arXiv subjects
Publications and source records attributed to Dangyang He.
Let $\mathcal{L}$ generate a symmetric contraction semigroup. We prove a holomorphic Hörmander calculus on $L^p(X;\ell^q)$ at the optimal angle $\max\{\arcsin|1-2/p|,\arcsin|1-2/q|\}$, with boundary Sobolev order $s>3$.
We determine the sharp $L^p$ ranges of the Riesz transform and the reverse Riesz inequality for a class of power-degenerate elliptic operators of Grushin type. When the degenerate variable has dimension at least two, the Riesz transform is bounded throughout the reflexive range. In dimension one, a capacity-driven transition occurs: weak degeneracy produces a sharp finite upper endpoint, whereas strong degeneracy separates the two half-lines and restores the full range. The reverse inequality displays a different sharp phenomenon. In the weakly degenerate one-dimensional regime it holds throughout the reflexive range except at a single Hardy-critical exponent, while in all remaining regimes it holds for every $1<p<\infty$.
We establish stability principles under compact gluing for the reverse square-root inequality and the Riesz transform. Below a common Sobolev dimension, the reverse estimate passes from finitely many model manifolds to their connected sum. A boundary-compatible Poisson parametrix also proves the Neumann reverse inequality for every $1<p<\infty$ for real symmetric uniformly elliptic divergence-form operators on exterior Lipschitz domains. For the forward transform, we glue a bounded-core model and an independent whole-space end model that agree with the coefficients on overlapping regions. A homogeneous Fredholm construction then resolves the form-valued error. The resulting range is controlled by the two model exponents, with the additional restriction $p<n$ for Dirichlet data. More broadly, our method illuminates the interplay between the bounded interior and exterior problems, and provides a flexible framework for studying other boundary value problems.
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.
Let $n,m\ge 1$, $α\in(0,1)$, and $β\ge 0$. For the Grushin-type operator \[ L=-\nabla_x\!\cdot\!\bigl(|x|^{2α}\nabla_x\bigr)+|x|^{2β}Δ_y \qquad \text{on } \mathbb R^n\times \mathbb R^m, \] we prove the isoperimetric inequality on the associated Grushin space. Equivalently, if \[ Q=\frac{n+m(β+1-α)}{1-α}, \] then \[ |Ω|^{\frac{Q-1}{Q}}\le C\,P(Ω) \] for every smooth bounded domain $Ω\subset \mathbb R^{n+m}$.
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.
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 $H=Δ+V_0/r^2$ be the associated Schrodinger operator, with $V_0\in C^\infty(Y)$ satisfying the positivity condition $Δ_Y+V_0+(d-2)^2/4>0$. 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 \[ \frac{d}{\min\big((d+4)/2+μ_0,\,d\big)} < p < \frac{d}{\max\big((d-2)/2-μ_0,\,0\big)}.\]
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.
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.
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 $dμ= |r|^{d_{1}-1}dr$ for $r \le -1$ and $dμ= r^{d_{2}-1}dr$ 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 Poincaré inequality. Our approach is based on careful estimates of the Riesz kernel and a method known as harmonic annihilation.
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.
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 Pólya.