Taylor polynomials and non-homogeneous blow-ups (Q1434150)

Let \(f:\mathbb{R}^n\to \mathbb{R}^k\) be of class \(C^h\) (\(h\geq 1\)) and let \(S\) be the graph of \(f\). Then the dilatated surface \(T_{h, \varepsilon}S\) (with respect to \(x_0 \in \mathbb{R}^n\)) one can define as the graph of \[ f_{h,\varepsilon} (u):=\frac{f(x_0+\varepsilon u)-P_{h-1}f(x_0+\varepsilon u)} {{\varepsilon}^h}, \] where \(u \in \mathbb{R}^n\) and \(P_{h-1}f\) indicates the Taylor polynomial of \(f\) at \(x_0\) of degree \(h-1\). Obviously, \(f_{h,\varepsilon}\) converges, uniformly on compact sets, to the monomial \(f_{h,0}\) of degree \(h\) in the Taylor expansion of \(f\) at \(x_0\) when \(\varepsilon \to 0\). Moreover, the Jacobian matrix of \(f_{h,\varepsilon}\) converges, uniformly on compact sets, to the Jacobian matrix of \(f_{h,0}\), and the convergence of \(n\)-dimensional Hausdorff measures on corresponding graphs is fulfilled. In the paper under review, the author investigates various types of convergence under suitable weakened conditions on \(f\). Let \(f:U\subset \mathbb{R}^n \to \mathbb{R}^k\) be of class \(C^{h-1}\) (\(h\geq 2\)) and there exists a map \(\Psi \in C^{h-1}(U,L(\mathbb{R}^n,\mathbb{R}^k))\) such that the set \(K:=\{x\in U\mid Df(x)=\Psi(x)\}\) has density one at \(x_0\). The role of \(f_{h,0}\) in this case should be play the map \(g:\mathbb{R}^n\to \mathbb{R}^k\) defined by \(g(u):=1/h!\langle \Psi^{(h-1)}(x_0)| u^h\rangle\). Under these conditions the author obtains some results which are unexpected in some sense. In particular, the author gives two special examples. The first (the second) example is the pair of functions \(f\) and \(\Psi\) such that \(f_{h,\varepsilon}\) does not converge (converges) to \(g\) almost everywhere and there is no convergence of \(n\)-dimensional Hausdorff measures on corresponding graphs (in both examples \(n=k=h-1=1\), \(U=\mathbb{R}\), \(\Psi :=0\), \(x_0=0\)).