This paper examines the question for global regularity for the Boussinesq equation with critical fractional dissipation (α,β) :α + β = 1. The main result states that the system admits global regular solutions for all (reasonably) smooth and decaying data, as long as α > 2/3. This improves upon some recent works [Q. Jiu, C. Miao, J. Wu and Z. Zhang, SIAM J. Math. Anal., 46:3426-3454, 2014] and [A. Stefanov and J. Wu, J. Anal. Math., 2015]. The main new idea is the introduction of a new, second generation Hmidi-Keraani-Rousset type, change of variables, which further improves the linear derivative in temperature term in the vorticity equation. This approach is then complemented by a new set of commutator estimates (in both negative and positive index Sobolev spaces!), which may be of independent interest.
