We consider the space L(u, v) of 2π-periodic real-valued functions which are integrable with respect to a sign sensitive weight (u, v). With some necessary hypothesis for this weight, L(u, v) is an asymmetric Banach space. After defining a convenient modulus of smoothness we introduce the corresponding space Lip α (u, v) and its subspace lip α (u, v) of Hölder (or Lipschitz) functions associated to this modulus. We prove these spaces are asymmetric Banach spaces too and use the result to study approximation problems.