Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that $$ \|f\|_{A^1} = \int_{\mathbb{D}} |f(z)|\, dA(z) < \infty, $$ where $dA$ is the normalized area measure on $\mathbb{D}$ (i.e., $\frac{1}{\pi}\, dx\, dy$). The Dirichlet space $D^1$ is the space of holomorphic functions such that the derivative is in $A^1$, i.e., $\|f'\|_{A^1} < \infty$.
It is well-known that the dual space of $A^1$, denoted $(A^1)^*$, can be identified with the Bloch space $\mathcal{B}$, which consists of holomorphic functions such that $(1 - |z|^2) |f'(z)|$ is bounded.
On the other hand, $A^1$ is isomorphic to $D^1$ via the map $J: D^1 \rightarrow A^1$ given by $J(f) = (zf)'$. This isomorphism is easy to see using the series representation of holomorphic functions.
While I have found a reference for the pairing between $(A^1)^*$ and $\mathcal{B}$ (see [1]), I haven't found a similar reference for the pairing between $(D^1)^*$ and $\mathcal{B}$. Intuitively, this should follow from the known facts, but I'm interested in whether this specific pairing is mentioned in any book or paper. It seems there is plenty of literature on Bergman spaces, but references for Dirichlet spaces are much harder to come by.
References
[1] Axler, S. (1988). Bergman spaces and their operators. In Surveys of some recent results in operator theory (Vol. 1, pp. 1-50). Longman Sci. Tech.
