Concentration phenomena for the fractional Q-curvature equation in dimension 3 and fractional Poisson formulas

We study the compactness properties of metrics of prescribed fractional Q-curvature of order 3 in R3. We will use an approach inspired from conformal geometry, seeing a metric on a subset of R3 as the restriction of a metric on R+4 with vanishing fourth-order Q-curvature. We will show that a sequence of such metrics with uniformly bounded fractional Q-curvature can blow up on a large set (roughly, the zero set of the trace of a non-positive bi-harmonic function phi in R+4), in analogy with a four-dimensional result of Adimurthi-Robert-Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest.