arXiv subjects

Pingxu Hu

Publications and source records attributed to Pingxu Hu.

4 recordsLinked to original sources

arXiv2026

Fractional Gagliardo--Nirenberg Inequalities: Pointwise Estimates,Sharp Asymptotics, and Optimal Target Spaces

We establish two pointwise estimates for fractional difference operators, tracking explicitly the dependence of the constants on the smoothness index $s\in(0,1)$. Using these, within the framework of ball Banach function spaces we obtain two fractional Gagliardo--Nirenberg inequalities, including the BMO endpoint case. Furthermore, we establish endpoint asymptotic results as $s\to0^+$ and $s\to1^-$, proving that the asymptotic factors appearing in these inequalities have optimal order. Under the additional assumption that the underlying function space is rearrangement invariant, we show that the optimal Gagliardo--Nirenberg target spaces are precisely those given by the Calder\'on--Lozanovski\u{\i} space. This completely characterizes the rearrangement invariant target spaces for which the corresponding Gagliardo--Nirenberg inequalities hold, thereby answering an open question posed by K. Le\'snik, T. Roskovec, and F. Soudsk\'y. These results can be applied to various function spaces; in particular, they are completely new in the off-diagonal and BMO cases.

math.CA↗

arXiv2026

A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications

In this article, we establish a nearly sharp localized weighted inequality related to Gagliardo and Sobolev seminorms, respectively, with the sharp $A_1$-weight constant or with the specific $A_p$-weight constant when $p\in (1,\infty)$. As applications, we further obtain a new characterization of Muckenhoupt weights and, in the framework of ball Banach function spaces, an inequality related to Gagliardo and Sobolev seminorms on cubes, a Gagliardo--Nirenberg interpolation inequality, and a Bourgain--Brezis--Mironescu formula. All these obtained results have wide generality and are proved to be (nearly) sharp. The original version of this article was published in [Adv. Math. 481 (2025), Paper No. 110537]. In this revised version, we correct an error appeared in Theorem 1.1 in the case where $p=1$, which was pointed out to us by Emiel Lorist.

math.FA↗

arXiv2025

Sharp Brezis--Seeger--Van Schaftingen--Yung Formulae for Higher-Order Gradients in Ball Banach Function Spaces

Let $X$ be a ball Banach function space on $\mathbb{R}^n$, $k\in\mathbb{N}$, $h\in\mathbb{R}^n$, and $\Delta^k_h$ denote the $k${\rm th} order difference. In this article, under some mild extra assumptions about $X$, the authors prove that, for both parameters $q$ and $\gamma$ in \emph{sharp} ranges which are related to $X$ and for any locally integrable function $f$ on ${\mathbb{R}^n}$ satisfying $|\nabla^k f|\in X$, $$ \sup_{\lambda\in(0,\infty)}\lambda \left\|\left[\int_{\{h\in\mathbb{R}^n:\ |\Delta_h^k f(\cdot)|>\lambda|h|^{k+\frac{\gamma}{q}}\}} \left|h\right|^{\gamma-n}\,dh\right]^\frac{1}{q}\right\|_X \sim \left\|\,\left|\nabla^k f\right|\,\right\|_{X} $$ with the positive equivalence constants independent of $f$. As applications, the authors establish the Brezis--Seeger--Van Schaftingen--Yung (for short, BSVY) characterization of higher-order homogeneous ball Banach Sobolev spaces and higher-order fractional Gagliardo--Nirenberg and Sobolev type inequalities in critical cases. All these results are of quite wide generality and can be applied to various specific function spaces; moreover, even when $X:= L^{q}$, these results when $k=1$ coincide with the best known results and when $k\ge 2$ are completely new. The first novelty is to establish a sparse characterization of dyadic cubes in level sets related to the higher-order local approximation, which, together with the well-known Whitney inequality in approximation theory, further induces a higher-order weighted variant of the remarkable inequality obtained by A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore; the second novelty is to combine this weighted inequality neatly with a variant higher-order Poincar\'e inequality to establish the desired upper estimate of BSVY formulae in weighted Lebesgue spaces.

math.FA↗

arXiv2022

New John--Nirenberg--Campanato-Type Spaces Related to Both Maximal Functions and Their Commutators

Let $p,q\in [1,\infty]$, $\alpha\in{\mathbb{R}}$, and $s$ be a non-negative integer. In this article, the authors introduce a new function space $\widetilde{JN}_{(p,q,s)_{\alpha}}(\mathcal{X})$ of John-Nirenberg-Campanato type, where $\mathcal{X}$ denotes $\mathbb{R}^n$ or any cube $Q_{0}$ of $\mathbb{R}^n$ with finite edge length. The authors give an equivalent characterization of $\widetilde{JN}_{(p,q,s)_{\alpha}}(\mathcal{X})$ via both the John-Nirenberg-Campanato space and the Riesz-Morrey space. Moreover, for the particular case $s=0$, this new space can be equivalently characterized by both maximal functions and their commutators. Additionally, the authors give some basic properties, a good-$\lambda$ inequality, and a John-Nirenberg type inequality for $\widetilde{JN}_{(p,q,s)_{\alpha}}(\mathcal{X})$.

math.FA↗