Improved direct and converse theorems in weighted Lorentz spaces

In the present work we prove the equivalence of the fractional modulus of smoothness to the realization functional and to the Peetre K-functional in weighted Lorentz spaces. Using this equivalence we obtain an improvement of the direct approximation theorem. Furthermore we prove the improved converse theorem in this space. Key words and phrases : Fractional modulus of smoothness, realization, weighted Lorentz spaces, Muckenhoupt weight, direct and converse theorem.