### An improved maximal inequality for 2D fractional order Schrödinger operators

The local maximal operator for the Schrödinger operators of order α > 1 is shown to be bounded from ${H}^{s}\left(\mathbb{R}\xb2\right)$ to L² for any s > 3/8. This improves the previous result of Sjölin on the regularity of solutions to fractional order Schrödinger equations. Our method is inspired by Bourgain’s argument in the case of α = 2. The extension from α = 2 to general α > 1 faces three essential obstacles: the lack of Lee’s reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable...