pure spacetime where we have the luxury of being able to solve the scalar wave-equation explicitly to see an explicit answer to the question.

Probe scalar in  {#s:adss}
----------------

We specialize our consideration to the pure geometry where $g_{\mu\nu} = \eta_{\mu\nu} $ and $g_{rr} = 1$ and one has enhanced Lorentz symmetry on the constant $r$ slices. The wave equation simplifies to $$\frac{1}{r^{d-1} }\, \frac{d}{dr} \, \left(r^{d+1} \, \frac{d}{dr} \Phi_{k}(r) \right) - \left(k^2 +m^2 \right) \Phi_{ k} = 0 
\label{mkgads}$$ This is well known to have solutions in terms of Bessel functions, but we will proceed to examine the behavior in a gradient expansion to set the stage for the real problem of interest later.

### The $k=0$ case

The translationally invariant solution of the massive Klein-Gordon equation in the bulk is[^14] $$\Phi(r) = \frac{\phi}{(2\nu)\,r^\Delta} +  r^{\Delta-d} J_\phi$$ where the dual primary has a scaling dimension $\Delta$ obeying $\Delta(\Delta-d)=m^2$ along with a source $J_\phi$ and a normalized vev[^15] $\phi$ defined via $$J_\phi \equiv \left[r^{d-\Delta}\, \Phi(r)\right]_{r\to\infty}$$ $$\phi \equiv \left[-r^{2\nu}\times r\partial_r \left( r^{d-\Delta}\Phi \right)\right]_{r\to\infty} = \left[-r^\Delta\left( r\partial_r \Phi -(\Delta-d)\Phi\right)\right]_{r\to\infty}$$ with $$\nu \equiv \Delta - \frac{d}{2} = \sqrt{\frac{d^2}{4} + m^2}$$ for convenience. For simplicity, we will assume $\Delta > \frac{d}{2} $ and choose 