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We prove that $\\mathcal T_\\alpha$ is injective on $C(\\mathbb D)\\cap L^\\infty(\\mathbb D)$ for $0<\\alpha<1$, and that $\\mathcal T_1$ is injective on $L^1(\\mathbb D)$. In contrast, for each $0<\\alpha<1$ there is an injective linear map from $C_c^\\infty((0,\\alpha))$ into the kernel of $\\mathcal T_\\alpha$ on $C^\\infty(\\mathbb D)$; every nonzero function in its image is necessarily unbounded near $\\partial\\mathbb D$. 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We prove that $\\mathcal T_\\alpha$ is injective on $C(\\mathbb D)\\cap L^\\infty(\\mathbb D)$ for $0<\\alpha<1$, and that $\\mathcal T_1$ is injective on $L^1(\\mathbb D)$. In contrast, for each $0<\\alpha<1$ there is an injective linear map from $C_c^\\infty((0,\\alpha))$ into the kernel of $\\mathcal T_\\alpha$ on $C^\\infty(\\mathbb D)$; every nonzero function in its image is necessarily unbounded near $\\partial\\mathbb D$. 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