论文标题
上半平面中加权谐波函数的独特定理
Uniqueness theorems for weighted harmonic functions in the upper half-plane
论文作者
论文摘要
我们考虑一类称为$α$ harmonic函数的开放上半平面中的加权谐波功能。特别令人感兴趣的是此类功能的唯一性问题，但在实际线路上遇到了Dirichlet边界价值，而无限属性则适当消失的条件。我们发现，与经典案例（$α\ neq0 $）相比，与经典的情况（$α\ neq0 $）相比，在上半平面中通常的谐波功能（$α= 0 $）相比，无穷大的消失条件更加轻松。这种二分法背后的原因是从经典二项式序列得出的某些多项式的零几何形状不同。 我们的发现为谐波函数理论提供了新的启示，为此，我们为此提供了沿着a）地球学的无限条件下消失的条件下的独特结果，而b）从起源中散发出来的射线。大地唯一性结果需要在两个不同的大地测量学上消失，这是最好的。射线唯一性结果涉及一种算术条件，我们通过引入可允许的角度功能的概念来分析。我们表明，算术条件在于，相对于自然部分秩序，角度的可允许函数的集合是最小的。
We consider a class of weighted harmonic functions in the open upper half-plane known as $α$-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case ($α\neq0$) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case ($α=0$) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. Our findings shed new light on the theory of harmonic functions, for which we provide uniqueness results under vanishing conditions at infinity along a) geodesics, and b) rays emanating from the origin. The geodesic uniqueness results require vanishing on two distinct geodesics which is best possible. The ray uniqueness results involves an arithmetic condition which we analyze by introducing the concept of an admissible function of angles. We show that the arithmetic condition is to the point and that the set of admissible functions of angles is minimal with respect to a natural partial order.
