arXiv subjects

Yirui Zhao

Publications and source records attributed to Yirui Zhao.

5 recordsLinked to original sources

arXiv2026

Weak-type characterizations of Sobolev and bounded variation spaces on metric measure spaces

Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincar\'e spaces with the Euclidean results of Brezis et al. [Anal. PDE 17 (2024), 943-979]. The main novelty is that the finiteness of a weak-type norm, which only refers to differences or mean oscillations of $f$ without assuming any smoothness a priori, already guarantees the membership of $f$ in the relevant Sobolev or BV space. This distinguishes our contribution from the recent work of F. Dai et al. [Adv. Math. 502 (2026), Paper No. 111153], where the related norm-equivalence was obtained under the a priori Lipschitz assumption on $f$. A key intermediate step in our approach is a new localized Bourgain-Brezis-Mironescu type characterization. More precisely, we prove that, if $p\in(1,\infty)$ and $\gamma\in\mathbb R\setminus\{0\}$, then, for any $f\in L^1_{\mathrm{loc}}(\mu)$, \begin{equation*}\tag{$*$} \|f\|_{\dot W^{1,p}(\mu)} \sim \|\rho^{-1}\phi^{-\gamma}F\|_{L^{p,\infty}(\phi^{\gamma p}V^{-1})}, \qquad F\in\{\Delta f,m_f\},\quad \phi\in\{\rho,V\}, \end{equation*} where the homogeneous Sobolev space $\dot{W}^{1,p}(\mu)$ is defined by the minimal $p$-weak upper gradient and, for any $x,y\in X$, we denote $V(x,y):=\mu(B(x,\rho(x,y)))$ and $\Delta f(x,y):=|f(x) - f(y)|$, and $m_f(x,y)$ is the mean oscillation of $f$ on the ball $B(x,\rho(x,y))$. For $p=1$, the equivalence $(*)$ holds after replacing $\|f\|_{\dot W^{1,1}(\mu)}$ by a bounded variation norm and restricting the parameters to the optimal ranges $\gamma\in(-\infty,-1)\cup(0,\infty)$ for $\phi=\rho$ or $\gamma\in (-\infty,-\frac1d)\cup(0,\infty)$ for $\phi=V$, where $d\in(0,\infty)$ is the lower dimension of $X$.

math.FA↗

arXiv2026

Optimal Weak-Type Estimates and Their Applications of Lifted Rough Maximal Operators

Let $n\in\mathbb N\cap[2,\infty)$ and $\Omega\in L^1(\mathbb S^{n-1})$ with $\Omega\not\equiv 0$. In this article, we introduce a new family of lifted rough maximal operators $\{\mathcal{M}_\theta^\Omega\}_{\theta\in(0,\infty)}$ in the upper-half plane and establish their optimal weak-type estimates. Specifically, we prove that, for any $p \in (1, \infty)$, the estimate, with the positive equivalence constants independent of $f$, \[ \sup_{\theta,\lambda\in(0,\infty)}\lambda^p \underset{{\mathcal M}^\Omega_\theta(f)(x,t) > \lambda t^\frac{\gamma}{p}} {\int_{\mathbb R^n}\int_0^\infty} t^{\gamma-1}\,dt\,dx \sim \|f\|_{L^p(\mathbb{R}^n)}^p \] holds for all $f\in L^p(\mathbb R^n)$ if and only if $\gamma\in\mathbb R\setminus\{0\}$. For the endpoint case $p=1$ and $\Omega \in L(\log L)(\mathbb{S}^{n-1})$, we prove that the above estimate holds if and only if $\gamma \in (-\infty, -n) \cup (0, \infty)$. As applications, we obtain weak-type estimates for generalized Poisson integrals without any logarithmic integrability assumptions, which gives an affirmative answer to the question posed by Sj\"ogren and Soria in page 228 of [Israel J. Math. 95 (1996)]. Moreover, although the operator $M^\ast_\Omega$, arising from the method of rotation of Calder\'on and Zygmund, is not of weak type $(1,1)$, we find that its lifted variant is weak type $(1,1)$. In addition, we establish a new characterization of Hardy spaces in terms of truncated rough singular integrals.

math.CA↗

arXiv2025

Gradient continuity estimates for elliptic equations of singular $p$-Laplace type with measure data

In this paper, we are concerned with elliptic equations of $p$-Laplace type with measure data, which is given by $-div\big(a(x)(|\nabla u|^2+s^2)^{\frac{p-2}{2}}\nabla u\big)=\mu$ with $p>1$ and $s\geq0$. Under the assumption that the modulus of continuity of the coefficient $a(x)$ in the $L^2$-mean sense satisfies the Dini condition, we prove a new comparison estimate and use it to derive interior and global gradient pointwise estimates by Wolff potential for $p\geq 2$ and Riesz potential for $1<p<2$, respectively. Our interior gradient pointwise estimates can be applied to a class of singular quasilinear elliptic equations with measure data given by $-div(A(x,\nabla u))=\mu$. We generalize the results in the papers of Duzaar and Mingione [Amer. J. Math. 133, 1093-1149 (2011)], Dong and Zhu [J. Eur. Math. Soc. 26, 3939-3985 (2024)], and Nguyen and Phuc [Arch. Rational Mech. Anal. (2023) 247:49], etc., where the coefficient is assumed to be Dini continuous. Moreover, we establish interior and global modulus of continuity estimates of the gradients of solutions.

math.AP↗

arXiv2024

Sharp Weighted Cohen--Dahmen--Daubechies--DeVore Inequality with Applications to (Weighted) Critical Sobolev Spaces, Gagliardo--Nirenberg Inequalities, and Muckenhoupt Weights

In this article, we establish a quantitative weighted variant of a far-reaching inequality obtained by A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore in 2003, whose dependence on the $A_p$-weight constant for any $p\in[1,\infty)$ is sharp. As applications, we obtain the almost characterization of the critical weighted Sobolev space in terms of wavelets, a sharp real interpolation between this weighted Sobolev space and weighted Besov spaces, and three new Gagliardo--Nirenberg type inequalities in the framework of ball Banach function spaces. Moreover, we apply this sharp weighted inequality to extend the famous Brezis--Seeger--Van Schaftingen--Yung formula in ball Banach function spaces, which gives an affirmative answer to the question in page 29 of [Calc. Var. Partial Differential Equations 62 (2023), Paper No. 234]. Notably, we further establish two new characterizations of Muckenhoupt weights related to the inequality of Cohen et al.\ and the formula of Brezis et al. The most novelty of this article exists in applying and further developing the good cube method introduced by Cohen et al.\ to trace the sharp dependences on weight constants.

math.CA↗

arXiv2022

Mixed-Norm Herz Spaces and Their Applications in Related Hardy Spaces

In this article, the authors introduce a class of mixed-norm Herz spaces, $\dot{E}^{\vecα,\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$, which is a natural generalization of mixed Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz-Thorin interpolation theorem on $\dot{E}^{\vecα,\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$. Applying these Riesz-Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the boundedness of the Hardy-Littlewood maximal operator and the Fefferman-Stein vector-valued maximal inequality on $\dot{E}^{\vecα,\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$. As applications, the authors develop various real-variable theory of Hardy spaces associated with $\dot{E}^{\vecα,\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$ by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of $\dot{E}^{\vecα,\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$ and the non-trivial constructions of auxiliary functions in the Riesz-Thorin interpolation theorem.

math.CA↗