Applied Math And Analysis Seminar
Tuesday, October 26, 2021, 3:15pm, Zoom
Daniel Eceizabarrena (University of Massachusetts Amherst)
Pointwise convergence over fractals for dispersive equations with homogeneous symbol
Abstract:
Let $P \in C^\infty(\mathbb R^n \setminus \{ 0 \} )$ be a real, homogeneous and non-singular symbol that defines the dispersive equation \begin{equation} i\, \partial_t u + P(D)u = 0, \qquad \qquad u(x,0) = f(x). \end{equation} Given $0 \leq \alpha \leq n$ and the Hausdorff measure $\mathcal H^\alpha$, for which $s >0$ do we have \begin{equation} \lim_{t \to 0} u(x,t) = f(x), \qquad \mathcal H^\alpha\text{-almost everywhere} \qquad \forall f \in H^s(\mathbb R^n) \quad ? \end{equation} This is a generalization of the famous problem proposed by Carleson for $\alpha = n$ and for $P(\xi) = |\xi|^2$, corresponding to the almost everywhere convergence of the free Schr\"odinger equation with respect to the Lebesgue measure. Recently, in the setting of the Lebesgue measure $\alpha = n$, An, Chu and Pierce adapted Bourgain's optimal counterexample to study the symbol \[ P_k(\xi) = \xi_1^k + \ldots + \xi_n^k, \qquad k \in \mathbb N, \quad k \geq 2 . \] In this talk, after introducing the basics of the problem, I will explain how one can generalize this counterexample to the fractal case \(\alpha < n\). For that, I will explain and use the Mass Transference Principle, a technique developed in the field of Diophantine approximation, as a tool to compute the Hausdorff dimension of the sets of divergence. Time permitting, I will also mention a couple of positive results that we obtain for general symbols $P$.

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