Let $U \subset \mathbb{R}^n$ be an open set. If $p, q$ are conjugate exponents, then the duality pairing $$ \langle -,- \rangle_{\mathsf{L}^q,\mathsf{L}^p}: \mathsf{L}^q(U;\mathbb{C}) \times \mathsf{L}^p(U;\mathbb{C}) \to \mathbb{C} $$ should be given by $\langle u,\varphi\rangle := \int_U \overline{u}\varphi$, generalizing the inner product on $\mathsf{L}^2(U;\mathbb{C})$. To allow for even more general "functions" in the first component, I can define a duality pairing $$ \langle -,- \rangle_{\mathscr{D}',\mathsf{C}_{\mathsf{c}}^\infty}: \mathscr{D}'(U;\mathbb{C}) \times \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{C}) \to \mathbb{C} $$ for distributions just by evaluation, $\langle T,\varphi \rangle := T(\varphi)$; so given a class-$\mathsf{L}^q$ (which is in particular of class-$\mathsf{L}^1_{\mathsf{loc}}$) function $u: U \to \mathbb{C}$, I can define the distribution $T_u: \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{C}) \to \mathbb{C}$ by $\varphi \mapsto \int_U \overline{u}\varphi$, and thus $\langle T_u,\varphi \rangle_{\mathscr{D}',\mathsf{C}_{\mathsf{c}}^\infty} = \langle u,\varphi \rangle_{\mathsf{L}^q,\mathsf{L}^p}$ for all $\varphi \in \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{C})$. Now, the $\alpha$-th weak derivative of $u$ should be a class-$\mathsf{L}^1_{\mathsf{loc}}$ function $v: U \to \mathbb{C}$ whose associated distribution $T_v$ is the $\alpha$-th distributional derivative of $T_u$, meaning we have $$ \left\langle T_u,\frac{\partial^\alpha \varphi}{\partial x^\alpha} \right\rangle = \int_U \overline{u}\frac{\partial^\alpha \varphi}{\partial x^\alpha} = \int_U \overline{v}\varphi = \langle T_v,\varphi \rangle $$ for every test function $\varphi \in \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{C})$. My question is, is this definition of the weak derivative correct? I know that when our functions are all real-valued instead of complex-valued, then this is of course the definition of the weak derivative, the complex-conjugations being irrelevant. But in any source in the literature that I can remember seeing, the complex-conjugation isn't part of the definition.
The only explanation that I can think of is that I'm using the "physicists' convention" of conjugate-linearity in the first component; if I required conjugate-linearity in the second, then I assume that it would be unnecessary to include complex-conjugation in the definition above, since (I think) $$ \left\langle T_u,\frac{\partial^\alpha \overline{\varphi}}{\partial x^\alpha} \right\rangle = \langle T_v, \overline{\varphi} \rangle \:\:\forall\:\: \varphi \in \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{C}) \iff \left\langle T_u,\frac{\partial^\alpha \varphi}{\partial x^\alpha} \right\rangle = \langle T_v,\varphi \rangle \:\:\forall\:\: \varphi \in \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{C}). $$ Any help (and possibly references using this definition) is really appreciated.