On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis

Let $f(x)$ be a function belonging to the Lebesgue class $L^p({\mathbb R}_+)$ on the semiaxis ${\mathbb R}_+=[0,+\infty)$, $1\le p\le 2$, and let $\widehat{f}$ be the Fourier–Walsh transform of the function $f$. In this paper, we give the solution of the following problem: if the function $f$ belongs to the dyadic Dini–Lipschitz class $\operatorname{DLip}_\oplus(\alpha,\beta,p;{\mathbb R}_+)$, $\alpha>0$, $\beta\in{\mathbb R}$, then for what values of $r$ can we guarantee that $\widehat{f}$ belongs to $L^r({\mathbb R}_+)$? The result obtained is an analog of the classical Titchmarsh theorem on the Fourier transform of functions from Lipschitz classes on ${\mathbb R}$.