On the dimension of divergence sets of dispersive equations (Q623320)

For the initial value problem of the linear Schrödinger equation \[ i\partial_t u-\triangle u =0, \quad u(\cdot , 0)\in H^s(\mathbb R^n), \] a classical problem is to identify the Sobolev exponent \(s\) such that \(\lim_{t\to 0}u(x,t)=u(x,0)\). In this paper the authors consider a refinement of this problem by regarding the Hausdorff dimension of the set where the convergence fails. The authors consider the operator \[ S^N_tf(x)=\frac 1{2\pi}\int e^{-(N^{-1}|\xi|)}\widehat f(\xi)e^{ix\cdot \xi -t|\xi|^m}d\xi, \] and let \(u(x,t)=\lim_{N\to \infty} S^N_tf(x)\). For \(m=2\), \(u(x,t)\) is the solution for the linear Schrödinger equation and for \(m=1\), \(u(x,t)\) is the solution for the wave equation. Let \(\mathcal M^\alpha(\mathbb B)\) be the set of all \(\alpha\)-dimensional probability measures supported in the unit ball in \(\mathbb R^n\), and for \(\mu \in \mathcal M^\alpha(\mathbb B^n)\), let \(c_\alpha(\mu) = \sup_{x\in \mathbb R^n,\,r>0} \frac {\mu (B(x,r))}{r^\alpha}\). Let \(\alpha_{m,n}\) be the infimum of \(\alpha\) such that \[ \Big\|\sup_{k\geq 1} \sup_{N\geq 1} \big|S^N_{t_k}f\big| \Big\|_{L^1(\mu)}\lesssim \sqrt{c_\alpha(\mu)}\|f\|_{H^s(\mathbb R^n)}, \tag{1} \] where \(\mu \in \mathcal M^\alpha(\mathbb B),f\in H^s(\mathbb R^n), (t_k) \in \mathbb R^{\mathbb N}\) with \(t_k \to 0\). By Forstman's lemma (1) implies that \[ \dim_H\{x\in \mathbb R^n : u(x,t_k) \nrightarrow u(x,0)\text{ as }k\to \infty \} \leq \alpha_{m,n}(s), \] for all \(u_0 \in H^s(\mathbb R^n)\) and every sequence \(t_k\to 0\). In this paper the authors derive bounds for \(\alpha_{m,n}(s)\). In section 3, they apply (a) the Kolmogorov-Seliverstov-Plessner method, (b) bilinear Fourier extension estimates and (c) weighted Fourier extension estimates to establish (1) to obtain upper bounds for \(\alpha_{m,n}(s)\). In section 4, they construct some examples such that (1) doesn't hold to get lower bounds for \(\alpha_{m,n}(s)\).