K-functionals and moduli of smoothness of functions defined on compact metric spaces

One proves that Peetreʹs K-functional for the couple (C(X,Lip(X)) and the first order modulus of continuity are equivalent for every Peano continuum. Some inequalities for the first order modulus of continuity and the mixed modulus of smoothness of functions defined on some compact metric spaces are proved and some possible applications are indicated.