Global Heat Kernel Estimates for $\Delta +{\Delta }^{\alpha /2}$ in Half-space-like domains

Abstract

Suppose that $d\ge 1$ and $\alpha \in (0,2)$. In this paper, by using probabilistic methods, we establish sharp two-sided pointwise estimates for the Dirichlet heat kernels of $\{\Delta +a^{\alpha }{\Delta }^{\alpha /2};a\in (0,1]\}$ on half-space-like $C^{1,1}$ domains in ${\mathbb{R}}^d$ for all time $t>0$. The large time estimates for half-space-like domains are very different from those for bounded domains. Our estimates are uniform in $a\in (0,1]$ in the sense that the constants in the estimates are independent of $a\in (0,1]$. Thus it yields the Dirichlet heat kernel estimates for Brownian motion in half-space-like domains by taking $a\to 0$. Integrating the heat kernel estimates in time $t$, we obtain uniform sharp two-sided estimates for the Green functions of $\{\Delta +a^{\alpha }{\Delta }^{\alpha /2};a\in (0,1]\}$ in half-space-like $C^{1,1}$ domains in ${\mathbb{R}}^d$.

References in corpus (2)

• Boundary Harnack inequalities for regional fractional Laplacian
• Minimal thinness for subordinate Brownian motion in half space

Cited by in corpus (1)

• Minimal thinness for subordinate Brownian motion in half space