  In this paper, we develop a sliding method for the fractional Laplacian. We
first obtain the key ingredients needed in the sliding method either in a
bounded domain or in the whole space, such as narrow region principles and
maximum principles in unbounded domains. Then using semi-linear equations
involving the fractional Laplacian in both bounded domains and in the whole
space, we illustrate how this new sliding method can be employed to obtain
monotonicity of solutions. Some new ideas are introduced. Among which, one is
to use Poisson integral representation of $s$-subharmonic functions in deriving
the maximum principle, the other is to estimate the singular integrals defining
the fractional Laplacians along a sequence of approximate maximum points by
using a generalized average inequality. We believe that this new inequality
will become a useful tool in analyzing fractional equations.
