Abstract
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 $n\geqslant 3$, $\lim _{t\rightarrow 0}e^{it\unicode[STIX]{x1D6E5}}f(x)$$=f(x)$ almost everywhere with respect to Lebesgue measure for all $f\in H^{s}(\mathbb{R}^{n})$ provided that $s>(n+1)/2(n+2)$. The proof uses linear refined Strichartz estimates. We also prove a multilinear refined Strichartz using decoupling and multilinear Kakeya.
Highlights
The solution to the free Schrodinger equation i ut − ∆u = 0, (x, t) ∈ Rn × R u(x, 0) = f (x), x ∈ Rn (1.1)c The Author(s) 2018
In [9], via polynomial partitioning developed in [11, 12] and linear and bilinear refined Strichartz, some new weighted restriction estimates were established, and as applications improved results were obtained for the Falconer distance set problem and the spherical average decay rates of the Fourier transform of fractal measures
We prove a multilinear refined Strichartz using decoupling and multilinear Kakeya
Summary
In [8], the almost sharp result s > 1/3 is obtained in the setting of Lebesgue measure, and the sharp Schrodinger maximal estimate in [8] implies directly the following generalized improvement: α2(s) 3 − 3s, 1/3 < s < 1/2. Linear and bilinear refined Strichartz were derived in [8] to solve the pointwise convergence problem in two dimensions. In [9], via polynomial partitioning developed in [11, 12] and linear and bilinear refined Strichartz, some new weighted restriction estimates were established, and as applications improved results were obtained for the Falconer distance set problem and the spherical average decay rates of the Fourier transform of fractal measures.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
