Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces
The real line \((-\infty,\infty)\) is denoted by \(R\) and the two-dimensional plane is denoted by \(R^2\). The Fourier transforms \(\mathcal F(f)\), \(\mathcal F(g)\), and Hilbert transforms \(H(f)\), \(H^{11}(g)\) of measurable functions \(f\), \(g\) on \(R\) and \(R^2\) are defined respectively by \[ \begin{aligned} & \mathcal F(f)(x)=(2\pi)^{-1/2}\int_Rf(t)e^{-ixu}du;\quad H(f)(x)=(1/\pi)(P .V.)\int_Rf(x-u)u^{-1}du; \\ \vspace{3pt} &\mathcal F(g)(x,y)=(2\pi)^{-1}\int_{R^2}g(u,v)e^{-i(xu+yv)}dudv;\\ \vspace{3pt} &H^{11}(g)(x,y)=\pi^2(P.V.)\int_{R^2}g(x-u,y-v)u^{-1}v^{-1}dudv.\end{aligned} \] If \(\varphi \in L^1(R)\) or \(\varphi\in L^1(R^2)\), then the Hausdorff operators \(\mathcal H_\varphi\) and \(\mathcal H_\varphi^{11}\) on the Hardy spaces \(H^1(R)\) and \(H^1(R^2)\) are defined by \(\mathcal F\mathcal H_\varphi(f)(x)=\int_R\mathcal F(f)(ux)\varphi(u)du\) and \(\mathcal F\mathcal H^{11}_\varphi(g)(x)=\int_{R^2}\mathcal F(g)(xu,yv)\varphi(u,v)dudv\), respectively. In the main results of this paper the authors derive necessary and sufficient conditions that the operator \(\mathcal H_\varphi\) is commutative with \(H\), and necessary and sufficient conditions that the operator \(\mathcal H_\varphi^{11}\) is commutative with \(H^{11}\). In particular, it is shown that \(H\mathcal H_\varphi(f)=\pm\mathcal H_\varphi H(f)\) for \(f\) in \(H^1(R)\) if and only if \(\varphi(x)=0\) (a.e.) for \(\pm x<0\); and \(H^{11}\mathcal H^{11}_\varphi(g)=\pm\mathcal H^{11}_\varphi H^{11}(g)\) for \(g\) in \(H^1(R^2)\) if and only if \(\varphi(x,y)=0\) for \(\pm xy<0\). Other results of the paper involve transforms \(\mathcal H^{01}_\varphi\), \(\mathcal H^{10}_\varphi\), defined in terms of integrals of one of the variables of two-variables functions. Reviewer's comment: It seems possible to express the operators \(\mathcal H_\varphi\) and \(\mathcal H_\varphi^{11}\) as `homogeneous convolutions'. In particular, by applying the inverse Fourier operator to the integral of the definition of \(\mathcal H_\varphi\) and applying the inversion formulae, it may be verified that \(\mathcal H_\varphi(f)(x)=\int_R|u|^{-1}f(x/u)\varphi(u)du\).
product Hardy space \(H^{11}(\mathbb R\times\mathbb R)\)
hybrid Hardy spaces \(H^{10}(\mathbb R^2)\) and \(H^{01}(\mathbb R^2)\)
scientific article; zbMATH DE number 1912074