What is the dual of $A\cap B$

Question

I encountered with some elliptic problem which admits a variational formulation in terms of space $X$ and I need to understand its dual. Suppose that $2<p<\infty$, $\Omega\subset {\mathbb R}^d$ where $d = 3$ and $|\Omega|<\infty$. $X = H^{2}_0(\Omega)\cap W^{1,p}(\Omega)$. What is the structure of $X^*$?

Answer 1

It's the direct sum of the duals of the two spaces, that is, $$ X^* = \{u + v : u \in H^{-2} , v \in W^{-1,q} \} . $$

Edit: Correction (see comments below)