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This paper gives the first affirmative answer to the question of the globalexistence of Boltzmann equations without angular cutoff in the$L^\infty$-setting. In particular, we show that when the initial data is closeto equilibrium and the perturbation is small in $L^2 \cap L^\infty$ with apolynomial decay tail, the Boltzmann equation has a global solution in theweighted $L^2\cap L^\infty$-space. In order to overcome the difficultiesarising from the singular cross-section and the low regularity, a De Giorgitype argument is crafted in the kinetic context with the help of the averaginglemma. More specifically, we use a strong averaging lemma to obtain suitable$L^p$-estimates for level-set functions. These estimates are crucial forconstructing an appropriate energy functional to carry out the De Giorgiargument. Similar as in \cite{AMSY}, we extend local solutions to global onesby using the spectral gap of the linearised Boltzmann operator. The convergenceto the equilibrium state is then obtained as a byproduct with relaxations shownin both $L^2$ and $L^\infty$-spaces.

De Giorgi argument for weighted $L^2 \cap L^\infty$ solutions to the non-cutoff Boltzmann equation