  In this paper, we consider the two-dimensional surface quasi-geostrophic
equation with fractional horizontal dissipation and fractional vertical thermal
diffusion. Global existence of classical solutions is established when the
dissipation powers are restricted to a suitable range. Due to the nonlocality
of these 1D fractional operators, some of the standard energy estimate
techniques no longer apply, to overcome this difficulty, we establish several
anisotropic embedding and interpolation inequalities involving fractional
derivatives. In addition, in order to bypass the unavailability of the
classical Gronwall inequality, we establish a new logarithmic type Gronwall
inequality, which may be of independent interest and potential applications.
