The rational homotopy of the K(2)-local sphere and the chromatic   splitting conjecture for the prime 3 and level 2
  We calculate the rational homotopy and the K(1)-local homotopy of the K(2)-local sphere at the prime 3 and level 2. We use this to verify the chromatic splitting conjecture in this case. 
Paul G. Goerss, Hans-Werner Henn, and Mark Mahowald
math.AT