arXiv subjects

Chenfeng Zhu

Publications and source records attributed to Chenfeng Zhu.

7 recordsLinked to original sources

arXiv2026

Real-Variable Theory of Hardy--Lorentz Spaces on Quasi-Ultrametric Spaces of Homogeneous Type with Reverse-Doubling Property

Let $(X,\mathbf{q},\mu)$ be an ultra-RD-space with upper dimension $n\in(0,\infty)$; i.e., it is a quasi-ultrametric space of homogeneous type whose measure $\mu$ satisfies an additional reverse doubling property. Let $\mathrm{ind\,}(X,\mathbf{q})\in(0,\infty]$ denote its lower smoothness index, as introduced by Mitrea et al. In this monograph, the authors first construct a new approximation of the identity on quasi-ultrametric spaces of homogeneous type, achieving a maximal degree of smoothness $0<\varepsilon\preceq\mathrm{ind\,}(X,\mathbf{q})$. This fundamental tool is then used to derive sharp homogeneous (as well as inhomogeneous) continuous/discrete Calder\'on reproducing formulae on ultra-RD-spaces. As applications, the authors establish Littlewood--Paley function characterizations for both Hardy spaces and Triebel--Lizorkin spaces on ultra-RD-spaces. The authors further introduce Hardy--Lorentz spaces $H^{p,q}_\ast(X)$ via the grand maximal function, with the sharp range $p\in(\frac{n}{n+\mathrm{ind\,}(X,\mathbf{q})},\infty)$ and $q\in(0,\infty]$, and provide their real-variable characterizations using radial/non-tangential maximal functions, (finite) atoms, molecules, and various Littlewood--Paley functions. Based on these characterizations, the authors prove a duality theorem between Hardy--Lorentz spaces and Campanato--Lorentz spaces, establish a real interpolation theorem for Hardy--Lorentz spaces, and derive boundedness results for Calder\'on--Zygmund operators on them. It should be emphasized that many of the main results in this monograph are indeed established in the more general setting of quasi-ultrametric spaces of homogeneous type.

math.FA↗

arXiv2025

Maz'ya--Shaposhnikova Representation of Quasi-Norms of Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type with Weak Reverse Doubling Property

Let $Y(\mathcal{X})$ be a ball quasi-Banach function space on the space of homogeneous type $(\mathcal{X},ρ,μ)$ satisfying some mild additional assumptions, $q\in(0,\infty)$, and $\dot{W}^{s,q}_Y(\mathcal{X})$ with $s\in(0,1)$ be the homogeneous fractional Sobolev space associated with $Y(\mathcal{X})$. In this article, we show that, for any $f\in Y(\mathcal{X})\cap\bigcup_{s\in(0,1)} \dot{W}^{s,q}_Y(\mathcal{X})$, \begin{align*} \|f\|_{Y(\mathcal{X})} &\lesssim\varliminf_{s \to 0^+} s^{\frac{1}{q}}\left\| \left\{\int_{\mathcal{X}} \frac{|f(\cdot)-f(y)|^q}{U(\cdot,y)[ρ(\cdot,y)]^{sq}} \, dμ(y) \right\}^{\frac{1}{q}}\right\|_{{Y(\mathcal{X})}}\\ &\leq \varlimsup_{s \to 0^+} s^{\frac{1}{q}}\left\|\left\{\int_{\mathcal{X}} \frac{|f(\cdot)-f(y)|^q}{U(\cdot,y)[ρ(\cdot,y)]^{sq}} \, dμ(y) \right\}^{\frac{1}{q}}\right\|_{{Y(\mathcal{X})}} \lesssim\|f\|_{Y(\mathcal{X})}, \end{align*} where $U(x,y):=\min\{μ(B(x,ρ(x,y))),\,μ(B(y,ρ(x,y)))\}$ for any $x,y\in\mathcal{X}$ and the implicit positive constants are independent of $f$, which is applied to ten specific ball quasi-Banach function spaces and hence is of wide generality. In particular, when $Y(\mathcal{X})=L^q(\mathbb{R}^n)$ with $q\in[1,\infty)$, the above formula is closely related to the celebrated result of Maz'ya and Shaposhnikova in 2002. We also establish the above representation formula on domains of $\mathcal{X}$. The main novelty lies in proposing two new concepts, namely the weak reverse doubling condition (for $\mathcal{X}$) and the weak measure density condition (for domains of $\mathcal{X}$), which are proved to be necessary in some sense. In addition, we find an interesting fact that, when the underlying space under consideration is bounded, the above Maz'ya--Shaposhnikova-type limit always tends to zero.

math.FA↗

arXiv2024

Parabolic Muckenhoupt Weights Characterized by Parabolic Fractional Maximal and Integral Operators with Time Lag

In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag. Then the authors introduce the uncentered parabolic fractional maximal operator with time lag and characterize its two-weighted boundedness (including the endpoint case) via these weights under an extra mild assumption (which is not necessary for one-weight case). The most novelty of this article exists in that the authors further introduce a new parabolic shaped domain and its corresponding parabolic fractional integral with time lag and, moreover, applying the aforementioned two-weighted boundedness of the uncentered parabolic fractional maximal operator with time lag, the authors characterize the (two-)weighted boundedness (including the endpoint case) of these parabolic fractional integrals in terms of the off-diagonal (two-weight) parabolic Muckenhoupt class with time lag; as applications, the authors further establish a parabolic weighted Sobolev embedding and a priori estimate for the solution of the heat equation. The key tools to achieve these include the parabolic Calderón--Zygmund-type decomposition, the chaining argument, and the parabolic Welland inequality which is obtained by making the utmost of the geometrical relation between the parabolic shaped domain and the parabolic rectangle.

math.AP↗

arXiv2023

Brezis--Seeger--Van Schaftingen--Yung-Type Characterization of Homogeneous Ball Banach Sobolev Spaces and Its Applications

Let $γ\in\mathbb{R}\setminus\{0\}$ and $X(\mathbb{R}^n)$ be a ball Banach function space satisfying some extra mild assumptions. Assume that $Ω=\mathbb{R}^n$ or $Ω\subset\mathbb{R}^n$ is an $(\varepsilon,\infty)$-domain for some $\varepsilon\in(0,1]$. In this article, the authors prove that a function $f$ belongs to the homogeneous ball Banach Sobolev space $\dot{W}^{1,X}(Ω)$ if and only if $f\in L_{\mathrm{loc}}^1(Ω)$ and $$ \sup_{λ\in(0,\infty)}λ\left\|\left[\int_{\{y\inΩ:\ |f(\cdot)-f(y)|>λ|\cdot-y|^{1+\fracγ{p}}\}} \left|\cdot-y\right|^{γ-n}\,dy \right]^\frac{1}{p}\right\|_{X(Ω)}<\infty, $$ where $p\in[1,\infty)$ is related to $X(\mathbb{R}^n)$. This result is of wide generality and can be applied to various specific Sobolev-type function spaces, including Morrey [Bourgain--Morrey-type, weighted (or mixed-norm or variable) Lebesgue, local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which is new even in all these special cases; in particular, it coincides with the well-known result of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung when $X(Ω):=L^q(\mathbb{R}^n)$ with $1<p=q<\infty$, while it is still new even when $X(Ω):=L^q(\mathbb{R}^n)$ with $1\leq p<q<\infty$.

math.FA↗

arXiv2023

Extension Theorem and Bourgain--Brezis--Mironescu-Type Characterization of Ball Banach Sobolev Spaces on Domains

Let $Ω\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\infty)$-domain with $\varepsilon\in(0,1]$, $X(\mathbb{R}^n)$ a ball Banach function space satisfying some extra mild assumptions, and $\{ρ_ν\}_{ν\in(0,ν_0)}$ with $ν_0\in(0,\infty)$ a $ν_0$-radial decreasing approximation of the identity on $\mathbb{R}^n$. In this article, the authors establish two extension theorems, respectively, on the inhomogeneous ball Banach Sobolev space $W^{m,X}(Ω)$ and the homogeneous ball Banach Sobolev space $\dot{W}^{m,X}(Ω)$ for any $m\in\mathbb{N}$. On the other hand, the authors prove that, for any $f\in\dot{W}^{1,X}(Ω)$, $$ \lim_{ν\to0^+} \left\|\left[\int_Ω\frac{|f(\cdot)-f(y)|^p}{ |\cdot-y|^p}ρ_ν(|\cdot-y|)\,dy \right]^\frac{1}{p}\right\|_{X(Ω)}^p =\frac{2π^{\frac{n-1}{2}}Γ(\frac{p+1}{2})}{Γ(\frac{p+n}{2})} \left\|\,\left|\nabla f\right|\,\right\|_{X(Ω)}^p, $$ where $Γ$ is the Gamma function and $p\in[1,\infty)$ is related to $X(\mathbb{R}^n)$. Using this asymptotics, the authors further establish a characterization of $W^{1,X}(Ω)$ in terms of the above limit. To achieve these, the authors develop a machinery via using a method of the extrapolation, two extension theorems on weighted Sobolev spaces, and some recently found profound properties of $W^{1,X}(\mathbb{R}^n)$ to overcome those difficulties caused by that the norm of $X(\mathbb{R}^n)$ has no explicit expression and that $X(\mathbb{R}^n)$ might be neither the reflection invariance nor the translation invariance. This characterization has a wide range of generality and can be applied to various Sobolev-type spaces, such as Morrey [Bourgain--Morrey-type, weighted (or mixed-norm or variable), local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, all of which are new.

math.FA↗

arXiv2023

Generalized Brezis--Van Schaftingen--Yung Formulae and Their Applications in Ball Banach Sobolev Spaces

Let $X$ be a ball Banach function space on $\mathbb{R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on both $X$ and the associate space of its convexification, the authors successfully recover the homogeneous ball Banach Sobolev semi-norm $\|\,|\nabla f|\,\|_X$ via the functional $$ \sup_{λ\in(0,\infty)}λ\left\|\left[\int_{\{y\in\mathbb{R}^n:\ |f(\cdot)-f(y)|>λ|\cdot-y|^{1+\fracγ{q}}\}} \left|\cdot-y\right|^{γ-n}\,dy\right]^\frac{1}{q}\right\|_X $$ for any distributions $f$ with $|\nabla f|\in X$, as well as the corresponding limiting identities with the limit for $λ\to\infty$ when $γ\in(0,\infty)$ or the limit for $λ\to0^+$ when $γ\in(-\infty,0)$, where $γ\in\mathbb{R}\setminus\{0\}$ and where $q\in(0,\infty)$ is related to $X$. In particular, some of these results are still new even when $X:=L^p(X)$ with $p\in[1,\infty)$. As applications, the authors obtain some fractional Sobolev-type and some fractional Gagliardo--Nirenberg-type inequalities in the setting of $X$. All these results are of quite wide generality and are applied to various specific function spaces, including Morrey, mixed-norm (or variable or weighted) Lebesgue, Lorentz, and Orlicz (or Orlicz-slice) spaces, some of which are new even in all these special cases. The novelty of this article is to use both the method of the extrapolation and the boundedness of the Hardy--Littlewood maximal operator on both $X$ and the associate space of its convexification to overcome the essential difficulties caused by the deficiency of both the translation and the rotation invariance and an explicit expression of the norm of $X$.

math.FA↗

arXiv2022

Parabolic Muckenhoupt Weights on Spaces of Homogeneous Type

This work discusses parabolic Muckenhoupt weights on spaces of homogeneous type, i.e.\ quasi-metric spaces with both a doubling measure and an additional monotone geodesic property. The main results include a characterization in terms of weighted norm inequalities for parabolic maximal operators, a reverse Hölder inequality, and a Jones-type factorization result for this class of weights. The connection between the space of parabolic bounded mean oscillation and parabolic Muckenhoupt weights is studied by applying a parabolic John--Nirenberg lemma. A Coifman--Rochberg-type characterization of the space of parabolic bounded mean oscillation in terms of parabolic maximal functions is also given. The main challenges in the parabolic theory are related to the time lag in the estimates. The results are motivated by the corresponding Euclidean theory and the regularity theory for parabolic variational problems on metric measure spaces.

math.AP↗