Singular points of Hölder asymptotically optimally doubling measures (Q5963587)

Let \(\mu\) be a nonzero Radon measure on \({\mathbb R}^n\) with support \(\Sigma\), \(\alpha>0\) and \(0<m\leq n\) is integer. The measure \(\mu\) is called \((\alpha, m)\)-Hölder asymptotically uniform measure if for any compact set \(K\subset\Sigma\) there exist a constant \(C_K\) and a radius \(r_0 > 0\) such that for \(0 < r\leq r_0\) and \(x \in K\) \[ \left|\frac{\mu(B(x, r)}{\omega_{m}r^m}-1 \right|\leq C_{K}r^{\alpha}, \] where \(B(x, r)\) is ball with center \(x\) and radius \(r\), and \(\omega_m\) is volume of the \(m\)-dimensional unit ball. The author studies measures of that kind. In particular, he shows that for all \(\alpha > 0\) there exists \(\beta = \beta(\alpha)>0\) with the following property: -- if \(\mu\) is an \((\alpha, 3)\)-Hölder asymptotically uniform measure on \({\mathbb R}^4\) and \(x \in \Sigma\) is a nonflat point, then there exists a neighborhood of \(x\) which is \(C^{1,\beta}\)-diffeomorphic to an open piece of cone \(x_{4}^2 = x_{1}^2 +x_{2}^2 +x_{3}^2\) containing the singular point 0.