element $z_0\!\in\!S$ a real subspace $E_{z_0}'\!\subset\!E_{z_0}$, let $$L_1^p\big(\Si;T^*\Si^{1,0}\!\otimes_{\C}\!E(\vr)\big)=\big\{
\eta\!\in\!L_1^p\big(\Si;E(S)\big)\!: \,
\Res_{z=z_0}\eta\!\in\!E_{z_0}'~\forall\,z_0\!\in\!S\big\}.$$\

By a $\Si$ we will mean a compact complex one-dimensional manifold with pairs of distinct points identified. In other words, =/\~, x\_i\^[(1)]{}\~x\_i\^[(2)]{}  i=1,…,m,for some smooth compact Riemann surface $\ti\Si$ and distinct points $x_i^{(1)},x_i^{(2)}\!\in\!\ti\Si$. The quotient map $$\si\!: \ti\Si\lra\Si$$ is determined by $\Si$ up to an isomorphism. We will denote by $$\Si_{\sing}\equiv\big\{\si(x_i^{(1)})\!:\,i\!=\!1,\ldots,m\big\}
\subset\Si
\qquad\hbox{and}\qquad
\ti\Si_{\sing}\equiv\big\{x_i^{(1)},x_i^{(2)}\!:\,i\!=\!1,\ldots,m\big\}
\subset\ti\Si$$ the subset of of $\Si$ and its preimage under $\si$, respectively. Let $\Si^*\!\subset\!\Si$ be the subspace of , i.e. the complement of $\Si_{\sing}$.\
If $Y$ is a smooth manifold and $\Si$ is a Riemann surface as above, an is an $L^p_1$-map $$\ti{u}\!:\ti\Si\lra Y \qquad\hbox{s.t.}\quad 
\ti{u}\big(x_i^{(1)}\big)=\ti{u}\big(x_i^{(2)}\big)~~\forall\,i=1,\ldots,m.$$ By a $E\!\lra\!\Si$, we will mean a topological complex vector bundle such that $\si^*E\!\lra\!\ti\Si$ is an $L^p_1$-complex vector bundle. Let $$\begin{split}
L_1^p(\Si;E)
&=\big\{\xi\!\in\!L_1^p(\ti\Si;\si^*E\big)\!:\,
\xi(x_i^{(1)})\!=\!\xi(x_i^{(2)})~\forall\,i\!=\!1,\ldots,m\big\};\\
L^p\big(\Si;T^*\Si^{0,1}\!\otimes\!_{\C}E\big)
&=L^p\big(\ti\Si;T^*\ti\Si^{0,1}\!\otimes\!_{\C}\si^*E\big).
\end{split}$$ If $S$ is a finite subset of $\Si^*$, let $\ti{S}\!=\!\si^{-1}(S)$ and define $$\label{dualsheafdfn_e}\begin{split}
L_1^p\big(\Si;\K_{\Si}\!\otimes_{\C}\!E(S)\big)
&=\Big\{\eta\!\in\!
L_1^p\big(\ti\Si;T^*\ti\Si^{1,0}\!\otimes_{\C}\!
\si^*E(\ti{S}\!\cup\!\ti\Si_{\sing})\big)\!:
\\ &\qquad\qquad\qquad
\sum_{\ti{z}_0\in\si^{-1}(z_0)}\!\!\!\!\!\!\Res_{z=\ti{z}_0}\eta(\ti{z}_0)\!=\!0~~\forall\,
z_0\!\in\!\Si_{\sing}\Big\},\\
L^p\big(\Si;T^*\Si^{0,1}\!\otimes_{\C}\!\K_{\Si}\!\otimes\!_{\C}E(S)\big)
&=L^p\big(\ti\Si;T^*\ti\Si^{0,1}\!\otimes_{\C}\!T^*\ti\Si^{1,0}\!\otimes_{\C}\!
\si^*E(\ti{S}\!\cup\!\ti\Si_{\sing})\big).
\end{split}$$ If $\vr$ is a function assigning to each element $z_0\!\in\!S$ a real subspace $E_{z_0}'\!\subset\!E_{z_0}$, let L\_1\^p(;\_\_E())={ L\_1\^p(;\_\_E(S)): \_[z=\^[-1]{}(z\_0)]{}E\_[z\_0]{}’ z\_0S}.Similarly, we define $$\begin{split}
L_1^p\big(\Si;E(-S)\big)
&=\big\{\xi\!\in\!L_1^p\big(\Si;E)\!:\,\xi(z_0)\!=\!0~\forall\,z_0\!\in\!S\big\},\\
L_1^p\big(\Si;E^*(-\vr)\big)
&=\big\{\xi\!\in\!L_1^p\big(\Si;E^*)\!:\,\xi(z_0)\!\in\!\Ann(E_{z_0}')
~\forall\,z_0\!\in\!S\big\},
\end{split}$$ where $\Ann(E_{z_0}')\!\subset\!\Hom_{\R}(E_{z_0},\R)$ is the annihilator of $E_{z_0}'\!\subset\!E_{z_0}$. The real 