Extending the notion of bounded variation, a function \(u \in L_c^1(\mathbb {R}^n)\) is of bounded fractional variation with respect to some exponent \(\alpha \) if there is a finite constant \(C \ge 0\) such that the estimate$$\begin{aligned} \biggl |\int u(x) \det D(f,g_1,\ldots ,g_{n-1})_x \, dx\biggr | \le C\text{ Lip }^\alpha (f) \text{ Lip }(g_1) \cdots \text{ Lip }(g_{n-1}) \end{aligned}$$holds for all Lipschitz functions \(f,g_1,\ldots ,g_{n-1}\) on \(\mathbb {R}^n\). Among such functions are characteristic functions of domains with fractal boundaries and Hölder continuous functions. We characterize functions of bounded fractional variation as a certain subspace of Whitney’s flat chains and as multilinear functionals in the setting of Ambrosio–Kirchheim currents. Consequently we discuss extensions to Hölder differential forms, higher integrability, an isoperimetric inequality, a Lusin type property and change of variables. As an application we obtain sharp integrability results for Brouwer degree functions with respect to Hölder maps defined on domains with fractal boundaries.

中文翻译：

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有界分数变化和分形电流的函数

扩展有界变化的概念，如果存在有限常数，则函数\（u \ in L_c ^ 1（\ mathbb {R} ^ n）\）对于某些指数\（\ alpha \）具有分数变化\（C \ ge 0 \）这样估计$$ \ begin {aligned} \ biggl | \ int u（x）\ det D（f，g_1，\ ldots，g_ {n-1}）_ x \，dx \ biggr | \ le C \ text {Lip} ^ \ alpha（f）\ text {Lip}（g_1）\ cdots \ text {Lip}（g_ {n-1}）\ end {aligned} $$适用于所有Lipschitz函数\ （F，G_1，\ ldots，G_ {N-1} \）上\（\ mathbb {R} ^ N \）。在这些函数中，包括具有分形边界的畴的特征函数和Hölder连续函数。我们将有界分数变化的函数表征为惠特尼平链的某个子空间，并在Ambrosio–Kirchheim电流的环境中表征为多线性函数。因此，我们讨论了Hölder微分形式的扩展，更高的可积性，等距不等式，Lusin类型属性和变量的变化。作为一个应用程序，我们获得了关于Brouwer度函数的清晰可积性结果，这些结果与在具有分形边界的域上定义的Hölder映射有关。