arXiv subjects

Pengcheng Tang

Publications and source records attributed to Pengcheng Tang.

11 recordsLinked to original sources

arXiv2026

Generalized Hilbert operators on Hardy spaces

Let $g\in H(\mathbb D)$, the generalized Hilbert operator $\mathcal H_g$ is defined by \[ \mathcal H_g(f)(z)=\int_0^1 f(t)g'(tz)dt,\ \ z\in \mathbb D\, \ \ f \in H(\mathbb D). \] Let $\mathcal R_p=\mathcal H(H^p)$ be the range of the classical Hilbert operator on Hardy space, equipped with the pullback norm, and let $(\mathcal R_p,H^p)$ denote the Hadamard multiplier space. For $1 2$, we prove that the multiplier space $(\mathcal R_p,H^p)$ is strictly contained in $H\left(p,\infty,\frac1{p'}\right)$. This shows that $g\in \Lambda(p,1/p)$ does not imply that $\mathcal H_g$ is bounded on $H^p$, giving a negative answer to the conjecture posed by Galanopoulos, Girela, Pel\'aez and Siskakis. In addition, we locate two previously known sufficient classes inside the multiplier space. This allow us obtain a complete coefficient characterization of $\mathcal H_g$ on $H^{p}$ for $g \in H(\mathbb D) $ with nonnegative decreasing Taylor coefficients. We then study the structure of $(\mathcal R_p,H^p)$. % It turns out that $(\mathcal{R}_p,H^p)$ contains all polynomials as well as Cauchy transforms. We show that the multiplier spaces $(\mathcal{R}_p,H^p)$ form a strictly increasing family with respect to the exponent $p$.

math.FA↗

arXiv2026

Operators of Hilbert type acting on some spaces of analytic functions

Let $H(\mathbb{D})$ be the space of all analytic functions in the unit disc $\mathbb{D}$. For $g\in H(\mathbb{D})$, the generalized Hilbert operator $\mathcal{H}_{g}$ is defined by $$\mathcal{H}_{g}(f)(z)=\int_{0}^{1}f(t)g'(tz)dt, \ \ z\in \mathbb{D}, f\in H(\mathbb{D}).$$ In this paper, we study the operator $\mathcal{H}_{g}$ acting on some spaces of analytic functions in $\mathbb{D}$. Specifically, we give a complete characterization of those $g\in H(\mathbb{D})$ for which the operator $\mathcal{H}_{g}$ is bounded (resp. compact) from the Dirichlet space $\mathcal{D}^{2}_{\alpha}$ to $\mathcal{D}^{2}_{\beta}$ for all possible indicators $\alpha,\beta \in \mathbb{R}$. We also study the action of the operator $\mathcal{H}_{g}$ on the space of bounded analytic functions $H^{\infty}$, which generalizes the known results for the classical Hilbert operator $\mathcal {H}$ acting on $H^{\infty}$. In particular, we consider the boundedness of the operator $\mathcal{H}_{g}$ with a symbol of non-negative Taylor coefficients, acting on logarithmic Bloch spaces and on Korenblum spaces. This work generalizes the corresponding results for the classical Hilbert operator.

math.FA↗

arXiv2025

Enhancing the Preference Extractor in Multi-turn Dialogues: From Annotating Disasters to Accurate Preference Extraction

Identifying user preferences in dialogue systems is a pivotal aspect of providing satisfying services. Current research shows that using large language models (LLMs) to fine-tune a task-specific preference extractor yields excellent results in terms of accuracy and generalization. However, the primary challenge stems from the inherent difficulty in obtaining high-quality labeled multi-turn dialogue data. Accurately tracking user preference transitions across turns not only demands intensive domain expertise and contextual consistency maintenance for annotators (termed \textbf{``Annotating Disaster''}) but also complicates model training due to error propagation in sequential dependency learning. Inspired by the observation that multi-turn preference extraction can be decomposed into iterative executions of one-turn extraction processes. We propose a novel dialogue data generation framework named \textbf{IterChat}. First, we construct a new data format that categorizes the dialogue data into attributed historical preferences and one-turn dialogues. This reduces the probability of annotation errors and improves annotation efficiency. Then, to generate a high-quality and diverse dialogue dataset, we adopt GPT4 to pre-define the preference slots in the target preference extractor task and then randomly sample the subset of the slots and their corresponding schema values to create the dialogue datasets. Experimental results indicate that fine-tuning or only few-shot prompting with the new dialogue format yields superior performance compared to the original multi-turn dialogues. Additionally, the new data format improves annotator efficiency with a win rate of 28.4\% higher than the original multi-turn dialogues.

cs.CL↗

arXiv2025

RecUserSim: A Realistic and Diverse User Simulator for Evaluating Conversational Recommender Systems

Conversational recommender systems (CRS) enhance user experience through multi-turn interactions, yet evaluating CRS remains challenging. User simulators can provide comprehensive evaluations through interactions with CRS, but building realistic and diverse simulators is difficult. While recent work leverages large language models (LLMs) to simulate user interactions, they still fall short in emulating individual real users across diverse scenarios and lack explicit rating mechanisms for quantitative evaluation. To address these gaps, we propose RecUserSim, an LLM agent-based user simulator with enhanced simulation realism and diversity while providing explicit scores. RecUserSim features several key modules: a profile module for defining realistic and diverse user personas, a memory module for tracking interaction history and discovering unknown preferences, and a core action module inspired by Bounded Rationality theory that enables nuanced decision-making while generating more fine-grained actions and personalized responses. To further enhance output control, a refinement module is designed to fine-tune final responses. Experiments demonstrate that RecUserSim generates diverse, controllable outputs and produces realistic, high-quality dialogues, even with smaller base LLMs. The ratings generated by RecUserSim show high consistency across different base LLMs, highlighting its effectiveness for CRS evaluation.

cs.HC↗

arXiv2024

Hilbert matrix operator on bound analytic functions

It is well known that the Hilbert matrix operator $\mathcal {H}$ is bounded from $H^{\infty}$ to the mean Lipschitz spaces $\Lambda^{p}_{\frac{1}{p}}$ for all $1 1}\Lambda^{p}_{\frac{1}{p}}$. We also provide explicit upper and lower bounds for the norm of the Hilbert matrix $\mathcal {H}$ acting from $H^{\infty}$ to $\Lambda^{1.*}_{1}$. Additionally, we also characterize the positive Borel measures $\mu$ such that the generalized Hilbert matrix operator $\mathcal {H}_{\mu}$ is bounded from $H^{\infty}$ to the Hardy space $H^{q}$. This part is a continuation of the work of Chatzifountas, Girela and Pel\'{a}ez [J. Math. Anal. Appl. 413 (2014) 154--168] regarding $\mathcal {H}_{\mu}$ on Hardy spaces.

math.FA↗

arXiv2023

Generalized Ces\`{a}ro-like operator from a class of analytic function spaces to analytic Besov spaces

Let $\mu$ be a finite positive Borel measure on $[0,1)$ and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n} \in H(\mathbb{D})$. For $0<\alpha<\infty$, the generalized Ces\`aro-like operator $\mathcal{C}_{\mu,\alpha}$ is defined by $$ \mathcal {C}_{\mu,\alpha}(f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)(n-k)!}a_k\right)z^n, \ z\in \mathbb{D}, $$ where, for $n\geq 0$, $\mu_n$ denotes the $n$-th moment of the measure $\mu$, that is, $\mu_n=\int_{0}^{1} t^{n}d\mu(t)$. For $s>1$, let $X$ be a Banach subspace of $ H(\mathbb{D})$ with $\Lambda^{s}_{\frac{1}{s}}\subset X\subset\mathcal {B}$. In this paper, for $1\leq p <\infty$, we characterize the measure $\mu$ for which $\mathcal{C}_{\mu,\alpha}$ is bounded(or compact) from $X$ into analytic Besov space $B_{p}$.

math.FA↗

arXiv2023

The Ces`aro-like operator on some analytic function spaces

Let $\mu$ be a finite positive Borel measure on the interval $[0, 1)$ and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n} \in H(\mathbb{D})$. The Ces\`aro-like operator is defined by $$ \mathcal {C}_{\mu} (f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}a_k\right)z^n, \ z\in \mathbb{D}, $$ where, for $n\geq 0$, $\mu_n$ denotes the $n$-th moment of the measure $\mu$, that is, $\mu_n=\int_{[0, 1)} t^{n}d\mu(t)$. Let $X$ and $Y$ be subspaces of $H( \mathbb{D})$, the purpose of this paper is to study the action of $\mathcal {C}_{\mu}$ on distinct pairs $(X, Y)$. The spaces considered in this paper are Hardy space $H^{p}(0<p\leq\infty)$, Morrey space $L^{2,\lambda}(0<\lambda\leq1)$, mean Lipschitz space, Bloch type space, etc.

math.FA↗

arXiv2022

Several Integral Estimates and Some Applications

In this paper, the authors first consider the bidirectional estimates of several typical integrals. As some applications of these integral estimates, the authors investigate the pointwise multipliers from the normal weight general function space $F(p,\mu,s)$ to the normal weight Bloch type space $\mathcal{B_{\nu}}(B_{n})$ on the unit ball $B_{n}$ of $\mathbb{C}^{n}$, where $\mu$ and $\nu$ are two normal functions on $[0,1)$. For the special normal function $\displaystyle{\mu(r)=(1-r^{2})^{\alpha}\log^{\beta}\frac{e}{1-r^{2}}}$ ($\alpha>0$, $-\infty<\beta<\infty$), the authors give the necessary and sufficient conditions of pointwise multipliers from $F(p,\mu,s)$ to $\mathcal{B_{\nu}}(B_{n})$ for all cases.

math.FA↗

arXiv2022

Ces\`{a}ro-like operator acting between Bloch type spaces

Let $\mu$ be a finite positive Borel measure on the interval $[0,1)$ and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n} \in H(\mathbb{D})$. The Ce\`{a}sro-like operator is defined by $$ \mathcal{C}_\mu(f)(z)=\sum^\infty_{n=0}\mu_n\left(\sum^n_{k=0}a_k\right)z^n, \ z\in \mathbb{D}, $$ where, for $n\geq 0$, $\mu_n$ denotes the $n$-th moment of the measure $\mu$, that is, $\mu_n=\int_{[0, 1)} t^{n}d\mu(t)$. In this paper, we characterize the measures $\mu$ for which $\mathcal{C}_\mu$ is bounded (compact) from one Bloch type space, $\mathcal {B}^{\alpha}$, into another one, $\mathcal {B}^{\beta}$.

math.FA↗

arXiv2022

Generalized integral type Hilbert operator acting on weighted Bloch space

Let $\mu$ be a finite Borel measure on $[0,1)$. In this paper, we consider the generalized integral type Hilbert operator $$\mathcal{I}_{\mu_{\alpha+1}}(f)(z)=\int_{0}^{1}\frac{f(t)}{(1-tz)^{\alpha+1}}d\mu(t)\ \ \ (\alpha>-1).$$ The operator $\mathcal{I}_{\mu_{1}}$ has been extensively studied recently. The aim of this paper is to study the boundedness(resp. compactness) of $\mathcal{I}_{\mu_{\alpha+1}}$ acting from the normal weight Bloch space into another of the same kind. As consequences of our study, we get completely results for the boundedness of $ \mathcal{I}_{\mu_{\alpha+1}}$ acting between Bloch type spaces, logarithmic Bloch spaces among others.

math.FA↗

arXiv2022

Generalized Hilbert operator acting on Bergman spaces

Let $\mu$ be a positive Borel measure on $[0,1)$. If $f \in H(\mathbb{D})$ and $\alpha>-1$, the generalized integral type Hilbert operator defined as follows: $$\mathcal{I}_{\mu_{\alpha+1}}(f)(z)=\int^1_{0} \frac{f(t)}{(1-tz)^{\alpha+1}}d\mu(t), \ \ \ z\in \mathbb{D} .$$ The operator $\mathcal{I}_{\mu_{1}}$ has been extensively studied recently. In this paper, we characterize the measures $\mu$ for which $\mathcal{I}_{\mu_{\alpha+1}}$ is a bounded (resp., compact) operator acting between the Bloch space $\mathcal {B}$ and Bergman space $ A^{p}$, or from $A^{p}(0 -1$.

math.FA↗