A log-weighted Moser inequality on the plane

We establish a sharp Moser type inequality with logarithmic weight in the nonradial mass-weighted Sobolev spaces, on the whole plane ℝ². We identify the sharp threshold for the uniform boundedness of the weighted Moser functional, which is still given by 4π: further, we prove the validity of the inequality also at the limiting sharp value 4π. Even if the increasing nature of the log weight prevents the application of any symmetrization tool, we prove our inequality in the general framework of Sobolev space, and not on radial subspaces, as in the available literature. The main strategy is a careful analysis of the behaviour of the normalized maximizing sequences.