This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions; the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium, to be precise. Together with the results of Parts I and II about the well-posedness of the Cauchy problem around the Maxwellian, we conclude this series with a satisfactory mathematical theory for the Boltzmann equation without angular cutoff.

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Let $ E \subset \mathbb{C} $ be a compact set, $ g:\mathbb{C} \to \mathbb{C} $ be a$K$-quasiconformal map, and let 0 <$t$< 2. Let $ {\mathcal{H}^t} $ denote$t$-dimensional Hausdorff measure. Then $$ {\mathcal{H}^t}(E) = 0\quad \Rightarrow \quad {\mathcal{H}^{t'}}\left( {gE} \right) = 0,\quad t' = \frac{{2Kt}}{{2 + \left( {K - 1} \right)t}}. $$