Area formulas in metric measure spaces under a weak Lipschitz-like condition
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: \[ \mathcal{J}_f^{\mathcal{V}} = \limsup\limits_{S\to x} \frac{\nu(f(S\cap A))}{\mu(S)}, \] where \(S\in\mathcal{V}\). Here the name of Vitali relation is justified by the following definition: A covering relation is a subset of \[ \{ (x,S):x\in S\subset X \}. \] Whenever \(\mathcal{C}\) is a covering relation and \(Z\subset X\), we let \[ \mathcal{C}(Z)=\{ S:(x, S)\in\mathcal{C} \text{ for some }x\in Z \}. \] We say that \(\mathcal{C}\) is fine at \(x\) if and only if \[ \inf\{dm(S):(x,S)\in\mathcal{C} \}=0. \] We say that a covering relation \(\mathcal{V}\) is a \(\mu\)-Vitali relation in \(X\) if \(\mathcal{V}(X)\) is a family of Borel sets, \(\mathcal{V}\) is fine at each \(x \in X\), and for any \(\mathcal{C}\subset\mathcal{V}\) and \(A\subset X\) such that \(\mathcal{C}\) is fine at each point of \(A\), the family \(\mathcal{C}(A)\) has a countable disjoint subfamily \(\mathcal{F}\) such that \[ \mu \left(A\diagdown\bigcup\limits_{S\in\mathcal{F}}S\right)=0. \] The next result deals with function defined in \(\mathbb{R}^n\) with values in a metric measure \(Y\): Let \(Y\) be any metric measure space. If \(f:\mathbb{R}^n\to Y\), \(n\leq Q\), satisfies the \(L\) condition in \(\mathbb{R}^n\) and the pull-back measure \(f^*\nu\) is finite on bounded sets, then it holds that \[ \int_A\mathcal{J}_f(x)d\mathcal{L}(x)=\int_Y\mathcal{N}(f,A,y)d\nu(y) \] for every measurable set \(A\subset\mathbb{R}^n\). Here \(\mathcal{N}(f,A,y)\) is a multiplicity function. The most general result is Theorem 12 including a general area formula for functions transforming a metric measure space into another metric measure space and satisfying the \(L\) condition and for \(n\)-rectifiable domains. The proof uses the decomposition of the space \(X\) obtained with the use of the Vitali relation and the new metric Jacobian \(\tilde{\mathcal{J}_f}(x)\).
scientific article; zbMATH DE number 7632544
scientific article