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

Let $\hat f_c$ be the Fourier cosine transform of $f$. Then, as proved for functions of class $L^p(\mathbb R_+)$ in Titchmarsh's book 'Introduction to the theory of Fourier integrals' (1937),
$$ \mathscr H(\hat f_c)=\widehat {\mathscr B(f)}_c, \qquad \mathscr B(\hat f_c)=\widehat {\mathscr H(f)}_c, $$
for the Hardy operator
$$ \mathscr H(f)(x)=\int _x^{+\infty }\frac {f(y)}y\,dy, \qquad x>0, $$
and the Hardy-Littlewood operator
$$ \mathscr B(f)(x)=\frac 1x\int _0^xf(y)\,dy, \qquad x>0. $$
In the present paper similar equalities are proved for functions of class $L^p(\mathbb R_+)$, $1<p\leqslant 2$, and the Walsh-Fourier transformation.