arXiv subjects

Dangyang He

Publications and source records attributed to Dangyang He.

10 recordsLinked to original sources

arXiv2026

Herz versus Fefferman: Symmetric and asymmetric Bochner--Riesz theory

We study Bochner--Riesz summability for a one-dimensional model of noncompact manifolds with ends. Each end has an effective Euclidean dimension, and these dimensions may differ from one end to another. The central point is the contrast between the symmetric and asymmetric cases. When the end dimensions agree, the model follows Herz's radial theory for the Euclidean Laplacian. When they are unequal, a Fefferman-type obstruction appears, analogous to the ball multiplier obstruction in higher-dimensional Fourier analysis, even though the model itself is one-dimensional. We give a complete characterisation of the $L^p$-boundedness of the corresponding spectral projections and Bochner--Riesz means. In the asymmetric case, the boundedness range contains an additional restriction depending on the difference between the end dimensions. This restriction is absent from Herz's radial model and shows that Bochner--Riesz summability on spaces with unequal ends is governed not only by the maximal end dimension, but also by the interaction between the ends.

math.AP↗

arXiv2026

Riesz transform and its related inequalities for degenerate elliptic operators of Grushin type

We study the $L^p$ boundedness of the Riesz transform and the reverse Riesz inequality for degenerate elliptic operators of Grushin type. We prove full-range $L^p$ boundedness of the Riesz transform when the degenerate variable has dimension at least two, and obtain the sharp range in the one-dimensional weakly degenerate case, including the endpoint obstruction. In the strongly degenerate one-dimensional regime, we recover full-range boundedness, revealing a striking transition in the behavior of the singular set. The proof develops a reverse H\"older theory for Grushin harmonic functions near the singular set. The main ingredients are explicit Poisson and Green kernel constructions adapted to the Friedrichs extension and a harmonic annihilation method which isolates the critical part of the Riesz kernel. These techniques illuminate the mechanism behind both the boundedness and unboundedness phenomena, and yield essentially sharp reverse Riesz inequalities for the same class of operators.

math.AP↗

arXiv2026

Isoperimetric Inequality for degenerate elliptic operators of Grushin type

Let $n,m\ge 1$, $\alpha\in(0,1)$, and $\beta\ge 0$. For the Grushin-type operator \[ L=-\nabla_x\!\cdot\!\bigl(|x|^{2\alpha}\nabla_x\bigr)+|x|^{2\beta}\Delta_y \qquad \text{on } \mathbb R^n\times \mathbb R^m, \] we prove the isoperimetric inequality on the associated Grushin space. Equivalently, if \[ Q=\frac{n+m(\beta+1-\alpha)}{1-\alpha}, \] then \[ |\Omega|^{\frac{Q-1}{Q}}\le C\,P(\Omega) \] for every smooth bounded domain $\Omega\subset \mathbb R^{n+m}$.

math.CA↗

arXiv2025

On the Reverse Inequality of Riesz transform on metric cone with potential

Let $M=(0,\infty)_r\times Y$ be a $d$-dimensional ($d\ge 3$) metric cone with metric $g=dr^2+r^2h$, where $(Y,h)$ is a closed Riemannian manifold. Let $H=\Delta+V_0/r^2$ be the associated Schrodinger operator, with $V_0\in C^\infty(Y)$ satisfying the positivity condition $\Delta_Y+V_0+(d-2)^2/4>0$. First, we complement previous results by proving Lorentz-type endpoint estimates for the Riesz transform $\nabla H^{-1/2}$: it is of restricted weak type at both endpoints of its $L^p$-boundedness range. Second, we establish the sharp reverse inequality $\|H^{1/2}f\|_{L^p}\le C\big(\|\nabla f\|_{L^p}+\|f/r\|_{L^p}\big)$ which holds if and only if \[ \frac{d}{\min\big((d+4)/2+\mu_0,\,d\big)} < p < \frac{d}{\max\big((d-2)/2-\mu_0,\,0\big)}.\]

math.AP↗

arXiv2025

On the Interplay Between Hodge Projections and Bounded Harmonic Functions on Manifolds with Ends

We investigate the $L^p$-boundedness of the Hodge projection in the setting of manifolds with ends. We examine its relationship to the Riesz transform and the space of bounded harmonic functions. In particular, we explore how the $L^p$-boundedness of the Hodge projection is connected to the structure of $L^2$ harmonic one-forms and, subsequently, to the space of bounded harmonic functions.

math.AP↗

arXiv2025

Isoperimetric Inequality on Manifolds with Quadratically Decaying Curvature

In this paper, we investigate the reverse improvement property of Sobolev inequalities on manifolds with quadratically decaying Ricci curvature. Specifically, we establish conditions under which the uniform decay of the heat kernel implies the validity of an isoperimetric inequality. As an application, we demonstrate the existence of isoperimetric sets in generalized Grushin spaces. Our approach is built on a weak-type Sobolev inequality, gradient estimates on remote balls, and a Hardy-type gluing technique. This method provides new insights into the deep connections between geometric and functional analysis.

math.FA↗

arXiv2025

On the Riesz transform and its reverse inequality on manifolds with quadratically decaying curvature

We study Riesz and reverse Riesz inequalities on manifolds whose Ricci curvature decays quadratically. First, we refine existing results on the boundedness of the Riesz transform by establishing a Lorentz-type endpoint estimate. Next, we explore the relationship between the Riesz and reverse Riesz transforms, proving that the reverse Riesz, Hardy, and weighted Sobolev inequalities are essentially equivalent. Finally, we apply our methods to Grushin spaces, which exhibit a quadratic decay in 'Ricci curvature', verifying that the reverse inequality holds for all $p\in (1,\infty)$ and that the Riesz transform is bounded on $L^p$ for $p\in (1,n)$. Our approach relies on an asymptotic formula for the Riesz potential combined with an extension of the so-called harmonic annihilation method.

math.AP↗

arXiv2025

Some Remarks on the Riesz and reverse Riesz transforms on Broken Line

In this note, we study both the Riesz and reverse Riesz transforms on broken line. This model can be described by $(-\infty, -1] \cup [1,\infty)$ equipped with the measure $d\mu = |r|^{d_{1}-1}dr$ for $r \le -1$ and $d\mu = r^{d_{2}-1}dr$ for $r\ge 1$, where $d_{1}, d_{2} >1$. For the Riesz transform, we show that the range of its $L^{p}$ boundedness depends solely on the smaller dimension, $d_{1} \wedge d_{2}$. Furthermore, we establish a Lorentz type estimate at the endpoint. In our subsequent investigation, we consider the reverse Riesz inequality by rigorously verifying the $L^{p}$ lower bounds for the Riesz transform for almost every $p\in (1,\infty)$. Notably, unlike most previous studies, we do not assume the doubling condition or the Poincar\'e inequality. Our approach is based on careful estimates of the Riesz kernel and a method known as harmonic annihilation.

math.CA↗

arXiv2024

Reverse Riesz Inequality on Manifolds with Ends

In our investigation, we focus on the reverse Riesz transform within the framework of manifolds with ends. Such manifolds can be described as the connected sum of finite number of Cartesian products $\mathbb{R}^{n_i} \times \mathcal{M}_i$, where $\mathcal{M}_i$ are compact manifolds. We rigorously establish the boundedness of this transform across all $L^p$ spaces for $1<p<\infty$. Notably, existing knowledge indicates that the Riesz transform in such a context demonstrates boundedness solely within a specific range of $L^p$ spaces, typically observed for $1<p<n_*$, where $n_*$ signifies the smallest dimension of the manifold's ends on a large scale. This observation serves as a significant counterexample to the presumed equivalence between the Riesz and reverse Riesz transforms. Our study illuminates the nuanced behaviour of these transforms within the setting of manifolds with ends, providing valuable insights into their distinct properties. Although the lack of equivalence has been previously noted in relevant literature, our investigation contributes to a deeper understanding of the intricate interplay between the Riesz and reverse Riesz transforms.

math.AP↗

arXiv2023

Endpoint Estimates For Riesz Transform And Hardy-Hilbert Type Inequalities

We consider a class of non-doubling manifolds $\mathcal{M}$ defined by taking connected sum of finite Riemannian manifolds with dimension N which has the form $\mathbb{R}^{n_i}\times \mathcal{M}_i$ and the Euclidean dimension $n_i$ are not necessarily all the same. In arXiv:1805.00132v3 [math.AP], Hassell and Sikora proved that the Riesz transform on $\mathcal{M}$ is weak type $(1,1)$, bounded on $L^{p}(\mathcal{M})$ for all $1<p<n^*$ where $n^* = \min_k n_k$ and is unbounded for $p \ge n^*$. In this note we show that the Riesz transform is bounded from Lorentz space $L^{n^* ,1}(\mathcal{M})$ to $L^{n^*,1}(\mathcal{M})$. This complete the picture by obtaining the end point results for $p=n^*$. Our approach is based on parametrix construction described in arXiv:1805.00132v3 [math.AP] and a generalisation of Hardy-Hilbert type inequalities first studied by Hardy, Littlewood and P\'olya.

math.AP↗