AI 中文总结
该数学研究针对变半径圆盘变换𝒯_α，证明其在不同函数空间上的单射性，回答了Hayman-Lingham提出的扎尔克曼面积积分问题，采用广义阿贝尔方程等方法完成证明。
AI 中文摘要
对于0<α≤1，定义(𝒯_α f)(z):=∫_{B(z,α(1-|z|))}f(ζ)dA(ζ)，其中z∈𝔻，dA为平面勒贝格测度。我们证明：当0<α<1时，𝒯_α在C(𝔻)∩L^∞(𝔻)上是单射；𝒯_1在L^1(𝔻)上是单射。相反，对每个0<α<1，存在从C_c^∞((0,α))到𝒯_α在C^∞(𝔻)上核的单射线性映射，其像中每个非零函数在∂𝔻附近必无界。基于面积测度解释，这些结果完整回答了Hayman-Lingham问题7.29（该问题归属于L. Zalcman），证明结合了广义阿贝尔方程、欧拉-泊松-达布能量论证与沃尔泰拉延拓。
英文摘要
For $0<α\leq1$, define $(\mathcal T_αf)(z) :=\int_{B(z,α(1-|z|))}f(ζ)\,dA(ζ)$ for $z\in\mathbb D$, where $dA$ is planar Lebesgue measure. We prove that $\mathcal T_α$ is injective on $C(\mathbb D)\cap L^\infty(\mathbb D)$ for $0<α<1$, and that $\mathcal T_1$ is injective on $L^1(\mathbb D)$. In contrast, for each $0<α<1$ there is an injective linear map from $C_c^\infty((0,α))$ into the kernel of $\mathcal T_α$ on $C^\infty(\mathbb D)$; every nonzero function in its image is necessarily unbounded near $\partial\mathbb D$. Under the area-measure interpretation, these results give a complete answer to Hayman--Lingham Problem~7.29, attributed there to L.~Zalcman. The proof combines generalized Abel equations, an Euler--Poisson--Darboux energy argument, and Volterra continuation.
Comments12 pages
