arXiv subjects
Publications and source records attributed to Xiumin Du.
We obtain $L^p$ estimates of the maximal Schrödinger operator in $\mathbb R^n$ using polynomial partitioning, bilinear refined Strichartz estimates, and weighted restriction estimates.
We show that if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 3$, then there is a point $x\in E$ such that the pinned distance set $Δ_x(E)$ has positive Lebesgue measure. This improves upon bounds of Du-Zhang and Du-Iosevich-Ou-Wang-Zhang in all dimensions $d \ge 3$. We also prove lower bounds for Hausdorff dimension of pinned distance sets when $\dim_H (E) \in (\frac{d}{2} - \frac{1}{4} - \frac{3}{8d+4}, \frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4})$, which improves upon bounds of Harris and Wang-Zheng in dimensions $d \ge 3$.
We obtain some sharp $L^p$ weighted Fourier restriction estimates of the form $\|Ef\|_{L^p(B^{n+1}(0,R),Hdx)} \lessapprox R^β\|f\|_2$, where $E$ is the Fourier extension operator over the truncated paraboloid, and $H$ is a weight function on $\mathbb R^{n+1}$ which is $n$-dimensional up to scale $\sqrt R$.
We prove some weighted refined decoupling estimates. As an application, we give an alternative proof of the following result on Falconer's distance set problem by the authors in a companion work: if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 4$, then there is a point $x\in E$ such that the pinned distance set $Δ_x(E)$ has positive Lebesgue measure. Aside from this application, the weighted refined decoupling estimates may be of independent interest.
We present a free Schrödinger solution studied by Barceló--Bennett--Carbery--Ruiz--Vilela and show why it can be viewed as a sharp example for the recently discovered refined decoupling theorem.
We study an extension of the Falconer distance problem in the multiparameter setting. Given $\ell\geq 1$ and $\mathbb{R}^{d}=\mathbb{R}^{d_1}\times\cdots \times\mathbb{R}^{d_\ell}$, $d_i\geq 2$. For any compact set $E\subset \mathbb{R}^{d}$ with Hausdorff dimension larger than $d-\frac{\min(d_i)}{2}+\frac{1}{4}$ if $\min(d_i) $ is even, $d-\frac{\min(d_i)}{2}+\frac{1}{4}+\frac{1}{4\min(d_i)}$ if $\min(d_i) $ is odd, we prove that the multiparameter distance set of $E$ has positive $\ell$-dimensional Lebesgue measure. A key ingredient in the proof is a new multiparameter radial projection theorem for fractal measures.
We show that if compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}$, where $d\geq 4$ is an even integer, then the distance set of $E$ has positive Lebesgue measure. This improves the previously best known result towards Falconer's distance set conjecture in even dimensions.
We show that certain $L^2$ space-time estimates for generalized density matrices which have been used by several authors in recent years to study equations of BBGKY or Hartree-Fock type, do not have non-trivial $L^pL^q$ generalizations.
For spherical and parabolic averages of the Fourier transform of fractal measures, we obtain new upper bounds on rates of decay by an "intermediate dimension" trick.
We show that, for $n\geq 3$, $\lim_{t \to 0} e^{itΔ}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is sharp and fully resolves a problem raised by Carleson. Our main theorem is a fractal $L^2$ restriction estimate, which also gives improved results on the size of divergence set of Schrödinger solutions, the Falconer distance set problem and the spherical average Fourier decay rates of fractal measures. The key ingredients of the proof include multilinear Kakeya estimates, decoupling and induction on scales.
We give new lower bounds for $L^p$ estimates of the Schrödinger maximal function by generalizing an example of Bourgain.
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 $n\geq 3$, $\lim_{t \to 0} e^{itΔ}f(x) = f(x)$ 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.
We prove some weighted Fourier restriction estimates using polynomial partitioning and refined Strichartz estimates. As application we obtain improved spherical average decay rates of the Fourier transform of fractal measures, and therefore improve the results for the Falconer distance set conjecture in three and higher dimensions.
To prove Fourier restriction estimate using polynomial partitioning, Guth introduced the concept of $k$-broad part of regular $L^p$ norm and obtained sharp $k$-broad restriction estimates. To go from $k$-broad estimates to regular $L^p$ estimates, Guth employed $l^2$ decoupling result. In this article, similar to the technique introduced by Bourgain-Guth, we establish an analogue to go from regular $L^p$ norm to its $(m+1)$-broad part, as the error terms we have the restricted $k$-broad parts ($k=2,\cdots,m$). To analyze the restricted $k$-broadness, we prove an $l^p$ decoupling result, which can be applied to handle the error terms and recover Guth's linear restriction estimates.
We show that $\lim_{t \to 0} e^{itΔ}f(x) = f(x)$ almost everywhere for all $f \in H^s (\mathbb{R}^2)$ provided that $s>1/3$. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.
Using Guth's polynomial partitioning method, we obtain $L^p$ estimates for the maximal function associated to the solution of Schrödinger equation in $\mathbb R^2$. The $L^p$ estimates can be used to recover the previous best known result that $\lim_{t \to 0} e^{itΔ}f(x)=f(x)$ almost everywhere for all $f \in H^s (\mathbb{R}^2)$ provided that $s>3/8$.