  In this paper, we consider the characterizations of the Lipschitz spaces and
homogeneous Lipschitz spaces associated to the biharmonic operator $\Delta^2.$
With this characterizations, we prove the boundedness of the Bessel potentials,
fractional integrals, fractional powers, Riesz transforms and multipliers of
Laplace transforms type associated to $\Delta^2$ on Lipschitz spaces and
homogeneous Lipschitz spaces. The proofs of these results need the language of
semigroups in an essential way.
