The primary purpose of this research was to significantly advance the rigorous mathematical theory of kinetic transport equations, using recent advances in techniques from harmonic analysis and geometric analysis. A particularly important aspect of the theory is the smoothing effect of velocity averages of the solution of the classical kinetic transport equation. This smoothing effect is most naturally captured in the scale of hyperbolic Sobolev spaces when the velocities belong to the unit sphere. A major achievement of the research supported by this grant is to provide a complete solution to the problem of understanding the precise smoothing effect of the velocity average on the sphere, for square-integrable initial data, and where the velocity average is measured in general mixed-norm Lebesgue spaces. The methods used were harmonic analysis techniques; in particular, we discovered a direct connection with the famous cone multiplier problem which has seen major advances in recent years. Furthermore, we proved that the smoothing effect is increased when considering radially symmetric initial data and we identified the optimal constants and extremal initial data in certain cases; these sharp estimates were established using the Funk-Hecke theorem from harmonic analysis. The results described here provide a major development in kinetic theory, particularly through the pioneering use of new techniques in this field.
Similar use of techniques from harmonic analysis also led to new progress on the smoothing effect for dispersive and wave-like equations.