arXiv subjects
Publications and source records attributed to Dilong Li.
This note presents a proof that the Riesz projection is contractive from $L^q$ to $L^{4(1-1/q)}$ for the case $1<q<\infty$. This fills in the previously unresolved parameter range of a conjecture of Brevig, Ortega-Cerd\`a, Seip, and Zhao.
In this paper, we investigate contractive projections, conditional expectations, and idempotent coefficient multipliers on the Hardy spaces $H^p(\mathbb{T})$ for $0<p<1$. For such values of $p$, we first establish a general extension theorem for contractive projections in a probability $L^p$-space. Combining this theorem with the study of conditional expectations on $H^p(\mathbb{T})$, we characterize a broad class of contractive projections on $H^p(\mathbb{T})$ that are of particular interest. Furthermore, we apply these results to give a complete characterization of contractive idempotent coefficient multipliers for the Hardy spaces $H^p(\mathbb{T}^d)$ on the $d$-dimensional torus for $0<p<1$ and $1\leq d\leq \infty$. This complements a remarkable result of Brevig, Ortega-Cerd\`{a}, and Seip characterizing such multipliers on $H^p(\mathbb{T}^d)$ for $1\leq p \leq \infty$.
Through the establishment of several extension theorems, we provide explicit expressions for all contractive projections and 1-complemented subspaces in the Hardy space $H^p(\mathbb{T})$ for $1\leq p<\infty$, $p\neq 2$. Our characterization leads to two corollaries: first, all nontrivial 1-complemented subspaces of $H^p(\mathbb{T})$ are isometric to $H^p(\mathbb{T})$; second, all contractive projections on $H^p(\mathbb{T})$ are restrictions of contractive projections on $L^p(\mathbb{T})$ that leave $H^p(\mathbb{T})$ invariant. The first corollary provides examples of prime Banach spaces \emph{in the isometric sense}, while the second answers a question posed by P. Wojtaszczyk in 2003.
Point cloud segmentation is one of the most important tasks in computer vision with widespread scientific, industrial, and commercial applications. The research thereof has resulted in many breakthroughs in 3D object and scene understanding. Previous methods typically utilized hierarchical architectures for feature representation. However, the commonly used sampling and grouping methods in hierarchical networks are only based on point-wise three-dimensional coordinates, ignoring local semantic homogeneity of point clusters. Additionally, the prevalent Farthest Point Sampling (FPS) method is often a computational bottleneck. To address these issues, we propose a novel 3D point cloud representation network, called Dynamic Clustering Transformer Network (DCTNet). It has an encoder-decoder architecture, allowing for both local and global feature learning. Specifically, we propose novel semantic feature-based dynamic sampling and clustering methods in the encoder, which enables the model to be aware of local semantic homogeneity for local feature aggregation. Furthermore, in the decoder, we propose an efficient semantic feature-guided upsampling method. Our method was evaluated on an object-based dataset (ShapeNet), an urban navigation dataset (Toronto-3D), and a multispectral LiDAR dataset, verifying the performance of DCTNet across a wide variety of practical engineering applications. The inference speed of DCTNet is 3.8-16.8$\times$ faster than existing State-of-the-Art (SOTA) models on the ShapeNet dataset, while achieving an instance-wise mIoU of $86.6\%$, the current top score. Our method similarly outperforms previous methods on the other datasets, verifying it as the new State-of-the-Art in point cloud segmentation.
In this paper, we consider the first Szeg\H{o} limit theorems on $d$-torus $\mathbb{T}^d$ for $1\leq d\leq +\infty$. It is shown that for any F{\o}lner sequence $\{\sigma_N\}$ of $\mathbb{Z}^d$ and $\varphi\in L^1_+(\mathbb{T}^d)$, it holds that $$ \lim_{N\rightarrow \infty}\left(\det T_{\sigma_N}\varphi\right)^{\frac{1}{|\sigma_N|}}=\exp\left(\int_{\mathbb{T}^d} \log\varphi~dm_{d}\right). $$ In the case $d=+\infty$, we are associated with multiplicative Toeplitz matrix $T \varphi=\{\widehat{\varphi}(j/i)\}_{i,j\in\mathbb{N}}$ and the most concerned non-F{\o}lner truncation, that is, $T_N \varphi=\{\widehat{\varphi}(j/i)\}_{1\leq i,j\leq N}$, where $\sigma_N=\{1,\dots,N\}$. It is shown that for each $\varphi\in L^\infty_{\mathbb{R}}(\mathbb{T^{\infty}})$ and $f\in C[\text{ess-inf} ~\varphi,~\text{ess-sup}~\varphi]$, the limit $\lim_{N\rightarrow \infty} \frac{1}{N}\mathrm{Tr} f \big(T_N \varphi\big)$ exsits. Moreover, it is proven that the limit $\lim_{N\rightarrow \infty}\left(\det T_N \varphi\right)^{\frac{1}{N}}$ exists for any $\varphi\in L^1_+(\mathbb{T}^\infty)$ with strictly positive essential infimum. These results are directly related to two problems posed by Nikolski and Pushnitski.