Let $M$ be a Riemann surface of genus >1, $g$ be an Hermitian metric on $M$. Let $E$ is a holomorphic negative line bundle over $M$, for example, the holomorhic tangent bundle of $M$. Let $h$ be an Hermitian metric on $E$. Suppose the Chern form of $h$ is negative.

Question: Is there a Poincar$\acute{e}$ inequality for $E$? More precisely, for $p\geq 1$,is there a constant $C=C(M,g,E,h,p)$ such that, for any smooth secion $s\in \Gamma(E)$, the following inequality holds, \begin{equation} \int|\bar{\partial}_{E}s|^{p}dV_{g}\geq C\int |s|^{p}dV_{g}. \end{equation} I can show that it is true for $p=2$, but no clue on $p\neq 2$.

Is there any reference on it?

Let $M$ be a Riemann surface of genus >1, $g$ be an Hermitian metric on $M$. Let $E$ is a holomorphic negative line bundle over $M$, for example, the holomorhic tangent bundle of $M$. Let $h$ be an Hermitian metric on $E$. Suppose the Chern form of $h$ is negative.

Question: Is there a Poincaré inequality for $E$? More precisely, for $p\geq 1$,is there a constant $C=C(M,g,E,h,p)$ such that, for any smooth secion $s\in \Gamma(E)$, the following inequality holds, \begin{equation} \int|\bar{\partial}_{E}s|^{p}dV_{g}\geq C\int |s|^{p}dV_{g}. \end{equation} I can show that it is true for $p=2$, but no clue on $p\neq 2$.

Is there any reference on it?