\(L^ q\)-mean limits for Taylor's expansion of Riesz potentials of functions in Orlicz classes
The author considers the Riesz potential \(U_\alpha f(x)\) of a non-negative function \(f(x)\) satisfying the conditions \[ \int_{\mathbb{R}^n}(1+|x|)^{\alpha-n}f(x) dx<\infty,\;\int_{\mathbb{R}^n}\Phi_p[f(x)] dx< \infty \] where \(\Phi_{p}(u)=u^p\varphi(u), 1<p<\infty\), with a function \(\varphi(u), u\in \mathbb{R}^1_+\), satisfying the conditions: i) \(\varphi\) is a positive non-decreasing on \(\mathbb{R}^1_+\), ii) there exists \(A_1>0\) such that \(A_1^{-1}\varphi(u)\leq \varphi(u^2)\leq A_1\varphi(u),\;u>0\). Under some assumptions on the weight function \(w(r)\), the author proves the \(L^q\)-mean convergence \[ \lim_{r\to 0}w(r)\left\{r^{-n}\int_{B(x_0,r)}|U_\alpha f(x)-P_{x_0}(x)|^q dx\right\}^\frac{1}{q}=0 \] for \(q\geq 0, \frac{1}{q}\geq\frac{1}{p}-\frac{\alpha}{n}, x_0\in \mathbb{R}^n\). Here \(P_{x_0}(x)\) is constructed in terms of the Taylor polynomial in \(x\) of the Riesz kernel \(|x-y|^{\alpha-n}\) at the point \(x_0\).
scientific article; zbMATH DE number 1021654