Geodesic
On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis
Matematičeskie zametki, Tome 108 (2020) no. 2, pp. 236-251 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 ${\mathbb R}$.
Keywords: dyadic harmonic analysis, Dini–Lipschitz classes, Fourier–Walsh transform. 
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     title = {On the {Fourier{\textendash}Walsh} {Transform} of {Functions} from {Dyadic} {Dini{\textendash}Lipschitz} {Classes} on the {Semiaxis}},
     journal = {Matemati\v{c}eskie zametki},
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     year = {2020},
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S. S. Platonov. On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis. Matematičeskie zametki, Tome 108 (2020) no. 2, pp. 236-251. http://geodesic.mathdoc.fr/item/MZM_2020_108_2_a7/

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