an $L^p_1$ complex vector bundle over a Riemann surface $(\Si,\fJ)$.

1.  A is a $\C$-linear map $$\dbar\!: L^p_1(\Si;E)\lra L^p\big(\Si;T^*\Si^{0,1}\!\otimes\!_{\C}E\big)$$ such that $$\dbar\big(f\xi)=(\dbar{f})\!\otimes\!\xi+f(\dbar\xi)
    \qquad\forall~f\!\in\!C^{\i}(\Si),~\xi\!\in\!\Ga(\Si;E).$$

2.  A (or ) on $(E,\fI)$ is a differential operator of the form D=+A: L\^p\_1(;E)L\^p(;T\^\*\^[0,1]{}\_E), where $\dbar$ is a $\dbar$-operator on $(E,\fI)$ and AL\^p(;\_(E,T\^\*\^[0,1]{}\_E)).\

If $\na$ is an affine connection in $(E,\fI)$, the operator L\^p\_1(;E)L\^p(;T\^\*\^[0,1]{}\_E), (+),is the usual $\dbar$-operator for a unique holomorphic structure in $(E,\fI)$. Furthermore, any $\C$-linear CR-operator is of the form \_ref[ClinCR\_e]{}.\
If $\Si$ and $N\!\lra\!Y$ are as above, an $L^p_1$-map $u\!:\Si\!\lra\!Y$ pulls back a smooth CR-operator $D$ on $N$ to a CR-operator $D_u$ on $u^*N\!\lra\!\Si$ as follows. Suppose $D$ is presented as in \_ref[CRsplit\_e0]{} with $\C$-anti-linear $A$ and $\na$ is a connection in $(N,\fI)$ inducing the corresponding $\dbar$-operator. Let $\ti{u}\!:\ti\Si\!\lra\!Y$ be the map corresponding to $u$ as in Section \[NRS\_subs\] and $$\ti\na\!: L^p_1(\ti\Si;\ti{u}^*N)\lra 
L^p(\ti\Si;T^*\ti\Si\!\otimes\!_{\R}\ti{u}^*N)$$ the connection induced by $\na$. Then, $$D_{\ti{u}}=\frac{1}{2}\Big(\ti\na+\fI\ti\na\circ\fJ\Big)
+A\circ\partial_J \ti{u},
\qquad\hbox{where}\quad
\partial_J \ti{u}=\frac{1}{2}\Big(du-J d\ti{u}\circ \fJ\Big),$$ is a generalized CR-operator on $\ti{u}^*(N,\fI)$; $D_{\ti{u}}$ is independent of the choice of $\na$ if $u$ is $(J,\fJ)$-holomorphic.\
Suppose $(Y,J)$ is an almost complex manifold and $D_Y$ is as above. If $(\Si,\fJ)$ is a Riemann surface and $u\!:\Si\!\lra\!Y$ is a $(J,\fJ)$-holomorphic $L^p_1$-map, then $D_{J;u}\!\equiv\!u^*D_Y$ is 