Variable-Radius Disk Transforms and an Area-Integral Problem of Zalcman

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Original abstract

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.

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• Official / arXiv page
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