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

Let (f) over cap(c) be the Fourier cosine transform of f. Then, as proved for functions of class L-2(R+) in Titchmarsh's book 'Introduction to the theory of Fourier integrals' (1937), H(f)(x) = integral(x)(+infinity) f(y)/ydy, x > 0, for the Hardy-Littlewood operator B(f)(x) = 1/x integral(0)(x) f(y) dy, x > 0. In the present paper similar equalities are proved for functions of class L-p(R+), 1 < p less than or equal to 2, and the Walsh-Fourier transformation.