  Using a classical technique introduced by Achi E. Brandt for elliptic
equations, we study a general class of nonlocal equations obtained as a
superposition of classical and fractional operators in different variables. We
obtain that the increments of the derivative of the solution in the direction
of a variable experiencing classical diffusion are controlled linearly, with a
logarithmic correction. From this, we obtain H\"older estimates for the
solution.
