Abstract:We establish optimal Lebesgue estimates for a class of generalized Radon transforms defined by averaging functions along polynomial-like curves. The presence of an essentially optimal weight allows us to prove uniform estimates, wherein the Lebesgue exponents are completely independent of the curves and the operator norms depend only on the polynomial degree. Moreover, our weighted estimates possess rather strong diffeomorphism invariance properties, allowing us to obtain uniform bounds for averages on curves satisfying natural nilpotency and nonoscillation hypotheses.

中文翻译：

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多项式曲线上加权平均值的端点 Lebesgue 估计

摘要：我们为一类由沿多项式曲线求平均函数定义的广义 Radon 变换建立了最优 Lebesgue 估计。本质上最优权重的存在使我们能够证明统一估计，其中 Lebesgue 指数完全独立于曲线并且算子范数仅取决于多项式次数。此外，我们的加权估计具有相当强的微分同胚不变性特性，使我们能够获得满足自然幂零和非振荡假设的曲线上的平均值的统一界限。