论文标题
Li-Yau和Harnack的不平等现象，通过离散的远程跳跃操作员（包括分数离散Laplacian）的曲率维度条件
Li-Yau and Harnack inequalities via curvature-dimension conditions for discrete long-range jump operators including the fractional discrete Laplacian
论文作者
论文摘要
我们考虑$ l u（x）= \ sum_ {y \ in \ mathbb {z}} k（x-y）\ big（u（y） - u（x） - u（x）\ big）$的运算符，带有对称的晶格，带有对称的晶格，具有集成的kernel $ k $。我们证明了几个结果表明，在内核上，操作员$ l $满足弯曲量的条件$cd_ñ（0，f）$（最近由两位作者最近引入），其中有一些$ cd $ function $ f $，在此期间，人们对$ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f。我们表明，$cd_υ（0，f）$意味着与操作员$ l $相关的热方程式的积极解决方案的li-yau不平等。 Li-Yau估计反过来导致了harnack不平等，我们也从中得出了热核的边界。我们的结果适用于包括分数离散Laplacian在内的广泛的运营商。
We consider operators of the form $L u(x) = \sum_{y \in \mathbb{Z}} k(x-y) \big( u(y) - u(x)\big)$ on the one-dimensional lattice with symmetric, integrable kernel $k$. We prove several results stating that under certain conditions on the kernel the operator $L$ satisfies the curvature-dimension condition $CD_Υ(0,F)$ (recently introduced by two of the authors) with some $CD$-function $F$, where attention is also paid to the asymptotic properties of $F$ (exponential growth at infinity and power-type behaviour near zero). We show that $CD_Υ(0,F)$ implies a Li-Yau inequality for positive solutions of the heat equation associated with the operator $L$. The Li-Yau estimate in turn leads to a Harnack inequality, from which we also derive heat kernel bounds. Our results apply to a wide class of operators including the fractional discrete Laplacian.
