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In this paper, we consider the characterizations of the Lipschitz spaces andhomogeneous 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 ofLaplace transforms type associated to $\Delta^2$ on Lipschitz spaces andhomogeneous Lipschitz spaces. The proofs of these results need the language ofsemigroups in an essential way.

Regularity property of some operators on Lipschitz spaces and homogeneous Lipschitz spaces related to biharmonic operator