  As a continuation of the previous work [40], in this paper we focus on the
Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations
with fractional Laplacian dissipation. We give an elementary proof of the
global regularity of the smooth solutions of the 2D Boussinesq equations with a
new range of fractional powers of the Laplacian. The argument is based on the
nonlinear lower bounds for the fractional Laplacian established in [12].
Consequently, this result significantly improves the recent works [12, 38, 40].
