Weak uncertainty principles on fractals
A pcf fractal \(K\) can be defined as a compact subset of \(\mathbb{R}^n\) that is invariant under an iterated function system of contractive linear maps \(\{F_i\}_{i=1}^N\). The boundary \(V_0\) of \(K\) consists of those fixed points of the \(F_i\) and the pcf condition is that \(K\) is connected and \[ F_i K \cap F_j K\subset F_i V_0\cap F_j V_0,\quad i\neq j. \] An iterate \(F_\omega K=F_{\omega_m}\circ \cdots \circ F_{\omega_1} K\) is called a cell at level \(m\). One also associates with the fractal a self-similar measure \(\mu =\sum \mu_i \mu\circ F_i^{-1}\) that depends on weights \(\mu_i\) and a self similar energy \(\mathcal{E}\) defined in terms of weights \(\{r_i\}_{i=1}^N\) extended to \(r_\omega =\prod r_{\omega_j}\) through the identity \[ \mathcal{E} (u,u)=\sum_m\sum_{| \omega| =m} r_\omega^{-1} \mathcal{E}(u\circ F_\omega,u\circ F_\omega). \] It is possible to define a Laplacian \(\Delta\) such that \[ -\mathcal{E}(u,u)=\int_K (\Delta u)^2 \, d\mu \] for suitable \(u\). \(\Delta\) has a precise meaning also as a limit as \(m\to\infty\) of Laplacians for graphs defined as boundaries of cells at level \(m\). Then, in analogy with the variance \({\text{var}}\,\mu=\frac{1}{2}\int\int (x-y)^2 \,d\mu(x)d\mu(y)\) for a probability measure on \(\mathbb{R}\), one defines \[ {\text{Var}}_d (| u| ^2)=\int\int_{K\times K} d_R(x,y)^{d-1}| u(x)| ^2| u(y)| ^2\, d\mu(x)\, d\mu(y) \] in which \(d\) is the Hausdorff dimension of \(K\) and \(d_R(x,y)=(\min\mathcal{E}(u,u):u(x)=0, u(y)=1)^{-1}\). The main result states that there is a constant \(C>0\) such that for all \(u\in {\text{Dom}}\,\mathcal{E}\) with \(\| u\| _{L^2(K)}=1\) and \({\text{Var}}_d(| u| ^2)\leq\frac{1}{2}\), \({\text{Var}}_d(| u| ^2)\mathcal{E}(u,u)\geq C\). In other words, the variance and energy of \(u\) cannot be made simultaneously small. Further aspects of the analogy between this inequality and the classical Heisenberg uncertainty inequality are discussed. Extensions to blowups of pcf fractals and to products of fractals are also provided.
scientific article; zbMATH DE number 5013624