A global proof of the weak Harnack inequality for parabolic equations in non-divergence form
The weak Harnack inequality is a fundamental result in the theory of fully nonlinear parabolic partial differential equations. It has several important consequences such as the H\"older continuity of derivatives of solutions to certain fully nonlinear parabolic equations. Classical proofs of the weak Harnack inequality rely on decay of measure estimates, which require delicate localization and covering arguments. We give a new proof based on a detailed study of particular envelopes of solutions to parabolic partial differential equations. This allows us to completely bypass the need of covering arguments. We also prove a $W^{2,\epsilon}$ estimate with an improved dependence of $\epsilon$ on the ellipticity ratio compared to previous results.