Area formulas in metric measure spaces under a weak Lipschitz-like condition
type: Item
instance of: scholarly article
title: Area formulas in metric measure spaces under a weak Lipschitz-like condition
author: Marcela Garriga
author: Pablo Ochoa
published in: Results in Mathematics
publication date: 2022-12-13T00:00:00Z
review text: The aim of the paper is to prove the area formula for functions between two metric measure spaces. Recall that \((X, d, \mu)\) is a metric measure space if \((X, d)\) is a separable metric space and \(\mu\) is a non-negative outer measure defined on all subsets of \(X\). In the whole paper the authors assume that \(\mu\) measures \(X\) and \(\nu\) measures \(Y\). They do not assume that a function \(f\) is continuous and consider functions fulfilling the following condition \(L\) (the very weak version of Lipschitz condition): Let \(f:X\to Y\) be given, where \(Y\) is any metric space. We say that \(f\) satisfies the \(L\) condition in \(X\) if for any \(a \in X\), there exist constants \(r_a>0\) and \(c(a)>0\) so that for any radius \(0<r<r_a\), we are able to find \(\mu\)-measurable sets \(A_r\subset B(a,r)\) satisfying: \[ d_Y(f(x),f(a))\leq c(a)d_X(x,a), \] for all \(x\in B(a,r) \ \backslash \ A_r\) and: \[ \mu(A_r)<r^Q\rho/2 \] for all \(r\). Here \(Q\) is the Ahlfors exponent in \(X\), which means that \(\frac{r^Q}{C}\leqslant\mu(B(x,r))\leqslant Cr^Q\) for some \(C\geqslant1\) and \(\varrho=\frac{1}{C\cdot 2^Q}\). The first result concerns function transforming a metric measure space into \(\mathbb{R}^n\) equipped with the Lebesgue measure: \textbf{Theorem}. If the function \(u:X\to\mathbb{R}^n\) satisfies the \(L\) condition on \(X\) and the pullback measure \(\bar{u}^*(\mu \times \mathcal{L})\) is finite on bounded sets, where \[ \bar{u}(x)=(x,u(x)), \quad x\in X \] is the graph mapping associated to \(u\), then the following area formula is valid \[ (\mu\times\mathcal{L})(\bar{u}(A))=\int_A \mathcal{J}_{\bar{u}}^{\nu}(x)d\mu(x) \] provided that \(A \subset X\) is a \(\mu\)-measurable set. Here \(\mathcal{J}_f^{\nu}\) is the metric Jacobian defined as follows:\\ Let \(\mathcal{V}\) be a Vitali relation in \(X\). Let \(f:A\to Y\) and \(x\in A\), \(A\subset X\) closed. The metric Jacobian of \(f\) on \(x\) is defined as follows: \[ \mathcal{J}_f^{\mathcal{V}}…
reviewed by: Władysław Wilczyński
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 26B10
Mathematics Subject Classification ID: 28A15
zbMATH DE Number: 7632544
zbMATH Keywords: geometric measure theory
zbMATH Keywords: area formula
zbMATH Keywords: N-Lusin property
zbMATH Keywords: metric Jacobian
zbMATH Keywords: rectifiability
MaRDI profile type: Publication
full work available at URL: s00025-022-01807-0
OpenAlex ID: W4312077954
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DOI: 10.1007/S00025-022-01807-0
description: scientific article; zbMATH DE number 7632544