POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES

Abstract

<jats:p>We obtain partial improvement toward the pointwise convergence problem of Schrödinger solutions, in the general setting of fractal measure. In particular, we show that, for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline1" /><jats:tex-math>$n\geqslant 3$</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline2" /><jats:tex-math>$\lim _{t\rightarrow 0}e^{it\unicode[STIX]{x1D6E5}}f(x)$</jats:tex-math></jats:alternatives></jats:inline-formula><jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline3" /><jats:tex-math>$=f(x)$</jats:tex-math></jats:alternatives></jats:inline-formula> almost everywhere with respect to Lebesgue measure for all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline4" /><jats:tex-math>$f\in H^{s}(\mathbb{R}^{n})$</jats:tex-math></jats:alternatives></jats:inline-formula> provided that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050509418000117_inline5" /><jats:tex-math>$s&gt;(n+1)/2(n+2)$</jats:tex-math></jats:alternatives></jats:inline-formula>. The proof uses linear refined Strichartz estimates. We also prove a multilinear refined Strichartz using decoupling and multilinear Kakeya.</jats:p>