AI 中文总结
受兰迪斯猜想启发，研究柯西-黎曼算子的无穷远唯一延拓性，证明有界位势下方程弱解的指数衰减阈值结论，并推广至径向衰减位势，还得到$L^2$及紧支集位势的相关结果。
AI 中文摘要
受关于拉普拉斯算子无穷远唯一延拓性的兰迪斯猜想启发，我们研究了柯西-黎曼算子的对应性质。我们证明，在无穷远邻域内满足 $\bar\backslash$partial u=Vu 且 $V\backslash$in $L^\backslash$infty 的所有弱解，若以大于 $2\backslash$|$V\backslash$|$_{L^\backslash$infty}$ 的速率指数衰减，则恒为零。该结论在常数 $2\backslash$|$V\backslash$|$_{L^\backslash$infty}$ 和所需指数衰减阶数两方面均是最优的。更一般地，我们建立了一大类径向衰减有界位势的无穷远唯一延拓性，其最优衰减率由位势的衰减性决定。我们还得到了 $L^2$ 位势以及紧支集位势在更弱无穷远假设下的相关唯一延拓结果。
英文摘要
Motivated by Landis's conjecture on unique continuation at infinity for the Laplacian, we study the corresponding property for the Cauchy-Riemann operator. We prove that every weak solution of $\bar\partial u=Vu$ on a neighborhood of infinity, with $V\in L^\infty$, vanishes identically if it decays exponentially at a rate greater than $ 2\|V\|_{L^\infty}$. This conclusion is sharp both in the constant $2\|V\|_{L^\infty}$ and in the order of exponential decay required. More generally, we establish unique continuation at infinity for a broad class of radially decaying bounded potentials, with optimal decay rates determined by the decay of the potential. We also obtain related unique continuation results for $L^2$ potentials and for compactly supported potentials under weaker assumptions at infinity.
Comments27 pages
