Let $a \in L^1 (\mathbb R^d)$ be a fixed function. Consider a linear operator $$ T_a: L^2 (\mathbb R^d) \to L^2 (\mathbb R^d), u \mapsto a * u. $$
By Young's inequality, $\|a*u\|_2 \le \|a\|_1 \|u\|_2$ for all $u \in L^2 (\mathbb R^d)$. Then $T_a$ is bounded. I'm trying to find the adjoint $T_a^*$ of $T_a$. Could you check if my reasoning is fine?
For $u, w \in L^2 (\mathbb R^d)$, we have $\langle T_a^* u, w \rangle = \langle u, T_a w \rangle$ and thus $\int (T_a^* u) w = \int u (T_a w)$. On the other hand, $$ \int u (T_a w) = \int u (a*w) = \int w(\check a *u), $$ where $\check a (x) := a(-x)$ for all $x \in \mathbb R^d$. It follows that $T_a^* u = \check a *u$.