for details. The vector-valued ${\mathcal {C}}^{k}$-spaces, ${\mathcal {C}}^{k}({\mathbb {R}}^{q},{\mathfrak {B}})$, are defined with respect to the Fréchet derivative. The vector-valued Sobolev space is defined by

$$\begin{aligned} W^{m,p}({\mathbb {R}}^{q},{\mathfrak {B}}):=\left\{ f\in L^{p}({\mathbb {R}}^{q},{\mathfrak {B}}):\partial ^{\alpha }f\in L^{p}({\mathbb {R}}^{q},{\mathfrak {B}}) \quad \forall \quad \left| \alpha \right| \le m \right\} \end{aligned}$$

(10.4)

where the derivatives of f are taken in the distribution sense i.e. weak derivatives.

Here we recall the definition of the Radon-Nikodym property and some results related to it. It turns out that the key property that ${\mathfrak {B}}$ must satisfy in order to have vector-valued Sobolev embeddings for $W^{m,p}({\mathbb {R}}^{q},{\mathfrak {B}})$ is the Radon-Nikodym property. For extended details the reader is referred to [23, chapter 2].

Definition 10.2

A Banach space ${\mathfrak {B}}$ has the Radon-Nikodym property if every Lipschitz continuous function $f:I\longrightarrow {\mathfrak {B}}$ is differentiable almost everywhere, where $I\subset {\mathbb {R}}$ is an arbitrary interval.

Proposition 10.3

Every reflexive space has the Radon-Nikodym property. In particular the spaces $L^{p}({\mathbb {R}}^{q})$ with $1<p<\infty$ and Hilbert spaces have the Radon-Nikodym property.

Corollary 10.4

The Sobolev embeddings in Theorem 10.1 are valid for the spaces $W^{m,p}({\mathbb {R}}^{q},L^{p}({\mathbb {R}}^{q}))$ with $1<p<\infty$ and $W^{m,p}({\mathbb {R}}^{q},{\mathfrak {H}})$ where ${\mathfrak {H}}$ is a Hilbert space.

As a consequence of these vector-valued results we have the following applications to cone and edge-Sobolev spaces.

Proposition 10.5

If $f\in {\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ (see (3.7)) and $s>\frac{m+1}{2}$ then there exists $C>0$ depending only on s and $\gamma$ such that we have the following estimate on $(0,1)\times {\mathcal {X}}$

$$\begin{aligned} \left| \partial ^{\alpha '}_{r}\partial ^{\alpha ''}_{\sigma }f(r,\sigma )\right| \le C\left\| f\right\| _{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })}r^{\gamma -\frac{m+1}{2}-\left| \alpha '\right| } \end{aligned}$$

(10.5)

for all $(r,\sigma )\in (0,1)\times {\mathcal {X}}$ and $\left| \alpha '\right| +\left| \alpha ''\right| \le [s-\frac{m+1}{2}] .$

Proof

We can work locally on ${\mathbb {R}}^{+}\times {\mathcal {U}}_{\lambda }$ where $\lbrace {\mathcal {U}}_{\lambda }\rbrace$ is a finite open covering of ${\mathcal {X}}$, $\lbrace \varphi _{\lambda }\rbrace$ is a subbordinate partition of unity and we consider $\omega \varphi _{\lambda }f$. For simplicity we write just f instead of $\omega \varphi _{\lambda }f$. At the end we take the smallest constant among those obtained for each element in the finite covering. Take $f\in {\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ then by (3.6) we have $S_{\gamma -\frac{m}{2}}f\in H^{s}({\mathbb {R}}^{1+m}).$ Therefore if $s>\frac{m+1}{2}$ by (10.3) we have $S_{\gamma -\frac{m}{2}}f\in L^{\infty }({\mathbb {R}}^{1+m})$ and

$$\begin{aligned} \sup _{(t,\sigma )\in {\mathbb {R}}^{1+m}}\left| \partial ^{\alpha }_{(t,\sigma )}(S_{\gamma -\frac{m}{2}}f)(t,\sigma )\right| \lesssim \left\| S_{\gamma -\frac{m}{2}}f\right\| _{H^{s}({\mathbb {R}}^{1+m})}\lesssim \left\| f\right\| _{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })} \end{aligned}$$

(10.6)

for all $\left| \alpha \right| \le [s-\frac{m+1}{2}]$. Now by definition (see (3.6))

$$\begin{aligned} (S_{\gamma -\frac{m}{2}}f)(t,x)=e^{-(\frac{1}{2}-(\gamma -\frac{m}{2}))t}f(e^{-t},x) \end{aligned}$$

with $r=e^{-t}$. Thus (10.5) follows immediately. $\square$

In general, if ${\mathfrak {B}}$ is a Banach space and $\lbrace \kappa _{\lambda }\rbrace _{\lambda \in {\mathbb {R}}^{+}}\in {\mathcal {C}}\left( {\mathbb {R}}^{+},{\mathcal {L}}({\mathfrak {B}})\right)$ is a continuous one-parameter group of invertible operators we have that there exist positive constants $K,{\mathfrak {c}}$ such that

$$\begin{aligned} \left\| \kappa _{\lambda }\right\| _{{\mathcal {L}}({\mathfrak {B}})} \le \left\{ \begin{array}{ll} K\lambda ^{{\mathfrak {c}}}&{}\quad \text { for } \lambda \ge 1 \\ K\lambda ^{-{\mathfrak {c}}}&{}\quad \text { for } 0<\lambda \le 1 \end{array} \right. . \end{aligned}$$

(10.7)

See [33, proposition 1.3.1] for details.

When ${\mathfrak {B}}={\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ and $(\kappa _{\lambda }f)(r,\sigma )=\lambda ^{\frac{m+1}{2}}f(\lambda r,\sigma )$ we can use 3.6 to compute $\left\| \kappa _{\lambda }\right\| _{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })}=\lambda ^{\gamma }$ (see [32, section 1.1]). By the proof of proposition 1.3.1 in [33] it is easy to see that the constant ${\mathfrak {c}}$ in (10.7) depends only on the weight $\gamma$. When ${\mathfrak {B}}={\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ we denote this constant by ${\mathfrak {c}}_{\gamma }$.

As a consequence of (10.7), we have the following continuous embeddings

$$\begin{aligned}&{\mathcal {W}}^{s,\gamma }({\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q}) \longrightarrow H^{s-{\mathfrak {c}}_{\gamma }}({\mathbb {R}}^{q},{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge })) \end{aligned}$$

(10.8)

$$\begin{aligned}&H^{s}({\mathbb {R}}^{q},{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }))\longrightarrow {\mathcal {W}}^{s+{\mathfrak {c}}_{\gamma },\gamma }({\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q}) \end{aligned}$$

(10.9)

for all $s\in {\mathbb {R}}$ where $H^{s}({\mathbb {R}}^{q},{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }))$ is the standard vector-valued Sobolev space with norm given by

$$\begin{aligned} \left\| f\right\| _{H^{s}({\mathbb {R}}^{q},{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }))}:=\left( \,\,\int \limits _{{\mathbb {R}}^q}[\eta ]^{2s}\left\| {\mathcal {F}}_{u\rightarrow \eta }f(\eta )\right\| ^{2}_{{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge })}d\eta \right) ^{\frac{1}{2}}. \end{aligned}$$

The reader is refer to [33, proposition 1.3.1 and remark 1.3.21] for details.

Proposition 10.6

If $g\in {\mathcal {W}}^{s,\gamma }(M)$ (see (3.9)) and $s>\frac{m+1+q}{2}+{\mathfrak {c}}_{\gamma }$ where ${\mathfrak {c}}_{\gamma }$ is the constant defined in (10.7), then there exists $C'>0$ depending only on s and $\gamma$ such that we have the following estimate on $(0,1)\times {\mathcal {X}}\times {\mathcal {E}}$

$$\begin{aligned} \left| \partial ^{\alpha '}_{r}\partial ^{\alpha ''}_{(\sigma ,u)}g(r,\sigma ,u)\right| \le C'\left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)}r^{\gamma -\frac{m+1}{2}-\left| \alpha '\right| } \end{aligned}$$

(10.10)

for all $(r,\sigma ,u)\in (0,1)\times {\mathcal {X}}\times {\mathcal {E}}$ and $\left| \alpha '\right| +\left| \alpha ''\right| \le [s-\frac{m+1}{2}].$

Proof

We work locally on $(0,1)\times {\mathcal {U}}_{\lambda } \times \Omega _{j}$ as in Proposition 4.6 Sect. 4.4. Take $g\in {\mathcal {W}}^{s,\gamma }(M)$. Again by (10.3) and $s>\frac{m+1}{2}$ we have that for each $u\in {\mathbb {R}}^{q}$

$$\begin{aligned} (S_{\gamma -\frac{m}{2}}g)(u)\in H^{s}({\mathbb {R}}^{1+m}) \end{aligned}$$

and

$$\begin{aligned} \left\| (S_{\gamma -\frac{m}{2}}g)(u)\right\| _{L^{\infty }({\mathbb {R}}^{m+1})}\lesssim \left\| g(u)\right\| _{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })}. \end{aligned}$$

(10.11)

Now (10.8) implies $g\in H^{s-{\mathfrak {c}}_{\gamma }}({\mathbb {R}}^{q},{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge }))$ and $s>\frac{q}{2}+{\mathfrak {c}}_{\gamma }$ together with (10.3) and Corollary 10.4 implies that we have a continuous embedding

$$\begin{aligned} H^{s-{\mathfrak {c}}_{\gamma }}({\mathbb {R}}^{q},{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge }))\hookrightarrow L^{\infty }({\mathbb {R}}^{q},{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })) \end{aligned}$$

as ${\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ is a Hilbert space, see Definition 3.2 in Sect. 3.2.

Consequently

$$\begin{aligned} \left\| g\right\| _{L^{\infty }({\mathbb {R}}^{q},{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge }))}&=\sup _{u\in {\mathbb {R}}^{q}}\left\{ \left\| g(u)\right\| _{{\mathcal {H}}^{s,\gamma }} \right\} \end{aligned}$$

(10.12)

$$\begin{aligned}&\lesssim \left\| g\right\| _{H^{s-{\mathfrak {c}}_{\gamma }}({\mathbb {R}}^{q},{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge }))} \end{aligned}$$

(10.13)

$$\begin{aligned}&\lesssim \left\| g\right\| _{{\mathcal {W}}^{s,\gamma }({\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q})}. \end{aligned}$$

(10.14)

Hence (10.11), (10.14) and the change of variable $r=e^{-t}$ implies (10.10) as in Proposition 10.5. $\square$

Appendix 2: Banach algebra property of edge-Sobolev spaces

In [8] Dreher and Witt used a variant of the edge-Sobolev spaces we use in this paper. In that paper they are interested in applications to weakly hyperbolic equations. Their spaces are defined on $(0,T)\times {\mathbb {R}}^{n}$. In this context they proved (proposition 4.1 in [8]) that their edge-Sobolev spaces have the structure of a Banach algebra. With some modifications and by using vector-valued Sobolev embeddings it is possible to extend their result to our edge-Sobolev spaces on M. This extension follows closely the proof of Witt and Dreher. For completeness we include the details of this extension in our context as we used the estimates (11.40) and (11.39) in Chapter 3 and 4.

Proposition 11.1

Let $f,g\in {\mathcal {W}}^{s,\gamma }(M)$ with $s\in {\mathbb {N}}$ and $s>\frac{q+m+3}{2}$. Then $fg\in {\mathcal {W}}^{s,2\gamma -\frac{m+1}{2}}(M)$ and we have the following estimate

$$\begin{aligned} \left\| fg\right\| _{{\mathcal {W}}^{s,2\gamma -\frac{m+1}{2}}(M)}\le C\left\| f\right\| _{{\mathcal {W}}^{s,\gamma }(M)}\left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)} \end{aligned}$$

(11.1)

with a constant $C>0$ depending only on s and $\gamma$.

Proof

By means of finite open covers and partitions of unity on ${\mathcal {X}}$ and ${\mathcal {E}}$ we need to estimate in terms of $\omega \varphi _{\lambda }\phi _{j}f$ and $\omega \varphi _{\lambda }\phi _{j}g$ as in Proposition 4.7 Sect. 4.4. To avoid unnecessary long expressions we will denote them simply by f and g. To save space in long expressions we use the notation $\hat{f}$ to denote the Fourier transformation with respect to the conormal variable $\eta$ i.e. $\hat{f}={\mathcal {F}}_{u\rightarrow \eta }f.$ We will estimate on an open set $(0,\varepsilon )\times {\mathcal {U}}_{\lambda }\times \Omega _{j}$. Then the global estimate is obtained by adding these terms. Take $f,g\in C^{\infty }_{0}(M)$. By definition of our edge-Sobolev (3.8) norm and by (3.6) we have

$$\begin{aligned}&\left\| fg\right\| ^{2}_{{\mathcal {W}}^{s,2\gamma -\frac{m+1}{2}}(M)} \approx \int \limits _{{\mathbb {R}}^{q}_{\eta }} [\eta ]^{2s}\left\| \kappa ^{-1}_{[\eta ]} {\mathcal {F}}_{u\rightarrow \eta }(fg)(\eta )\right\| _{{\mathcal {H}}^{s,2\gamma -\frac{m+1}{2}} ({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})}^{2}d\eta \end{aligned}$$

(11.2)

$$\begin{aligned}&\quad \approx \sum \limits _{\left| \alpha \right| \le s} \int \limits _{{\mathbb {R}}^{q}_{\eta }} \int \limits _{{\mathbb {R}}^{}_{t}}\int \limits _{{\mathbb {R}}^{m}_{\sigma }}[\eta ]^{2s-(m+1)}\Big |\partial ^{\alpha }\big (e^{-(\frac{m+1}{2}-\gamma )2t}{\mathcal {F}}_{u\rightarrow \eta }(fg)([\eta ]^{-1}e^{-t}, \end{aligned}$$

(11.3)

$$\begin{aligned}&\qquad \sigma ,\eta )\big )\Big |^{2} d\sigma dt d\eta \end{aligned}$$

(11.4)

Here we will estimate the term with $\alpha =0$. The estimates on the other terms $\alpha \ne 0$ are similar. For each term in (11.4) we have

$$\begin{aligned}&\int \limits _{{\mathbb {R}}^{q}_{\eta }}\int \limits _{{\mathbb {R}}^{}_{t}}\int \limits _{{\mathbb {R}}^{m}_{\sigma }} [\eta ]^{2s-(m+1)}\left| e^{-(\frac{m+1}{2}-\gamma )2t} {\mathcal {F}}_{u\rightarrow \eta }(fg)\right| ^{2}d\eta \end{aligned}$$

(11.5)

$$\begin{aligned}&\quad = \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }}e^{-(\frac{m+1}{2}-\gamma )4t} \left\| [\eta ]^{s-\frac{m+1}{2}}{\mathcal {F}}_{u\rightarrow \eta }(fg)\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}^{2}d\sigma dt. \end{aligned}$$

(11.6)

Now, the hypothesis $s>\frac{q+m+3}{2}$ allows us to use lemma 4.6 in [8]. Basically, this lemma implies that for fixed $(t,\sigma )$ we have the following estimate

$$\begin{aligned}&\left\| \Lambda (\eta )\widehat{fg(t,\sigma )}(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}\le \left\| f(t,\sigma )(u)\right\| _{L^{\infty }({\mathbb {R}}^{q}_{u})}\cdot \left\| \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}\\&\quad +\left\| g(t,\sigma )(u)\right\| _{L^{\infty }({\mathbb {R}}^{q}_{u})}\cdot \left\| \Lambda (\eta ) \hat{f}(r,\sigma )(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}\\&\quad +C_{0}\left\| \Lambda (\eta )\hat{f}(t,\sigma )(u)\slash [\eta ]\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}\cdot \left\| \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })} \end{aligned}$$

with $C_0>0$ and $\Lambda (\eta )=[\eta ]^{s-\frac{m+1}{2}}$.

Applying this estimate to (11.6) we have

$$\begin{aligned}&\left( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }} e^{-(\frac{m+1}{2}-\gamma )4t}\left\| [\eta ]^{s-\frac{m+1}{2}} {\mathcal {F}}_{u\rightarrow \eta }(fg)\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}^{2}d\sigma dt \right) ^{\frac{1}{2}} \end{aligned}$$

(11.7)

$$\begin{aligned}&\quad \le \Bigg ( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }} \bigg (e^{-(\frac{m+1}{2}-\gamma )2t}\left\| f(t,\sigma )(u)\right\| _{L^{\infty }({\mathbb {R}}^{q}_{u})} \cdot \left\| \Lambda (\eta ) \hat{g}(t,x)(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}+\quad \end{aligned}$$

(11.8)

$$\begin{aligned}&\qquad + e^{-(\frac{m+1}{2}-\gamma )2t}\left\| g(t,\sigma )(u)\right\| _{L^{\infty } ({\mathbb {R}}^{q}_{u})}\cdot \left\| \Lambda (\eta ) \hat{f}(t,\sigma )(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })} \end{aligned}$$

(11.9)

$$\begin{aligned}&\qquad + e^{-(\frac{m+1}{2}-\gamma )2t} C_{0}\left\| \Lambda (\eta )\hat{f}(t,\sigma )(u)\slash [\eta ]\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })} \end{aligned}$$

(11.10)

$$\begin{aligned}&\qquad \cdot \left\| \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| _{L^{2} ({\mathbb {R}}^{q}_{\eta })}\Bigg )^{2} d\sigma dt \Bigg )^{\frac{1}{2}}. \end{aligned}$$

(11.11)

By the Minkowski inequality we have that (11.7) is less or equal to the following terms

$$\begin{aligned}&\Bigg ( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }}e^{-(\frac{m+1}{2}-\gamma )4t}\left\| f(t,\sigma ) (u)\right\| ^{2}_{L^{\infty }({\mathbb {R}}^{q}_{u})}\cdot \left\| \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| ^{2}_{L^{2}({\mathbb {R}}^{q}_{\eta })}d\sigma dt \Bigg )^{\frac{1}{2}} \end{aligned}$$

(11.12)

$$\begin{aligned}&\quad + \Bigg ( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }}e^{-(\frac{m+1}{2}-\gamma )4t}\left\| g(t,\sigma ) (u)\right\| ^{2}_{L^{\infty }({\mathbb {R}}^{q}_{u})}\cdot \left\| \Lambda (\eta ) \hat{f}(t,\sigma )(\eta )\right\| ^{2}_{L^{2}({\mathbb {R}}^{q}_{\eta })}d\sigma dt \Bigg )^{\frac{1}{2}} \end{aligned}$$

(11.13)

$$\begin{aligned}&\quad + \Bigg ( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }}e^{-(\frac{m+1}{2}-\gamma )4t} C_{0}\left\| \Lambda (\eta )\hat{f}(t,\sigma )(u)\slash [\eta ]\right\| ^{2}_{L^{2}({\mathbb {R}}^{q}_{\eta })} \end{aligned}$$

(11.14)

$$\begin{aligned}&\quad \cdot \left\| \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| ^{2}_{L^{2}({\mathbb {R}}^{q}_{\eta })} d\sigma dt \Bigg )^{\frac{1}{2}}, \end{aligned}$$

(11.15)

hence, by the inequality in (11.7), we have

$$\begin{aligned}&\left( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }}e^{-(\frac{m+1}{2}-\gamma )4t} \left\| [\eta ]^{s-\frac{m+1}{2}}{\mathcal {F}}_{u\rightarrow \eta }(fg)\right\| _{L^{2} ({\mathbb {R}}^{q}_{\eta })}^{2}d\sigma dt \right) ^{\frac{1}{2}} \end{aligned}$$

(11.16)

$$\begin{aligned}&\quad \le \left\| e^{-(\frac{m+1}{2}-\gamma )t}f(t,\sigma ,u)\right\| ^{}_{_{L^{\infty } ({\mathbb {R}}^{m+1}\times {\mathbb {R}}^{q}_{u})}} \end{aligned}$$

(11.17)

$$\begin{aligned}&\qquad \cdot \left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| ^{}_{_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}} \end{aligned}$$

(11.18)

$$\begin{aligned}&\qquad + \left\| e^{-(\frac{m+1}{2}-\gamma )t}g(t,\sigma ,u)\right\| ^{}_{_{L^{\infty }({\mathbb {R}}^{m+1}\times {\mathbb {R}}^{q}_{u})}} \end{aligned}$$

(11.19)

$$\begin{aligned}&\qquad \cdot \left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta ) \hat{f}(t,\sigma )(\eta )\right\| ^{}_{_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}} \end{aligned}$$

(11.20)

$$\begin{aligned}&\qquad + C_{0}\left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta )\hat{f}(t,\sigma )(u)\slash [\eta ]\right\| ^{}_{_{L^{\infty }({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}} \end{aligned}$$

(11.21)

$$\begin{aligned}&\qquad \cdot \left\| e^{-(\frac{m+1}{2}-\gamma )t} \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| ^{}_{_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}}. \end{aligned}$$

(11.22)

The edge-Sobolev norm of f and g written as in (11.6) implies that

$$\begin{aligned} \left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| ^{}_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}\le \left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)} \end{aligned}$$

(11.23)

and

$$\begin{aligned} \left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta ) \hat{f}(t,\sigma )(\eta )\right\| ^{}_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}\le \left\| f\right\| _{{\mathcal {W}}^{s,\gamma }(M)}, \end{aligned}$$

(11.24)

hence we only need to deal with the $L^{\infty }$ terms.

To analyze the $L^{\infty }$ terms recall that by hypothesis $s>\frac{q}{2}$ so we have the standard continuous Sobolev embedding $H^{s}({\mathbb {R}}^{q})\hookrightarrow L^{\infty }({\mathbb {R}}^{q})$. Consequently for fixed $(t,\sigma )$ we have

$$\begin{aligned} \left\| e^{-(\frac{m+1}{2}-\gamma )t}g(t,\sigma )(\eta ) \right\| ^{2}_{L^{\infty }({\mathbb {R}}^{q}_{u})}&\lesssim \left\| e^{-(\frac{m+1}{2}-\gamma )t}g(t,\sigma )(\eta ) \right\| ^{2}_{H^{s}({\mathbb {R}}^{q}_{u})} \end{aligned}$$

(11.25)

$$\begin{aligned}&=\left\| \langle \eta \rangle ^{s} e^{-(\frac{m+1}{2}-\gamma )t}\hat{g}(t,\sigma )(\eta ) \right\| ^{2}_{L^{2}({\mathbb {R}}^{q}_{\eta })}, \end{aligned}$$

(11.26)

therefore

$$\begin{aligned}&\left\| e^{-(\frac{m+1}{2}-\gamma )t}g(t,\sigma )(\eta ) \right\| ^{2}_{L^{\infty }({\mathbb {R}}^{m+1} \times {\mathbb {R}}^{q}_{u})} \end{aligned}$$

(11.27)

$$\begin{aligned}&\quad \lesssim \left\| \langle \eta \rangle ^{s} e^{-(\frac{m+1}{2}-\gamma )t}\ hat{g}(t,\sigma )(\eta ) \right\| ^{2}_{L^{\infty }({\mathbb {R}}^{m+1}, L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.28)

$$\begin{aligned}&\quad \lesssim \left\| \langle \eta \rangle ^{s} e^{-(\frac{m+1}{2}-\gamma )t} \hat{g}(t,\sigma )(\eta ) \right\| ^{2}_{W^{s,2}({\mathbb {R}}^{m+1}, L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.29)

$$\begin{aligned}&\quad =\sum _{\left| \beta \right| \le s} \left\| \langle \eta \rangle ^{s} \partial ^{\beta }_{(t,\sigma )}\Big ( e^{-(\frac{m+1}{2}-\gamma )t}\hat{g}(t,\sigma ) (\eta ) \Big )\right\| ^{2}_{L^{2}({\mathbb {R}}^{m+1}, L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.30)

$$\begin{aligned}&\quad \le \left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)}^{2}. \end{aligned}$$

(11.31)

In (11.29) we have used the vector-valued version of the standard Sobolev embedding (see Sect. 10). In the same way we obtain

$$\begin{aligned}&\left\| e^{-(\frac{m+1}{2}-\gamma )t}f(t,\sigma )(\eta ) \right\| ^{2}_{L^{\infty } ({\mathbb {R}}^{m+1}\times {\mathbb {R}}^{q}_{u})} \end{aligned}$$

(11.32)

$$\begin{aligned}&\quad \lesssim \left\| f\right\| _{{\mathcal {W}}^{s,\gamma }(M)}^{2}. \end{aligned}$$

(11.33)

Then (11.31) and (11.33) implies that (11.18) and (11.20) are bounded by

$$\begin{aligned} C(s,\gamma ) \left\| f\right\| _{{\mathcal {W}}^{s,\gamma }(M)}^{2} \left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)}^{2}. \end{aligned}$$

(11.34)

Thus the only term remaining is (11.22).

Again using the vector-valued Sobolev embedding we have

$$\begin{aligned}&\left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta )\hat{f}(t,\sigma ) (u)\slash [\eta ]\right\| ^{2}_{L^{\infty }({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.35)

$$\begin{aligned}&\quad \lesssim \left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta )\hat{f} (t,\sigma )(u)\slash [\eta ]\right\| ^{2}_{W^{s,2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.36)

$$\begin{aligned}&\quad =\sum _{\left| \beta \right| \le s}\left\| \frac{\Lambda (\eta )}{ [\eta ]} \partial ^{\beta }_{(t,x)}\bigg ( e^{-(\frac{m+1}{2}-\gamma )t}\hat{f}(t,\sigma ) (u)\bigg )\right\| ^{2}_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.37)

$$\begin{aligned}&\quad \lesssim \left\| f\right\| ^{2}_{{\mathcal {W}}^{s,\gamma }(M)} \end{aligned}$$

(11.38)

as $\left| \frac{\Lambda (\eta )}{[\eta ]}\right| ^{2}=[\eta ]^{2s-(m+1)}\cdot [\eta ]^{-2}\lesssim [\eta ]^{2s-(m+1)}.$ Thus (11.38) and (11.23) implies that (11.22) is bounded by (11.34). $\square$

Corollary 11.2

If $s\in {\mathbb {N}}$ with $s>\frac{q+m+3}{2}$ and $\gamma \ge \frac{m+1}{2}$ then the edge Sobolev space ${\mathcal {W}}^{s,\gamma }(M)$ is a Banach algebra under point-wise multiplication i.e. given $f,g\in {\mathcal {W}}^{s,\gamma }(M)$ we have

$$\begin{aligned} \left\| fg\right\| _{{\mathcal {W}}^{s,\gamma }(M)}\le C'\left\| f\right\| _{{\mathcal {W}}^{s,\gamma }(M)}\left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)} \end{aligned}$$

(11.39)

with a constant $C'$ depending only on s and $\gamma$.

Proof

$\square$

By (11.1) we have $fg\in {\mathcal {W}}^{s,2\gamma -\frac{m+1}{2}}$. Note that $\gamma \ge \frac{m+1}{2}$ if and only if $2\gamma -\frac{m+1}{2}\ge \gamma$ from which the corollary follows immediately.

Corollary 11.3

Let $f,g\in {\mathcal {W}}^{s,\gamma }(M)$ such that $s\in {\mathbb {N}}$ with $s>\frac{q+m+3}{2}$ and $\gamma >\frac{m+1}{2}$. Then

$$\begin{aligned} f g\in {\mathcal {W}}^{s,\gamma +\beta }(M) \end{aligned}$$

(11.40)

for $\beta >0$ given by $\beta =\gamma -\frac{m+1}{2}$.

Proof

By (11.1) we have $fg\in {\mathcal {W}}^{s,2\gamma -\frac{m+1}{2}}$. Moreover $\gamma >\frac{m+1}{2}$ implies $2\gamma -\frac{m+1}{2}=\gamma +\beta$ with $\beta =\gamma -\frac{m+1}{2}>0$.