Generalized Liouville differentiation, truncated hyerpsingular integrals and \(K\)-functionals (Q1434193)

Let \({\mu: [0,\infty)\to [0,\infty)}\) be a slowly increasing, continuous function. Define a generalized radial Liouville differentiation \(D^\mu\) on the Schwartz space \(S\) via the Fourier transformation \[ D^\mu={F}^{-1}[\mu(| \xi| ){\widehat f}], \qquad f\in S({\mathbb R}^n). \] Denote \[ L_\mu^p({\mathbb R}^n):=\{f\in L_\mu^p({\mathbb R}^n) : D^\mu f\in L_\mu^p({\mathbb R}^n)\},\quad p\geq 1. \] The choice \({\mu(t)=t^\alpha,\;\alpha>0}\) leads to standard Riesz potential spaces, \({\mu(t)=(1+t^2)^{\alpha/2}}\) to Bessel potential spaces. The \(K\)-functional is defined as \[ K(t,f; L^p({\mathbb R}^n), L_\mu^p({\mathbb R}^n)):=\inf_{g\in L_\mu^p}(\| f-g\| _p+t\| D^\mu g\| _p). \] Notation \({A(f,t)\lesssim B(f,t)}\) means that there exists a constant \({C>0}\), independent of \(f\) and \(t\), such that \({A(f,t)\lesssim B(f,t)}\) for all \(f\) and \(t\). Notation \({A(f,t)\approx B(f,t)}\) means that \({A(f,t)\lesssim B(f,t)}\) and \({B(f,t)\lesssim A(f,t)}\). The main result of the paper is the following. Let a function \(\mu\) satisfy the following three conditions. (A1) \(\mu(0)=0\), \(\mu\) is nondecreasing with \({\lim_{t\to\infty}\mu(t)=\infty}\). (A2) There exists \({\rho>0}\) such that \({t^{-\rho}\mu(t)}\) is nonincreasing. (A3) \({\mu\in C^{n^*+1}(0,\infty)}\), \({n^*:=[(n-1)/2]+1}\) with \({t^k| \mu^{(k+1)}(t)| \lesssim\mu'(t)}\), \({0\leq k\leq n^*}\). Let \(s\) be an integer such that \({2s>\rho}\). Then, for all \({t>0}\), all \({f\in L^p({\mathbb R}^n)}\), and \({1<p<\infty}\), \[ K\left(\frac{1}{\mu(t^{-1})},f; L^p({\mathbb R}^n), L_\mu^p({\mathbb R}^n)\right) \approx \frac{1}{\mu(t^{-1})}\left\| \int_{\| h\| >t}\Delta_h^{2s}f\left[\mu\left(\frac{2}{| h| }\right) -\mu\left(\frac{1}{| h| }\right)\right]\frac{dh}{| h| ^n}\right\| _p,\tag{1} \] where \(\Delta_h^{2s}\) is the central difference of order \(2s\). The authors also consider anisotropic Riesz potentials. In the definitions of these potentials the symbol \(| \xi| ^\alpha\) is replaced by an anisotropic one; for details see the survey paper of \textit{E.~Stein} and \textit{S.~Wainger} [Bull. Am. Math. Soc. 84, 1239--1295 (1978; Zbl 0393.42010)]. The authors define an anisotropic Liouville differentiation and the corresponding \(K\)-functional. Then they obtain the representation of \(K\)-functional similar to \((1)\). This extends results of \textit{O. I. Kuznetsova} and \textit{R. M.Trigub} [Sov. Math., Dokl. 21, 374--377 (1980); translation from Dokl. Akad. Nauk SSSR 251, 34--36 (1980; Zbl 0472.42001)] to the anisotropic case.