Dimension of divergence sets for dispersive equation

Abstract

Consider the generalized dispersive equation defined by

\{\begin{array}{cc}{\mathrm{i}\partial }_{t}u+\phi \sqrt{-\mathrm{\Delta }})u=0,& (x,t)\in {ℝ}^n\times ℝ,\\ u(x,0)=f\left(x\right),& f\in \mathit{\varphi }\left({ℝ}^n\right),\end{array}(*)

where \phi (\sqrt{-\mathrm{\Delta }}) is a pseudo-differential operator with symbol \phi (\left|\xi \right|). In the present paper, assuming that \phi satisfies suitable growth conditions and the initial data in H^s\left({ℝ}^n\right), we bound the Hausdorff dimension of the sets on which the pointwise convergence of solutions to the dispersive equations (*) fails. These upper bounds of Hausdorff dimension shall be obtained via the Kolmogorov-Seliverstov-Plessner method.

Keywords

Dispersive equation / Hausdor_ dimension / maximal operator

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