5 papers
Stability of the Riesz-type inequalities under gluing
Dangyang He
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…
Sharp Riesz and reverse Riesz inequalities for degenerate elliptic operators of Grushin type
Dangyang He
We determine the sharp ranges of the Riesz transform and the reverse Riesz inequality for a class of power-degenerate elliptic operators of Grushin type. When the degenerate…
Isoperimetric Inequality for degenerate elliptic operators of Grushin type
Dangyang He
Let , , and . 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…
Isoperimetric Inequality on Manifolds with Quadratically Decaying Curvature
Dangyang He
In this paper, we investigate the reverse improvement property of Sobolev inequalities on manifolds with quadratically decaying Ricci curvature. Specifically, we establish conditio…
On the Riesz transform and its reverse inequality on manifolds with quadratically decaying curvature
Dangyang He
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 b…