arXiv subjects

Linfei Zheng

Publications and source records attributed to Linfei Zheng.

4 recordsLinked to original sources

arXiv2026

From generalized Poincar\'e to Poincar\'e-Sobolev inequalities via self-improving methods

We establish several improvements to the main results of [PR19] and [CP21], refining the seminal self-improving method for generalized Poincar\'e inequalities from [FPW98, MP98]. These results, together with various related applications, stem from a general self-improving property for functions satisfying the local inequality $$\frac{1}{|Q|}\int_Q |f(x)-f_Q|\,dx \le a(Q)$$ for all cubes $Q\subset\mathbb{R}^n$. The functional $a$ is assumed to obey a specific discrete geometric summability condition. By restricting our focus to axis-parallel cubes in $\mathbb{R}^n$, this geometric setting allows us to obtain sharper estimates than those available in more general metric measure spaces.

math.CA↗

arXiv2026

The bilinear cone multiplier on $\mathbb{R}^2\times \mathbb{R}^2$

In this paper, we study the bilinear cone multiplier operator in two dimensions. We establish $L^{p_1}\times L^{p_2}\to L^{p}$ boundedness for a regularized version of this operator over a broad range of exponents satisfying the H\"older scaling condition. Our approach is based on a decomposition of the bilinear operator into square functions associated with linear cone multipliers and their variants. We derive pointwise bounds for these square functions via suitable strong maximal function estimates, and obtain sharp $L^4$ bounds using geometric methods originating in the work of C\'ordoba and Carbery. The combination of these estimates yields the $L^p$ boundedness for the bilinear cone multiplier.

math.CA↗

arXiv2024

New Sparse Domination and Weighted Estimates for Fractional Operators Beyond Calder\'on-Zygmund Theory

Let $L$ be a closed, densely defined operator on $L^2(\mathbb{R}^n)$ satisfying suitable $L^p-L^q$ off-diagonal estimates of order $\kappa > 0$. This paper aims to investigate the two-weight estimate and the Bloom weighted estimate for the fractional operator $L^{-\alpha/\kappa}$ with $0 < \alpha < n$ through the method of sparse domination. Our assumptions on the operators are minimal, and our result applies to a wide range of differential operators. As a byproduct, we also establish a new sparse domination criterion for a general class of fractional operators, including the classical fractional integral.

math.CA↗

arXiv2023

Weak type $A_p$ estimate for bilinear Calder\'on-Zygmund operators

In this paper, we investigate the boundedness of bilinear Calder\'on-Zygmund operators $T$ from ${L^{p_1}\left(w_1\right)} \times {L^{p_2}\left(w_2\right)}$ to ${L^{p,\infty}\left(v_{\vec{w}}\right)}$ with the stopping time method, where $1 / p = 1 / p_1 + 1 / p_2$ , $1 < p_1, p_2 < \infty$ and $\vec{w}$ is a multiple $A_{\vec{P}}$ weight. Specifically, we studied the exponent $\alpha$ of $A_{\vec{P}}$ constant in formula $$\|T(\vec{f})\|_{L^{p,\infty}\left(v_{\vec{w}}\right)} \leqslant C_{m, n, \vec{P}, T}[\vec{w}]_{A_{\vec{P}}}^{\alpha}\left\|f_1\right\|_{L^{p_1}\left(w_1\right)}\left\|f_2\right\|_{L^{p_2}\left(w_2\right)}.$$ Surprisingly, we show that when $p \geqslant \frac{3+\sqrt{5}}{2}$ or $\min\{p_1,p_2\} > 4$, the index $\alpha$ in the above equation can be less than $1$, which is different from the linear scenario.

math.CA↗