some special classes of functionals, this state-dependent boundary condition has a very simple bulk interpretation as a Dirichlet boundary condition imposed on an intermediate surface, implying that we can trade the non-locality of the boundary sources into local behavior at some lower radius.

We just have one further question to answer before we declare victory: how do we in practice determine this special set of sources in various holographic setups? For the general backgrounds we can formally write the solution to the wave equations in terms of integrals over the Dirichlet data convolved with suitable ‘Dirichlet bulk to boundary propagators’, ${\cal K}_\text{source}$ and ${\cal K}_\text{vev}$. The former propagates the information contained in $\hat{J}_\phi$ to the boundary source, while the latter allows determination of the contribution from the vev $\hat{\phi}$ on $\Sigma_D$, i.e., formally $$J_\phi(x)  = \int d^dx' \, \left\{{\cal K}_\text{source}(r_D, x; x') \, \hat{J}_\phi(x') + {\cal K}_\text{vev}(r_D, x; x') \, \hat{\phi}(x')\right\}$$ Note that implicit in our definition of these Dirichlet bulk to boundary propagators is the information of the boundary condition in the interior of the geometry and the necessary counter-terms. While it is possible to work this out in more specific geometries, such as a Schwarzschild-AdS$_{d+1}$ spacetime to see 