  We investigate regularity and a priori estimates for Fokker-Planck and
Hamilton-Jacobi equations with unbounded ingredients driven by the fractional
Laplacian of order $s\in(1/2,1)$. As for Fokker-Planck equations, we establish
integrability estimates under a fractional version of the Aronson-Serrin
interpolated condition on the velocity field and Bessel regularity when the
drift has low Lebesgue integrability with respect to the solution itself. Using
these estimates, through the Evans' nonlinear adjoint method we prove new
integral, sup-norm and H\"older estimates for weak and strong solutions to
fractional Hamilton-Jacobi equations with unbounded right-hand side and
polynomial growth in the gradient. Finally, by means of these latter results,
exploiting Calder\'on-Zygmund-type regularity for linear nonlocal PDEs and
fractional Gagliardo-Nirenberg inequalities, we deduce optimal $L^q$-regularity
for fractional Hamilton-Jacobi equations.
