Tianjun Shen | Search arXiv

arXiv2026

Riesz transform on manifolds with mixed ends for $1<q<2$

Let $M_1$, $\cdots$, $M_\ell$ be complete manifolds of the same dimension, where $2\le \ell\in\mathbb{N}$. Suppose that each $M_i$ satisfies two side Gaussian bounds. If some of these manifolds are parabolic and there exists a constant $1\le n_i\le 2$ such that for some $x_i\in M_i$ with $1 \le r \le R < \infty$, $$c_i\left(\frac{R}{r}\right)^{n_i}\le \frac{Vol_{M_i}(B(x_i,R))}{Vol_{M_i}(B(x_i,r))}\le C_i\left(\frac{R}{r}\right)^{n_i},$$ by assuming that all the manifolds satisfy the relative connectedness of the annuli ($RCA$) condition introduced by Grigor'yan and Saloff-Coste, we show that the Riesz transform $\nabla \mathcal L^{-1/2}$ is bounded on $L^q(M)$ for each $1<q<2$ on the gluing manifold $M=M_1 \# M_2 \# \cdots \# M_\ell$.

math.AP↗

arXiv2025

Heat kernel estimates, fractional Riesz transforms and applications on exterior domains

In this paper, we derive sharp two side heat kernel estimate on exterior $C^{1,1}$ domains in the plane, and sharp upper heat kernel bound on exterior $C^{1,\mathrm{Dini}}$ domains in $\mathbb{R}^n$, $n\ge 2$. Estimates for Green's function and Riesz potentials on exterior domains in the plane are also presented. Based on the heat kernel estimates, we show the boundedness of the fractional Riesz transforms on exterior $C^{1,\mathrm{Dini}}$ domains in $\mathbb{R}^n$, $n\ge 2$. Some further applications to product and chain rules and nonlinear Schrödinger equation are also presented.

math.CA↗