pairings in \_ref[Aadj\_e]{} extend to pairings $$\begin{split}
L_1^p\big(\Si;E\big)\!\otimes\!
L^p\big(\Si;T^*\Si^{0,1}\!\otimes_{\C}\!\K_{\Si}\!\otimes_{\C}\!E^*(S)\big)
&\lra\R,\\
L^p\big(\Si;T^*\Si^{0,1}\!\otimes_{\C}\!E\big)\!\otimes\!
L_1^p\big(\Si;\K_{\Si}\!\otimes_{\C}\!E^*(S)\big) 
&\lra\R.
\end{split}$$ Furthermore, the equality in \_ref[Aadj\_e]{} holds for all $\eta\in L_1^p\big(\Si;\K_{\Si}\!\otimes_{\C}\!E^*(S)\big)$.

Generalized Cauchy-Riemann operators {#CR_subs}
------------------------------------

\[CR\_dfn\] Let $(Y,J)$ be an almost complex manifold and $(N,\fI)\!\lra\!(Y,J)$ a smooth vector bundle.

1.  A is a $\C$-linear map $$\dbar\!: \Ga(Y;N)\lra \Ga^{0,1}(Y;N)\equiv
    \Ga\big(Y;T^*Y^{0,1}\!\otimes\!_{\C}N\big)$$ such that $$\dbar\big(f\xi)=(\dbar{f})\!\otimes\!\xi+f(\dbar\xi)
    \qquad\forall~f\!\in\!C^{\i}(Y),~\xi\!\in\!\Ga(Y;N).$$

2.  A (or ) on $(N,\fI)$ is a differential operator of the form D=+A: (Y;N)\^[0,1]{}(Y;N),where $\dbar$ is a $\dbar$-operator on $(N,\fI)$ and $$A\in \Ga\big(Y;\Hom_{\R}(N,T^*Y^{0,1}\!\otimes_{\C}\!N)\big).$$\

If $\na$ is an affine connection in $(N,\fI)$, the operator (Y;N)\^[0,1]{}(Y;N), (+J),is a $\dbar$-operator on $(N,\fI)$. Furthermore, any $\C$-linear CR-operator on $(N,\fI)$ is a $\dbar$-operator, and any $\dbar$-operator on $(N,\fI)$ is of the form \_ref[ClinCR\_e0]{} for some (not unique) connection $\na$ in $(N,\fI)$. In particular, $A$ in the decomposition \_ref[CRsplit\_e0]{} can be assumed to be $\C$-anti-linear.\
Let $\na^J$ be the $J$-linear connection in $TY$ obtained from a Levi-Civita connection $\na$ on $Y$ and $A_Y(\cdot,\cdot)$ the of $J$: $$\begin{split}
\na^J_{\xi_1}\xi_2&=\frac{1}{2}\Big(\na_{\xi_1}\xi_2-J\na_{\xi_1}(J\xi_2)\Big)\\
A_Y(\xi_1,\xi_2)&=\frac{1}{4}\Big([\xi_1,\xi_2]+J[\xi_1,J\xi_2]+J[J\xi_1,\xi_2]
-[J\xi_1,J\xi_2]\Big)
\end{split}
\qquad\forall\,\xi_1,\xi_2\!\in\!\Ga(Y;TY).$$ We identify $A_Y$ with the element $$A_Y\in \Ga\big(Y;\Hom_{\R}(TY,T^*Y^{0,1}\!\otimes_{\C}\!TY)\big),
\qquad v\lra A_Y(\cdot,v).$$ Then, $$\dbar_Y\!\equiv \frac{1}{2}\Big(\na^J\xi+J\na^J\circ J\Big), \,
D_Y\!\equiv\! \dbar_Y\!+\!A_Y\!:
\Ga(Y;TY)\lra \Ga^{0,1}(Y;TY)$$ are a $\dbar$-operator on $TY$ and a smooth CR-operator on $TY$, respectively.

\[CR\_dfn2\] Let $(E,\fI)$ be 