AN UPBOUND OF HAUSDORFF'S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRÖDINGER OPERATOR ON Hs($\mathbb{R}^n$)

doi:10.1007/s10473-021-0412-z

Abstract

Abstract: Given $n\geq2$ and $\alpha > \frac 12$, we obtained an improved upbound of Hausdorff's dimension of the fractional Schrödinger operator; that is, $$ \sup\limits_{f\in H^s(\mathbb{R}^n)}\dim _H\left\{x\in\mathbb{R}^n:\ \lim_{t\rightarrow0}e^{{\rm i}t(-\Delta)^\alpha}f(x)\neq f(x)\right\}\leq n+1-\frac{2(n+1)s}{n}%\ \ \text{under}\ \ \frac{n}{2(n+1)} < s\leq\frac{n}{2} $$ for $\frac{n}{2(n+1)} < s\leq\frac{n}{2}$.

Key words: The Carleson problem, divergence set, the fractional Schrödinger operator, Hausdorff dimension, Sobolev space

CLC Number:

42B37

Dan LI, Junfeng LI, Jie XIAO. AN UPBOUND OF HAUSDORFF'S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRÖDINGER OPERATOR ON H^s($\mathbb{R}^n$)[J].Acta mathematica scientia,Series B, 2021, 41(4): 1223-1249.

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AN UPBOUND OF HAUSDORFF'S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRÖDINGER OPERATOR ON H^s($\mathbb{R}^n$)

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