arXiv subjects
Publications and source records attributed to Guoqing Zhan.
We prove the boundedness of multilinear Hilbert transforms along moment curves by establishing uniform estimates for both translated and untranslated multilinear paraproducts and by applying a Sobolev smoothing inequality. As a byproduct of our method, we obtain the same bounds for multilinear maximal functions along moment curves.
For any integer $n \geq 2$, let $(m_{1},\ldots,m_{n})$ be a strictly increasing $n$-tuple of positive integers. We show that any subset $A\subset [N]^n$ of density at least $(\log N)^{-c}$ contains a nontrivial configuration of the form \begin{equation*} \boldsymbol{x},\boldsymbol{x}+r^{m_{1}}\boldsymbol{e_{1}},\ldots,\boldsymbol{x}+r^{m_{n}}\boldsymbol{e_{n}}, \end{equation*} where $c=c(n,m_{1},\ldots,m_{n} )$ is a positive constant. This quantitative multidimensional Szemerédi theorem extends a recent two-dimensional result of Peluse, Prendiville, and Shao concerning the configuration of the form $(x,y),(x+r,y),\left(x,y+r^{2}\right)$. The theorem is obtained as a consequence of an effective ``popular'' version.
Pólya's conjecture on the eigenvalues of the Laplacian has been one of the core problems in spectral geometry. Building upon the recent breakthrough works on Pólya's conjecture for balls and annuli by Filonov, Levitin, Polterovich and Sher, we study several aspects of Pólya's conjecture for balls and cylinders: by refining the purely analytical portion of the proof in [2] for the Neumann Pólya's conjecture for the disk, we extend the regime of the spectral parameter that can be established without computer assistance; we obtain improvement of Pólya's conjecture for disks and balls; we obtain improvement of Pólya's conjecture for cylinders and confirm the Neumann Pólya's conjecture for cylinders in $\mathbb{R}^3$. As a supplementary effort, we study Weyl's law for cylinders.