$H^0(X, \omega_X(A))\neq 0$, then, for every $e\in\mathbb Z_{>0}$, the trace map $${{\operatorname{Tr}}}_X^e(A):H^0(X, \omega_X(p^eA))\to H^0(X, \omega_X(A))$$ is a non-zero map.

If $g(X)\geq 1$ where $g(X)$ is the genus of $X$, then the assertion follows from Theorem \[curve-trace-nonzero\]. Thus, we may assume that $X\simeq \mathbb P^1$. Since $\mathbb P^1$ is $F$-split, the trace map is surjective.

In characteristic zero, we obtain the following result by the Kodaira vanishing theorem. In positive characteristic, we obtain the following result by the trace map of Frobenius (Corollary \[curve-trace-cor\]).

\[surface-restriction\] Let $X$ be a smooth projective surface and let $C$ be a smooth prime divisor. Let $A$ be an ample [$\mathbb{Z}$]{}-divisor on $X$. If $H^0(C, K_C+A)\neq0$, then the natural restriction map $$H^0(X, K_X+C+A)\to H^0(C, K_C+A)$$ is a non-zero map.

By Lemma \[trace-diagram\], we obtain the following commutative diagram [$$\begin{CD}
H^0(X, K_X+C+p^eA)@>>> H^0(C, K_{C}+p^eA)@>>> H^1(X, K_X+p^eA)\\
@VV{{\operatorname{Tr}}}_{X, C}^e(A) V @VV {{\operatorname{Tr}}}_{C}^e(A) V\\
H^0(X, K_X+C+A)@>>> H^0(C, K_C+A).
\end{CD}$$]{} Then, the assertion follows from Corollary \[curve-trace-cor\].

In characteristic zero, in the above situation, the restriction map is surjective by the Kodaira vanishing theorem. But, in positive characteristic, the restriction map is not surjective in general.

\[cex-surjective\] There exists a smooth projective surface $X$, a smooth prime divisor $H$ on 