$S$ is schwarz space, $S'$ is the dual space of $S$. $F$ is fourier transform operator. Let $S_h$ be the space that consists of schwarz functions $u$ satisfy: $\partial^{\alpha}F(u)(0)=0$ for all indexs $\alpha$ (with the subspace topology), and $S_{h}'$ be the dual space of $S_h$. Let $J$ be the restriction map: $u\in S' \Rightarrow Ju=u|_{S_h}$. Thanks to Hahn Banach theorem and single point support distribution theory we see $J$ is surjective, $Ker J=P$, which is the polynomial space.That is, We have $S_h'$=$S'/P$.
If we define the space $C^{\infty}_{0h}$ as the subspace of $C^{\infty}_0$ that satisfy $\displaystyle\int u=0\, (F(u)(0)=0)$ for all elements(with subspace topology), and space $D_h'$ that is the dual space of $C^{\infty}_{0h}$. What can we say about $D_h'$?I Guess it may be the space of $D'/P_0$, which $D$ is distribution space, $P_0$ is constant space. But I can't prove it.Can someone help me prove or deny this conclusion? Thanks in advance.