论文标题
在具有绿色潜力的DunklSchrödinger半群上
On Dunkl Schrödinger semigroups with Green bounded potentials
论文作者
论文摘要
在$ \ mathbb r^n $上配备了标准化的根系$ r $，多重功能$ k（α）> 0 $和关联的度量 $ dw（\ Mathbf x）= \ prod_ {α\ in r} | \ langle \ mathbf x，α\ rangle |^{k（α）} \，d \ mathbf x，$$，我们考虑一个dunklschrödingeroperator $ lap $ lap = -u = -ulace unk unk $ lap = -ulace unk unk $ unk $ unk $ ung unk $Δ_ $ v \ in l^1 _ {\ rm loc}（dw）$是无负的潜力。令$ h_t（\ mathbf x，\ mathbf y）$和$ k^{\ {v \}} _ t（\ mathbf x，\ mathbf y）$表示dunkl热内核和分别由$ -l $生成的半元组的整体内核。我们证明$ k^{\ {v \}} _ t（\ mathbf x，\ mathbf y）$满足以下热量内核的下限：有常数$ c，c> 0 $ $ c> 0 $ h_ {ct} x，\ mathbf y）$$，仅当$ \ sup _ {\ mathbf x \ in \ mathbb r^n} \ int_0^\ infty \ infty \ infty \ int _ {\ mathbb r^n} v（\ mathbf y） x，\ sqrt {t}）））^{ - 1} e^{ - \ | \ | \ | \ | \ Mathbf x- \ Mathbf y \ |^2/t} \，dw（\ mathbf y）\，dt <\ infty，dt <\ infty，$ b（$ b） $ \ mathbf x \ in \ Mathbb {r}^n $和半径$ \ sqrt {t} $。
On $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k(α) > 0$, and the associated measure $$ dw(\mathbf x)=\prod_{α\in R}|\langle \mathbf x,α\rangle|^{k(α)}\, d\mathbf x, $$ we consider a Dunkl Schrödinger operator $L=-Δ_k+V$, where $Δ_k$ is the Dunkl Laplace operator and $V\in L^1_{\rm loc} (dw)$ is a non-negative potential. Let $h_t(\mathbf x,\mathbf y)$ and $k^{\{V\}}_t(\mathbf x,\mathbf y)$ denote the Dunkl heat kernel and the integral kernel of the semigroup generated by $-L$ respectively. We prove that $k^{\{V\}}_t(\mathbf x,\mathbf y)$ satisfies the following heat kernel lower bounds: there are constants $C, c>0$ such that $$ h_{ct}(\mathbf x,\mathbf y)\leq C k^{\{V\}}_t(\mathbf x,\mathbf y)$$ if and only if $$ \sup_{\mathbf x\in\mathbb R^N} \int_0^\infty \int_{\mathbb R^N} V(\mathbf y)w(B(\mathbf x,\sqrt{t}))^{-1}e^{-\|\mathbf x-\mathbf y\|^2/t}\, dw(\mathbf y)\, dt<\infty, $$ where $B(\mathbf x,\sqrt{t})$ stands for the Euclidean ball centered at $\mathbf x \in \mathbb{R}^N$ and radius $\sqrt{t}$.
