An analogue of a theorem of Titchmarsh for Walsh-Fourier transformations

Let f-hatc be the Fourier cosine transform of f. Then, as proved for functions of class Lp(R+) in Titchmarsh's book 'Introduction to the theory of Fourier integrals' (1937), the Hardy operator and the Hardy-Littlewood operator can be defined. In the present paper similar equalities are proved for functions of class Lp(R+), 1<p≤2, and the Walsh-Fourier transformation