On the binary analogs of the Hardy and Hardy--Littlewood operators (Q1975816)

Given \(x\in\mathbb R^+\) and an integer \(n\) such that \(2^n\leq x <2^{n+1}\), put \[ H_{\omega}(f)(x)= \sum_{m=n+1}^{\infty}2^{-m}\int_{2^m}^{2^{m+1}} f(y) dy,\quad B_{\omega}(f)(y)=2^{-n} \int_{0}^{2^{n}}f(y) dy,\; \] where \(f\in L_p({\mathbb R}^+)\) with \(1\leq p<\infty\). The operators \(H_{\omega}\) and \(B_{\omega}\) are binary analogs of the Hardy and Hardy-Littlewood operators. Recall the definition of the Walsh transform. Given \(x\in\mathbb R^+\) and a positive integer \(n\), put \(x_n\equiv [2^n x]\pmod 2\) and \(x_{-n}\equiv [2^{1-n} x]\pmod 2\), i.e., by definition, the numbers \(x_n\) and \(x_{-n}\) belong to \(\{0,1\}\). We can define the functions \[ t(x,y)=\sum_{n=1}^{\infty}(x_ny_{-n}+x_{-n}y_n),\quad X(x,y)=(-1)^{t(x,y)} \] with \(x,y\in\mathbb R^+\). The Walsh transform is defined by the equality \[ \widehat{f}(x)=\int_{{\mathbb R}^+}X(x,y)f(y) dy. \] The main result of the article is the following analog of the Titchmarsh theorem: Theorem. Let \(f\in L_p({\mathbb R}^+)\). Then \[ \widehat{H}_{\omega}(f)=B_{\omega}(\widehat{f})\quad (1\leq p\leq 2),\qquad \widehat{B}_{\omega}(f)=H_{\omega}(\widehat{f})\quad (1< p\leq 2). \]