  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.
