  By using Fourier's transform and Fefferman-Stein's theorem, we investigate
the $L^p$-maximal regularity of nonlocal parabolic and elliptic equations with
singular and non-symmetric L\'evy operators, and obtain the unique strong
solvability of the corresponding nonlocal parabolic and elliptic equations,
where the probabilistic representation plays an important role. In particular,
a characterization for the domain of pseudo-differential operators of L\'evy
type with singular kernels is given in terms of the Bessel potential spaces. As
a byproduct, we show that a large class of non-symmetric L\'evy operators
generates an analytic semigroup in $L^p$-space. Moreover, as applications, we
prove a Krylov's estimate for stochastic differential equation driven by Cauchy
processes (i.e. critical diffusion processes), and also obtain the
well-posedness to a class of quasi-linear first order parabolic equation with
critical diffusion. In particular, critical Hamilton-Jacobi equation and
multidimensional critical Burger's equation are uniquely solvable and the
smooth solutions are obtained.
