Lagrangians, SO(3)-Instantons and Mixed Equation

Abstract

The mixed equation, defined as a combination of the anti-self-duality equation in gauge theory and Cauchy–Riemann equation in symplectic geometry, is studied. In particular, regularity and Fredholm properties are established for the solutions of this equation, and it is shown that the moduli spaces of solutions to the mixed equation satisfy a compactness property which combines Uhlenbeck and Gormov compactness theorems. The results of this paper are used in a sequel to study the Atiyah–Floer conjecture.

Appendices

Appendix A: Elliptic regularity of bundle-valued 1-forms

In this appendix, first we review some well-known results about regularity of the Laplace–Beltrrami operator. Then we consider slight variations to the case of bundle valued maps. Throughout this section, M denotes a compact Riemannian manifold possibly with boundary. In this appendix, for any Riemannian manifold M and differential k-forms α and β on M, we slightly diverge from our notation in (1.20), and denote the inner product of α and β by

$$ \int _{M}\langle \alpha ,\beta \rangle . $$

For any real number r>1, we also write r^∗ for the conjugate of r which satisfies

$$ \frac{1}{r}+\frac{1}{r^{*}}=1. $$

The following lemma is a standard fact about the Laplace–Beltrrami operator (see [GT13, Theorems 9.14 and 9.15], [ADN59, Theorem 15.2] and [Weh042, Chaps. 3 and Appendix D].)

Lemma A.1

Let k be a non-negative integer and p>1 be a real number. Let u be an $L^{p}_{k}$ function on M.

• (i) If k≥1, suppose there is an $L^{p}_{k-1}$ function F on M such that for any smooth function φ with φ|_∂M=0, we have

  $$ \int _{M} \langle u, \Delta \varphi \rangle = \int _{M} \langle F, \varphi \rangle . $$

  (A.2)

  Then u is in $L^{p}_{k+1}(M)$, and there is a constant C, independent of u, such that

  $$ |\!|u|\!|_{L^{p}_{k+1}(M)}\leq C(|\!|F|\!|_{L^{p}_{k-1}(M)}+|\!|u|\!|_{L^{p}(M)}). $$

  (A.3)

  In the case that k=0, the assumption (A.2) has to be replaced with

  $$ |\int _{M} \langle u, \Delta \varphi \rangle |\leq \kappa |\!| \varphi |\!|_{L^{p^{*}}_{1}(M)}, $$

  (A.4)

  and the conclusion (A.3) has to be modified to:

  $$ |\!|u|\!|_{L^{p}_{1}(M)}\leq C(\kappa +|\!|u|\!|_{L^{p}(M)}). $$

  (A.5)

• (ii) If k≥1, suppose there are functions F and G on M such that for any smooth function φ with ∂_νφ|_∂M=0 we have:

  $$ \int _{M} \langle u, \Delta \varphi \rangle = \int _{M} \langle F, \varphi \rangle +\int _{\partial M} \langle G, \varphi \rangle . $$

  (A.6)

  If F and G are respectively in $L^{p}_{k-1}(M)$ and $L^{p}_{k}(M)$, then u is in $L^{p}_{k+1}(M)$. Furthermore, there is a constant C, independent of u, such that

  $$ |\!|u|\!|_{L^{p}_{k+1}(M)}\leq C(|\!|F|\!|_{L^{p}_{k-1}(M)}+|\!|G|\!|_{L^{p}_{k}(M)}+| \!|u|\!|_{L^{p}(M)}). $$

  (A.7)

  In the case that k=0, the assumption (A.6) has to be replaced with:

  $$ |\int _{M} \langle u, \Delta \varphi \rangle |\leq \kappa |\!| \varphi |\!|_{L^{p^{*}}_{1}(M)}, $$

  (A.8)

  and the conclusion (A.9) has to be modified to:

  $$ |\!|u|\!|_{L^{p}_{1}(M)}\leq C(\kappa +|\!|u|\!|_{L^{p}(M)}). $$

  (A.9)

We recall the following definition from Sect. 3.1 about some functions spaces associated to the sections of a vector bundle.

Definition A.10

Suppose U is a (possibly non-compact) manifold with boundary and E is a vector bundle over U. Then the space of smooth sections of E with compact support are denoted by Γ_c(U,E). The space of compactly supported sections of E, which vanish on the boundary of E, are denoted by Γ_τ(U,E). Suppose a connection A₀ is fixed on E. Then Γ_ν(U,E) is the space of all compactly supported sections s of E such that the covariant derivative of s in the normal directions to the boundary of U vanish.

The following Lemma is a slightly generalized version of [Weh051, Lemma A.2].

Lemma A.11

Let k be a positive integer, and r>1 be a real number. Let M be a compact n-manifold with boundary and a Riemannian metric g, U be an open subset of M, and K be an open subspace of U whose closure in U is compact. Let E be an SO(3)-vector bundle over M equipped with a smooth connection A₀. Let σ be a smooth vector field on U. Let Γ_∘(U,E) be one of the spaces Γ_τ(U,E) or Γ_ν(U,E), where Γ_ν(U,E) is defined using A₀. Then there is a constant C such that the following holds. Let

$$ \begin{aligned} &f\in L^{r}_{k}(U,E), \hspace{.5cm} \alpha ,\xi \in L^{r}_{k}(U,\Lambda ^{1}(M)\otimes E), \\ &\zeta \in L^{r}_{k-1}(U,\Lambda ^{1}(M)\otimes E), \hspace{.5cm} \omega \in L^{r}_{k}(U,\Lambda ^{2}(M)\otimes E), \end{aligned} $$

and for any ϕ∈Γ_c(U,E), ψ∈Γ_∘(U,E) we have

$$ \int _{M}\langle \alpha , d_{A_{0}} \phi \rangle =\int _{M} \langle f, \phi \rangle , $$

(A.12)

$$ \int _{M}\langle \alpha , d_{A_{0}}^{*}d_{A_{0}}(\psi \cdot \iota _{ \sigma }g) \rangle = \int _{M}\langle \omega , d_{A_{0}} (\psi \cdot \iota _{\sigma }g)\rangle +\int _{M}\langle \zeta , \psi \cdot \iota _{ \sigma }g \rangle +\int _{\partial M}\langle \xi , \psi \cdot \iota _{ \sigma }g \rangle . $$

(A.13)

Then α(σ) is an element of $L^{r}_{k+1}(K)$ and we have:

$$ |\!| \alpha (\sigma )|\!|_{L^{r}_{k+1}(K)}\leq C (|\!|f|\!|_{L_{k}^{r}(U)}+| \!|\xi |\!|_{L_{k}^{r}(U)} +|\!|\zeta |\!|_{L_{k-1}^{r}(U)}+|\!| \omega |\!|_{L_{k}^{r}(U)}+ |\!| \alpha |\!|_{L_{k}^{r}(U)}). $$

Proof

Without loss of generality, we may assume that U is a precompact open subset of the half-space

$$ \mathbb{H}^{n}:=\{(x_{1},\dots ,x_{n})\in \mathbf{R}^{n}\mid x_{1} \geq 0\}, $$

E is trivialized over U and the connection A₀ is given by a 1-form with values in R³. We will denote this 1-form with A₀, too. We may pick this trivialization in a way that the normal covariant derivative with respect to the connection A₀ agrees with the ordinary derivative. That is to say, Γ_ν(U,R³) defined with respect to A₀ is the space of all compactly supported sections η of R³ such that ∂_νη vanishes along the boundary.

Fix a function ρ:M→R which is supported in U and is equal to 1 on K. Then we show that there are compactly supported maps F and G from U to R³ such that for any η∈Γ_∘(U,R³) we have

$$ \int _{M} \langle \rho \alpha (\sigma ), \Delta \eta \rangle = \int _{M} \langle F, \eta \rangle +\int _{\partial M} \langle G, \eta \rangle $$

(A.14)

and F, G respectively have finite $L^{r}_{k-1}$, $L^{r}_{k}$ norms.

First we claim that

$$\begin{aligned} \int _{M} \langle \rho \alpha (\sigma ), \Delta \eta \rangle =&-\int _{M} \rho \langle \alpha ,d \iota _{\sigma }d\eta \rangle - \int _{M} \langle \alpha , d^{*}(\rho \iota _{\sigma }g \wedge d\eta )\rangle - \int _{M}\rho \operatorname{div}(\sigma ) \langle \alpha , d\eta \rangle \\ &-\int _{M} \rho \langle B_{g}\alpha , d\eta \rangle -\int _{M} \langle \iota _{\sigma }(d\rho \wedge \alpha ), d\eta \rangle , \end{aligned}$$

(A.15)

where B_g is defined by firstly taking the Lie derivative $\mathcal {L}_{\sigma }g$ of the Riemannian metric g and then requiring B_g to satisfy the following identity for any pair of 1-forms β and β′:

$$ \mathcal {L}_{\sigma }(g)(\beta ,\beta ')=\langle B_{g} \beta ,\beta ' \rangle. $$

To see (A.15), we pick a sequence {α_i}_i∈N of smooth 1-forms on U with values in R³ such that α_i vanishes in a neighborhood of $U\cap \partial {\mathbb{H}}^{n}$ and the sequence {α_i} is L^r-convergent to α. Then the left hand side of (A.15) is equal to

$$\begin{aligned} &\lim _{i\to \infty}\int _{M} \langle \rho \alpha _{i}(\sigma ),d^{*}d \eta \rangle \\ &\qquad =\lim _{i\to \infty} \int _{M} \langle d\iota _{\sigma }( \rho \alpha _{i}),d\eta \rangle =\lim _{i\to \infty}\left [ \int _{M} \langle \mathcal {L}_{\sigma }(\rho \alpha _{i}),d \eta \rangle - \langle \mathcal {\iota}_{\sigma }d(\rho \alpha _{i}),d \eta \rangle \right ] \\ &\qquad =\lim _{i\to \infty}\left [-\int _{M} \langle \rho \alpha _{i}, \mathcal {L}_{\sigma }d \eta \rangle - \int _{M}\operatorname{div}( \sigma ) \langle \rho \alpha _{i},d \eta \rangle -\int _{M}\mathcal {L}_{ \sigma}(g)(\rho \alpha _{i}, d \eta )\right . \\ &\quad \qquad{} \left . -\int _{M}\langle d\alpha _{i}, \rho \iota _{\sigma }g \wedge d\eta \rangle -\int _{M}\langle \iota _{\sigma }(d\rho \wedge \alpha _{i}), d\eta \rangle \right ] \\ &\qquad =\lim _{i\to \infty}\left [-\int _{M} \rho \langle \alpha _{i},d \iota _{\sigma }d\eta \rangle - \int _{M} \rho \operatorname{div}( \sigma ) \langle \alpha _{i}, d\eta \rangle -\int _{M} \rho \langle B_{g} \alpha _{i}, d\eta \rangle \right . \\ &\quad \qquad {}\left . -\int _{M}\langle \alpha _{i}, d^{*}(\rho \iota _{\sigma }g \wedge d\eta )\rangle -\int _{M}\langle \iota _{\sigma }(d\rho \wedge \alpha _{i}), d\eta \rangle \right ]. \end{aligned}$$

(A.16)

Now by taking the limit in (A.16) we obtain the desired identity.

The assumption (A.12) can be used to rewire the first term in the right hand side of (A.15) as

$$\begin{aligned} \int _{M} \rho \langle \alpha ,d \iota _{\sigma }d\eta \rangle =& \int _{M} \langle \alpha ,d_{A_{0}}(\rho \iota _{\sigma }d\eta ) \rangle -\int _{M} \langle \alpha ,\rho [A_{0},\iota _{\sigma }d \eta ] \rangle - \int _{M} \langle \alpha ,\mathcal {(}d\rho )\cdot ( \iota _{\sigma }d\eta ) \rangle \\ =&\int _{M} \langle \rho f-*[\rho \alpha ,*A_{0}]-*(\alpha \wedge *d \rho ), \iota _{\sigma }d\eta \rangle \\ =&\int _{M} \langle \left (\rho f-*[\rho \alpha ,*A_{0}]-*(\alpha \wedge *d\rho )\right )\iota _{\sigma }g, d\eta \rangle \\ =&\int _{M} \langle d^{*}\mathopen{}\left ( \left (\rho f-*[\rho \alpha ,*A_{0}]-*(\alpha \wedge *d\rho )\right )\iota _{\sigma }g \right )\mathclose{},\eta \rangle \\ &+\int _{\partial M} \langle *_{n-1}*(\left (\rho f-*[\rho \alpha ,*A_{0}]-*( \alpha \wedge *d\rho )\right )\iota _{\sigma }g), \eta \rangle , \end{aligned}$$

(A.17)

where ∗_n−1 in the last line denotes the Hodge operator on ∂M.

We rewrite the second term in the right hand side of (A.15) as

$$\begin{aligned} \int _{M}\langle \alpha , d^{*}(\rho \iota _{\sigma }g \wedge d\eta ) \rangle =&\int _{M}\langle \alpha , d^{*}( \eta d(\rho \iota _{ \sigma }g))\rangle -\int _{M}\langle \alpha , d^{*}d(\rho \iota _{ \sigma }g \eta )\rangle \\ =&\int _{M}\langle d \alpha , \eta d(\rho \iota _{\sigma }g) \rangle - \int _{\partial M}\langle *_{n-1}(\alpha \wedge *d(\rho \iota _{ \sigma }g)), \eta \rangle \\ &-\int _{M}\langle \alpha , d_{A_{0}}^{*}d_{A_{0}}(\iota _{\sigma }g \rho \eta )\rangle + (-1)^{n-1}\int _{M} \langle \alpha , *[A_{0},*d( \iota _{\sigma }g \rho \eta )]\rangle \\ &+\int _{M}\langle \alpha , d^{*}[A_{0},\iota _{\sigma }g \rho \eta ] \rangle +(-1)^{n-1}\int _{M} \langle \alpha , *[A_{0},*[A_{0},\iota _{ \sigma }g \rho \eta ]]\rangle . \end{aligned}$$

Therefore, we can use (A.13), to write

$$\begin{aligned} \int _{M}\langle \alpha , d^{*}(\rho \iota _{\sigma }g \wedge d\eta ) \rangle =&\int _{M}\langle *(d \alpha \wedge *d(\rho \iota _{\sigma }g)), \eta \rangle -\int _{\partial M}\langle *_{n-1}(\alpha \wedge *d( \rho \iota _{\sigma }g)), \eta \rangle \\ &-\int _{M}\langle \omega ,d_{A_{0}}(\iota _{\sigma }g \rho \eta ) \rangle -\int _{M}\langle \zeta ,\iota _{\sigma }g \rho \eta \rangle - \int _{\partial M}\langle \xi ,\iota _{\sigma }g \rho \eta \rangle \\ &+(-1)^{n-1}\int _{M} \langle \alpha , *[A_{0},*d(\iota _{\sigma }g \rho \eta )]\rangle +\int _{M}\langle \alpha , d^{*}[A_{0},\iota _{ \sigma }g \rho \eta ]\rangle \\ &+(-1)^{n-1}\int _{M} \langle \alpha , *[A_{0},*[A_{0},\iota _{ \sigma }g \rho \eta ]]\rangle . \end{aligned}$$

(A.18)

Finally, the last three terms of (A.15) are equal to

$$\begin{aligned} &{-}\int _{M}\langle d^{*}\left [(B_{g}+\operatorname{div}(\sigma )) \rho \alpha -\iota _{\sigma }(\alpha \wedge d\rho )\right ], \eta \rangle \\ &\qquad -\int _{\partial M}\langle *_{n-1}*[(B_{g}+\operatorname{div}( \sigma ))\rho \alpha -\iota _{\sigma }(\alpha \wedge d\rho )],\eta \rangle. \end{aligned}$$

(A.19)

By applying further integration by parts to the expressions in (A.18), we can find F and G satisfying (A.14), which are respectively in $L^{r}_{k-1}$ and $L^{r}_{k}$, and satisfy

$$ |\!|F|\!|_{L^{r}_{k-1}}+|\!|G|\!|_{L^{r}_{k}(U)}\leq C'(|\!|f|\!|_{L_{k}^{r}(U)}+| \!|\omega |\!|_{L_{k}^{r}(U)}+ |\!|\zeta |\!|_{L_{k-1}^{r}(U)}+|\!| \xi |\!|_{L_{k}^{r}(U)}+ |\!| \alpha |\!|_{L_{k}^{r}(U)}) $$

for some constant C′ depending only on A₀, g, σ, U and K. Therefore, Lemma A.1 (part (i) or (ii) depending on whether ∘=τ or ν) implies that

$$ |\!| \rho \alpha (\sigma )|\!|_{L^{r}_{k+1}(U)} \leq C (|\!|f|\!|_{L_{k}^{r}(U)}+|\!|\omega |\!|_{L_{k}^{r}(U)} +|\!| \zeta |\!|_{L_{k-1}^{r}(U)}+|\!|\xi |\!|_{L_{k}^{r}(U)}+ |\!| \alpha | \!|_{L_{k}^{r}(U)}). $$

This inequality proves the desired claim. □

The following lemma is an extension of the previous lemma to the case that k=0.

Lemma A.20

Let r, M, K, U, σ, E and A₀ be as in Lemma A.11. Let ∘ be either τ and ν. There is a constant C such that the following holds. Let α be an L^r section of Λ¹⊗E over the open subset U of M such that for any ϕ∈Γ_c(U,E) and ψ∈Γ_∘(U,E):

$$ | \int _{M}\langle \alpha , d_{A_{0}} \phi \rangle |\leq C_{1} |\!| \phi |\!|_{L^{r^{*}}(U)}, \hspace{1cm} | \int _{M}\langle \alpha , d_{A_{0}}^{*}d_{A_{0}}(\psi \cdot \iota _{ \sigma }g) \rangle |\leq C_{2} |\!|\psi |\!|_{L_{1}^{r^{*}}(U)}. $$

(A.21)

Then α(σ) belongs to $L^{r}_{1}(K)$ and

$$ |\!| \alpha (\sigma )|\!|_{L^{r}_{1}(K)}\leq C(C_{1}+C_{2}+|\!| \alpha |\!|_{L^{r}(U)}). $$

(A.22)

Proof

In the following C is a constant independent of α which might increase from each line to the next one. As in the proof of Lemma A.11, we can show that α satisfies (A.15). In particular, we have

$$\begin{aligned} |\int _{M} \langle \rho \alpha (\sigma ), \Delta \eta \rangle |\leq & \hspace{2mm} |\int _{M} \rho \langle \alpha ,d \iota _{\sigma }d\eta \rangle |+ | \int _{M}\langle \alpha , d^{*}(\rho \iota _{\sigma }g \wedge d\eta ) \rangle |+ |\int _{M}\rho \operatorname{div}(\sigma ) \langle \alpha , d\eta \rangle | \\ &+|\int _{M} \rho \langle B_{g}\alpha , d\eta \rangle |+ |\int _{M} \langle \iota _{\sigma }(d(\rho )\wedge \alpha ), d\eta \rangle |. \end{aligned}$$

(A.23)

The first term on the left hand side of the above inequality can be estimated as in (A.17):

$$\begin{aligned} |\int _{M} \rho \langle \alpha ,d \iota _{\sigma }d\eta \rangle | \leq &|\int _{M} \langle \alpha ,d_{A_{0}}(\rho \iota _{\sigma }d \eta ) \rangle |+|\int _{M} \langle \alpha ,\rho [A_{0},\iota _{ \sigma }d \eta ] \rangle |+ |\int _{M} \langle \alpha ,\mathcal {(}d \rho )\cdot (\iota _{\sigma }d\eta ) \rangle | \\ \leq &C(C_{1}+|\!| \alpha |\!|_{L^{r}(U)})|\!|\eta |\!|_{L^{r^{*}}_{1}(U)}. \end{aligned}$$

(A.24)

To. obtain the second inequality, we use the first assumption in (A.21). Next, we find an upper bound for the second term in (A.23) using the second inequality in (A.21) following an argument similar to the previous lemma:

$$\begin{aligned} |\int _{M}\langle \alpha , d^{*}(\rho \iota _{\sigma }g \wedge d\eta ) \rangle | \leq &|\int _{M}\langle \alpha , d^{*}( \eta d(\rho \iota _{ \sigma }g))\rangle |+ |\int _{M}\langle \alpha , d_{A_{0}}^{*}d_{A_{0}}( \iota _{\sigma }g \rho \eta )\rangle | \\ &+|\int _{M} \langle \alpha , *[{A_{0}},*d(\iota _{\sigma }g \rho \eta )]\rangle |+ |\int _{M}\langle \alpha , d^{*}[{A_{0}},\iota _{ \sigma }g \rho \eta ]\rangle | \\ &+|\int _{M} \langle \alpha , *[{A_{0}},*[{A_{0}},\iota _{\sigma }g \rho \eta ]]\rangle | \\ \leq &C(C_{2}+|\!| \alpha |\!|_{L^{r}(U)})|\!|\eta |\!|_{L^{r^{*}}_{1}(U)}. \end{aligned}$$

(A.25)

It is straightforward to bound the remaining three terms in (A.23) with $C|\!| \alpha |\!|_{L^{r}(U)}|\!|\eta |\!|_{L^{r^{*}}_{1}(U)}$. Consequently, Lemma A.1 implies that α(σ) is in $L^{r}_{1}(K)$ and (A.22) holds. □

Lemma A.26

Let k be a non-negative integer and r>1 is a real number. Suppose M is a Riemannian manifold possibly with boundary. Suppose Σ is a closed surface and F is an SO(3)-bundle over Σ. Suppose β={β_x}_x∈M is a smooth family of connections on F parametrized by M. Suppose f is an $L^{r}_{k}$ section of the bundle T^∗Σ⊗F over Σ×M. If k≥1, suppose there are $L^{r}_{k}$ sections ζ₁ and ζ₂ of the pullback of F over Σ×M such that for any smooth section ξ of the pullback of F over Σ×M, we have

$$ \int _{M\times \Sigma} \langle f, d_{\beta} \xi \rangle = \int _{M \times \Sigma} \langle \zeta _{1}, \xi \rangle , \hspace{1cm} \int _{M\times \Sigma} \langle f, *_{\Sigma }d_{\beta }\xi \rangle = \int _{M\times \Sigma} \langle \zeta _{2}, \xi \rangle , $$

(A.27)

where d_βξ denotes the section of T^∗Σ⊗F over Σ×M given by the exterior derivatives of ξ in the Σ direction with respect to the family of connections β. Then $\nabla ^{\beta}_{\Sigma }f$, the covariant derivative of f in the Σ direction with respect to β, is in $L^{r}_{k}$, and there is a constant C, independent of f, such that:

$$ |\!|\nabla ^{\beta}_{\Sigma }f|\!|_{L^{r}_{k}(M\times \Sigma )}\leq C(| \!|\xi _{1}|\!|_{L^{r}_{k}(M\times \Sigma )}+|\!|\xi _{2}|\!|_{L^{r}_{k}(M \times \Sigma )}+|\!|f|\!|_{L^{r}_{k}(M\times \Sigma )}). $$

(A.28)

In the case that k=0, the assumption (A.27) has to be replaced with

$$ |\int _{\Sigma \times X} \langle f, d_{\beta} \xi \rangle |+ |\int _{ \Sigma \times X} \langle f,*_{\Sigma }d_{\beta}\xi \rangle | \leq \kappa |\!|\xi |\!|_{L^{r^{*}}(\Sigma \times X)}. $$

(A.29)

In this case, $\nabla ^{\beta}_{\Sigma }f$ belongs to L^r(X×Σ) and

$$ |\!|\nabla ^{\beta}_{\Sigma }f|\!|_{L^{r}(X\times \Sigma )}\leq C( \kappa +|\!|f|\!|_{L^{r}(X\times \Sigma )}). $$

(A.30)

Lemma A.26 can be regarded as the family version of A.1 where we also replace the degree two elliptic operator Δ with the degree one operator $d_{\beta}\oplus d_{\beta}^{*}$. This proposition in the case that F is the trivial bundle and β is the trivial family of connections is proved in [Weh051, Lemma 2.9]. Clearly, this implies the lemma for the case that F is trivial and B is arbitrary. The proof in the case that F is non-trivial is similar.

Appendix B: Regularity of holomorphic curves in a Banach space

Suppose B is a Banach space and M is a compact Riemannian manifold. In this appendix, we are interested in maps from M to B. For 1<p<∞ and any non-negative integer k, we can define the Sobolev norm $|\!|\cdot |\!|_{L^{p}_{k}}$ on the space of such maps in the usual way. The completion of space of smooth maps from M to B with respect to this Sobolev norm is denoted by $L^{p}_{k}(M,B)$. As an example, let B=L^p(N) for a compact manifold N. Any function in C^∞(M×N), determines an element of L^p(M,B). In fact, the space of smooth functions on M×N is dense in L^p(M,B) (see [Weh041] and [Lip14]). This gives us the following identifications of Sobolev spaces:

$$ L^{p}(M,L^{p}(N))=L^{p}(N,L^{p}(M))=L^{p}(M\times N). $$

More generally, C^∞(M×N) is dense in $L^{p}_{k}(M,L^{p}(\Sigma ))$ for any non-negative integer k, and we have (see [Weh041, Lip14]):

$$ \begin{aligned} &L^{p}_{k}(M\times N)=L^{p}_{k}(M,L^{p}(N))\cap L^{p}_{k}(N,L^{p}(M)), \\ &L^{p}_{k}(M, L^{p}(N))=L^{p}(N,L^{p}_{k}(M)). \end{aligned} $$

(B.1)

For the rest of this appendix, we fix B_p to be a Banach space that can be identified with a closed subspace of the space L^p(N) for a closed manifold N. In particular, the intersection B_q:=B_p∩L^q(N) with q>p determines a closed subspace of L^q(N). For q<p, B_q is the closure of B_p in L^q(N).

Lemma B.2

[Weh041] and [Lip14]

Suppose M is a Riemannian manifold with boundary. Let k be a non-negative integer and p>1 be a real number. Let $u\in L^{p}_{k}(M,B_{p})$. Then the same claims as in parts (i) and (ii) of Lemma A.1 hold if we assume that F, G and φ are B_p-valued.

Sketch of the Proof.

Without loss of generality, we can assume that B_p=L^p(N). Using the identifications in (B.1), we can regard u as an L^p map from N to the Banach space $L^{p}_{k}(M)$. Next, we can apply the properties of the Laplacian operator acting on $L^{p}_{k}(M)$ to obtain the desired conclusions. For more details, we refer the reader to [Weh041, Lemma 2.1] and [Lip14, Sect. 3.3]. □

The proof of the following proposition about regularity of Banach valued Cauchy–Riemann equation can be found in [Weh041, Theorem 1.2] and [Lip14, Lemmas 27 and 28]. In this proposition, B_p denotes the direct sum B_p⊕B_p. This space admits an obvious complex structure J₀ given by

$$ \mathcal {J}_{0}(v_{0},v_{1})=(-v_{1},v_{0}). $$

(B.3)

The subspace $\mathcal {L}:=0\oplus B_{p}$ defines a completely real subspace of B_p with respect to $\mathcal {J}_{0}$.

Proposition B.4

Suppose U is a bounded open subspace of

$$ {\mathbb{H}}^{2}:=\{(s,\theta )\in \mathbf{R}^{2}\mid s \geq 0\}, $$

and U_∂ denotes the intersection ${\mathbb{H}}^{2}\cap U$. Suppose $\mathcal {J}:\mathbf{B}_{p}\to {\mathrm{End}}(\mathbf{B}_{p},\mathbf{B}_{p})$ is a smooth family of complex structures such that $\mathcal {J}(x)=\mathcal {J}_{0}$ for $x\in \mathcal {L}$. For p>2 and k≥2, suppose u:U→B_p is an $L^{p}_{k}$ map that satisfies

$$ \partial _{\theta }u-\mathcal {J}(u)\partial _{s} u=z\in L^{p}_{k}(U, \mathbf{B}_{p}), $$

(B.5)

and the boundary condition

$$ u|_{U_{\partial}}\subset \mathcal {L}. $$

(B.6)

Then for any open subspace K⊂U, whose closure in U is compact, the map u is in $L^{p}_{k+1}(K)$. Moreover, there is a constant C, depending only on K, such that

$$ |\!|u|\!|_{L^{p}_{k+1}(K)}\leq C(|\!|z|\!|_{L^{p}_{k}(U)}+|\!|u|\!|_{L^{p}}(U)). $$

(B.7)

If u_i:U→B_p is a sequence of $L^{p}_{k}$ map that satisfies

$$ \partial _{\theta }u_{i}-\mathcal {J}(u_{i})\partial _{s} u_{i}=z_{i} \in L^{p}_{k}(U,\mathbf{B}_{p}), $$

(B.8)

such that u_i and z_i are respectively $L^{p}_{k}$-convergent to u and z, then u_i restricted to K is $L^{p}_{k+1}$-convergent to the restriction of u to K. In the case that k=1, similar results hold if we replace $L^{p}_{k+1}$ with $L^{p/2}_{k+1}$.

Sketch of the proof

For k≥2, suppose u is a map that satisfies (B.5) and (B.6). We apply $\partial _{\theta}+\mathcal {J}(u)\partial _{s}$ to (B.5). Then we have:

$$ \partial _{s}^{2} u+\partial _{\theta}^{2} u=\mathcal {J}(u)\partial _{s}( \mathcal {J}(u))\partial _{s}u+ \partial _{\theta}(\mathcal {J}(u)) \partial _{s}u +\partial _{\theta }z+\mathcal {J}(u)\partial _{s}z. $$

(B.9)

Using the assumptions k≥2, $u\in L^{p}_{k}$ and $z\in L^{p}_{k}$, we can conclude that the left hand side of the above identity is an element of $L^{p}_{k-1}$. The maps u and z can be written as (u₀,u₁) and (z₀,z₁) with respect to the decomposition of B_p. The boundary condition (B.6) implies that $u_{0}|_{U_{\partial}}=0$ and $\partial _{s}u_{1}|_{U_{\partial}}=z_{0}|_{U_{\partial}}$. Therefore, we can invoke Lemma B.2 to verify the claim. To be a bit more detailed, we use the assumption k≥2 to conclude that the products of two $L^{p}_{k-1}$ functions are still in $L^{p}_{k-1}$. In the case that k=1, the products of two L^p(U,B_p) functions is in L^p/2(U,B_p), which in turn is a subspace of L^p/2(U,B_p/2). That allows us to use the same argument to prove the claim in this case. The sequential versions of these claims can be also treated similarly. □

We need a slight improvement of Proposition B.4 to the case k=0 [Lip14, Lemma 29].

Proposition B.10

Suppose U is given as in Proposition B.4. Suppose $\mathcal {J}:\mathbf{B}_{p}\to {\mathrm{End}}(\mathbf{B}_{p},\mathbf{B}_{p})$ is a smooth family of complex structures such that $\mathcal {J}(x)=\mathcal {J}_{0}$ for $x\in \mathcal {L}$ and for any x∈B_p, the space $\mathcal {L}$ is totally real with respect to $\mathcal {J}(x)$, i.e., $\mathbf{B}_{p}=\mathcal {L}\oplus \mathcal {J}(x)\mathcal {L}$. For p>2, let u:U→B_p be in $L^{p}_{1}$. Suppose q>p and u is also an L^q map from U to B_q. Suppose u satisfies

$$ \partial _{\theta }u-\mathcal {J}(u)\partial _{s} u=z\in L^{q}(U, \mathbf{B}_{q}), $$

(B.11)

and the boundary condition (B.6). Then u is an $L^{q}_{1}$ map from U to B_q and

$$ |\!|u|\!|_{L^{q}_{1}}\leq C(|\!|z|\!|_{L^{q}}+|\!|u|\!|_{L^{q}}). $$

(B.12)

Moreover, if u_i:U→B_p are $L^{q}_{1}$ solutions of

$$ \partial _{\theta }u_{i}-\mathcal {J}(u_{i})\partial _{s} u_{i}=z_{i} \in L^{q}(U,\mathbf{B}_{q}), $$

(B.13)

such that u_i is convergent to u in $L^{p}_{1}\cap L^{q}$ and z_i is convergent to z in L^q, then u_i is convergent to u in $L^{q}_{1}$.

Proof

Given p>2 and any bounded domain Ω in R² with smooth boundary, let $L^{p}_{1}(\Omega ,\mathbf{B}_{p})_{\partial}$ be the space of $L^{p}_{1}$ maps u:Ω→B_p such that the restriction of u to the boundary is in $\mathcal {L}$. Then the Cauchy–Riemann operator

$$ \partial _{\theta }-\mathcal {J}_{0}\partial _{s} :L^{p}_{1}(\Omega , \mathbf{B}_{p})_{\partial }\to L^{p}(\Omega ,\mathbf{B}_{p}) $$

(B.14)

is a surjective bounded operator with kernel being constant maps to $\mathcal {L}$. This can be seen in the same way as in Lemma B.2.

Now suppose x∈∂U and $D_{r}(x)=B_{r}(x)\cap {\mathbb{H}}^{2}$ is contained in U. Suppose Ω_r is the region given by rounding the corners of D_r(x) such that it is contained in D_r(x) and it contains D_r/2(x). Since $\mathcal {J}(u):U \to {\mathrm{End}}(\mathbf{B}_{p},\mathbf{B}_{p})$ is continuous and $\mathcal {J}(x)=J_{0}$, the operator $\partial _{\theta }-\mathcal {J}(u)\partial _{s}:L^{p}_{1}(\Omega _{r}, \mathbf{B}_{p})_{\partial }\to L^{p}(\Omega _{r},\mathbf{B}_{p})$ is surjective with kernel being constant maps to $\mathcal {L}$ if r is small enough. This holds because the operator $\partial _{\theta }-\mathcal {J}(u)\partial _{s}$ is a deformation of the operator in (B.14) by a bounded operator of small norm for small values of r. We assume that r is chosen such that the same claim holds if we replace q with p. Now let ρ:Ω_r→R be a smooth bump function that vanishes on the complement of D_r/2(x) and equals 1 on D_r/3(x). Then our assumption implies that ρu is an element of $L^{p}_{1}(\Omega _{r},\mathbf{B}_{p})_{\partial}$ and

$$ \partial _{\theta}(\rho u) -\mathcal {J}(u)\partial _{s}(\rho u)=\rho z+ \partial _{\theta}(\rho ) u -\mathcal {J}(u)\partial _{s}(\rho ) u $$

is in L^q. Thus there is $u'\in L^{q}_{1}(\Omega ,\mathbf{B}_{q})_{\partial}$ such that

$$ \partial _{\theta }u' -\mathcal {J}(u)\partial _{s}u'=\rho z+\partial _{ \theta}(\rho ) u -\mathcal {J}(u)\partial _{s}(\rho ) u. $$

This implies that u′−ρu is a constant map to $\mathcal {L}$. In particular, the restriction of u to D_r/3(x) is in $L^{q}_{1}(\Omega ,\mathbf{B}_{q})_{\partial}$. For an interior point x, we may apply a similar argument to show that the restriction of u to a neighborhood of x in is $L^{q}_{1}(\Omega ,\mathbf{B}_{q})_{\partial}$. The only new point that we need is that we can find an isomorphism T:B_p→B_p such that $T^{-1} \mathcal {J}(x) T=\mathcal {J}_{0}$. In fact, we may take T to be the linear map that sends (v₀,v₁)∈B_p⊕B_p to $(v_{0},0)+\mathcal {J}(x)(v_{1},0)$. Since $\mathcal {L}$ is totally with respect to $\mathcal {J}(x)$, T is an isomorphism. □