Avikainen showed that, for any $p,q \in [1,\infty)$, and any function $f$ of bounded variation in $\mathbb{R}$, it holds that $\mathbb{E}[|f(X)-f(\widehat{X})|^{q}] \leq C(p,q) \mathbb{E}[|X-\widehat{X}|^{p}]^{\frac{1}{p+1}}$, where $X$ is a one-dimensional random variable with a bounded density, and $\widehat{X}$ is an arbitrary random variable. In this article, we will provide multi-dimensional versions of this estimate for functions of bounded variation in $\mathbb{R}^{d}$, Orlicz--Sobolev spaces, Sobolev spaces with variable exponents, and fractional Sobolev spaces. The main idea of our arguments is to use the Hardy--Littlewood maximal estimates and pointwise characterizations of these function spaces. We apply our main results to analyze the numerical approximation for some irregular functionals of the solution of stochastic differential equations.

翻译：Avikinen 显示,对于任何美元,q 和美元[1,\\ infty]$[,1,\\ infty]$(美元)中受约束变量的任何函数,美元[\mathbb{R}],它认为,$\mathbb{E}[ ⁇ f(X)-f(@loyhat{X}){qqq}]\leq C(p,q)\mathbb{(E}}[{{X-\bloyhat{X}}}}{frac{{{1\\\\\\\\\\+1美元,美元是具有约束密度的一维随机变量,而$(全域){X}则是一个任意随机随机变量。在本文章中,我们将提供这一受约束变量功能的函数的函数的多维值估计值版本,在$\\\\\thb{R}$、Orlicz-Soblevblevlev 空间、有可变异的外观的Spoalalalalolevle 空间。我们的论点主要想法是使用硬-Latt-lood-lood adwood colood coal ad ad 和点解这些函数的公式的公式的模型对等式模型的分析。