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

TL;DR

The paper develops sharp heat-kernel bounds for exterior domains, obtaining two-sided planar bounds and upper bounds in $C^{1,\mathrm{Dini}}$ settings, and uses these to derive Green’s function and Riesz-potential estimates. Leveraging harmonic weights and Doob transforms, the authors establish boundedness of fractional Riesz transforms on exterior $C^{1,\mathrm{Dini}}$ domains and derive Hardy inequalities, kernel-difference bounds, and main fractional Sobolev estimates. These tools allow explicit product/chain rules and enable analysis of nonlinear Schrödinger equations on exterior domains, including local well-posedness results. The work extends prior results from convex exterior obstacles to general exterior $C^{1,\mathrm{Dini}}$ domains and provides a cohesive framework for potential theory and dispersive PDEs in exterior geometries.

Abstract

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.

Figures (1)

• Figure 1: Example of the cube in the two-dimensional case

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Remark 1.6
• Corollary 1.7
• Corollary 1.8
• Theorem 1.9
• Proposition 2.1