Logarithmic integrals, Sobolev spaces and Radon transform in the plane (Q1431397)

It is well known that the Hermite polynomial \(H_n\) and Hermite functions \(\varphi_n\) are defined, respectively, by \[ \sum^\infty_{n=0} H_n(x){t^n\over n!}= \exp(2tx- t^2)\;\text{ and }\;\varphi_n(x):= H_n(x)\exp(- x^2/2), \] and the system \(\{\varphi_n: n= 0,1,2,\dots\}\) is orthogonal in \(L^2(\mathbb{R})\). The following results are proved: (i) The system \(\{\varphi_n\}\) can be divided into three subsystems, each of them is orthogonal in the Sobolev space \(H_{(1)} (\mathbb{R})\). (ii) The logarithmic integral \(F\) of a function \(f\) in the real Hardy space \({\mathcal H}^1(\mathbb{R})\) is absolutely continuous and \(F'(x)= -\widetilde f(x)\) for almost all \(x\in\mathbb{R}\), where \(f\) is the Hilbert transform of \(f\). (iii) For odd \(n\), the logarithmic integral \(\Phi_n\) of \(\varphi_n\) belongs to \(H_{(1)}(\mathbb{R})\) and \[ \|\Phi_n\|^2_{(1)}= \int_{\mathbb{R}} \Biggl|{\varphi_n(x)\over x}\Biggr|^2 dx+ \|\varphi_n\|^2. \] (iv) For odd \(n\), the logarithmic integral \(\Psi_n\) of the function \(\psi_n(x):= H_n(x)\exp(- x^2)\) belongs to \(H_{(1)} (\mathbb{R})\) and \(\|\Psi_n\|^2_{(1)}= \sqrt{2\pi}n(2n- 3)\).