the linearization of the $\dbar_J$-operator on the space of $L^p_1$-maps from $\Si$, with complex structure fixed, to $Y$; see [@McS Section 3.1]. If in addition, $(Y,J)$ is an almost complex submanifold of an almost complex manifold $(X,J)$, then $$D_{J;u}\!\equiv\!D_{J;u}^Y\!\equiv\!u^*D_Y\!:
L^p_1(\Si;u^*TY)\lra L^p\big(\Si;T^*\Si^{0,1}\!\otimes\!_{\C}u^*TY\big)$$ is the restriction of $$D_{J;u}^X\!\equiv\!u^*D_X\!:
L^p_1(\Si;u^*TX)\lra L^p\big(\Si;T^*\Si^{0,1}\!\otimes\!_{\C}u^*TX\big).$$ Thus, $D_{J;u}^X$ induces a CR-operator $$D_{J;u}^{NY}\!:
L^p_1(\Si;u^*NY)\lra L^p\big(\Si;T^*\Si^{0,1}\!\otimes\!_{\C}u^*NY\big),$$ where $NY\!\equiv\!TX|_Y/TY$ is the complex normal bundle of $Y$ in $X$.\
The next lemma extends Serre duality from $\dbar$-operators to CR-operators. If $D$ is as in \_ref[CRdfn\_e]{}, let $$D^*=\dbar-A^*\!: L^p_1(\Si;\K_{\Si}\!\otimes\!_{\C}E^*)\lra 
L^p(\Si;T^*\Si^{0,1}\!\otimes_{\C}\K_{\Si}\!\otimes\!_{\C}E^*);$$ see \_ref[Aadj\_e]{} and \_ref[dualsheafdfn\_e]{} for notation. If $S\!\subset\!\Si$ is a finite subset of smooth points of $\Si$ and $\vr$ is a function assigning to $z_0\!\in\!S$ a complex subspace of $E_{z_0}^*$, $D^*$ extends to an operator $$D_{\vr}^*\!: L^p_1\big(\Si;\K_{\Si}\!\otimes\!_{\C}E^*(\vr)\big)\lra 
L^p\big(\Si;T^*\Si^{0,1}\!\otimes_{\C}\K_{\Si}\!\otimes\!_{\C}E^*(S)\big);$$ see \_ref[dualsheafdfn\_e2]{}. Let $D_{\vr}$ be the restriction of $D$ to the closed subspace $L^p_1(\Si;E(-\vr))$ of $L^p_1(\Si;E)$.

\[serre\_lmm\] Let $D$ be a CR-operator on a complex vector bundle $(E,\fI)$ over a Riemann surface $(\Si,\fJ)$. If $S$ is a finite subset of smooth points of $\Si$ and $\vr$ is a function assigning to $z_0\!\in\!S$ a real subspace of $E_{z_0}^*$, the homomorphism D\_\_(D\_\^\*,), ,is an isomorphism.

[*Proof:*]{} If $\Si$ is smooth and $S\!=\!\eset$, this is [@IvSh Lemma 2.3.2]. 