Boundary-to-bulk maps for AdS causal wedges and RG flow

We consider the problem of defining spacelike-supported boundary-to-bulk propagators in AdS$_{d+1}$ down to the unitary bound $\Delta=(d-2)/2$. That is to say, we construct the `smearing functions' $K$ of HKLL but with different boundary conditions where both dimensions $\Delta_+$ and $\Delta_-$ are taken into account. More precisely, we impose Robin boundary conditions, which interpolate between Dirichlet and Neumann boundary conditions and we give explicit expressions for the distributional kernel $K$ with spacelike support. This flow between boundary conditions is known to be captured in the boundary by adding a double-trace deformation to the CFT. Indeed, we explicitly show that using $K$ there is a consistent and explicit map from a Wightman function of the boundary QFT to a Wightman function of the bulk theory. In order to accomplish this we have to study first the microlocal properties of the boundary two-point function of the perturbed CFT and prove its wavefront set satisfies the microlocal spectrum condition. This permits to assert that $K$ and the boundary two-point function can be multiplied as distributions.