Let $\mu$ be an $\alpha$-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform $\widehat{\mu}$. More precisely, if $\mathbb{H}$ is a truncated hyperbolic paraboloid in $\mathbb{R}^d$ we study the optimal $\beta$ for which $$\int_{\mathbb{H}} |\hat{\mu}(R\xi)|^2 \, d \sigma (\xi)\leq C(\alpha, \mu) R^{-\beta}$$ for all $R > 1$. Our estimates for $\beta$ depend on the minimum between the number of positive and negative principal curvatures of $\mathbb{H}$; if this number is as large as possible our estimates are sharp in all dimensions.

中文翻译：

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双曲面分形测度的傅里叶衰减

令 $\mu$ 是 $\alpha$ 维概率测度。我们证明了傅立叶变换 $\widehat{\mu}$ 的双曲线平均值衰减率的新上限和下限。更准确地说，如果 $\mathbb{H}$ 是 $\mathbb{R}^d$ 中的截断双曲抛物面，我们研究最优 $\beta$，其中 $$\int_{\mathbb{H}} |\hat {\mu}(R\xi)|^2 \, d \sigma (\xi)\leq C(\alpha, \mu) R^{-\beta}$$ 对于所有 $R > 1$。我们对 $\beta$ 的估计取决于 $\mathbb{H}$ 的正负主曲率数量之间的最小值；如果这个数字尽可能大，我们的估计在所有方面都是尖锐的。