On some interpolation inequalities between Hölder and Lebesgue's spaces

We present a simple proof of some interpolation inequalities between Hölder and Lebesgue’s spaces. As an example, to demonstrate the simplicity of their applications to nonlinear PDE, we give also a simple proof of an a-priory estimate in a smooth Hölder class for a solution to some quasilinear parabolic initial boundary value problem.