  The paper is devoted to the study of slightly supercritical active scalars
with nonlocal diffusion. We prove global regularity for the surface
quasi-geostrophic (SQG) and Burgers equations, when the diffusion term is
supercritical by a symbol with roughly logarithmic behavior at infinity. We
show that the result is sharp for the Burgers equation. We also prove global
regularity for a slightly supercritical two-dimensional Euler equation. Our
main tool is a nonlocal maximum principle which controls a certain modulus of
continuity of the solutions.
