论文标题
倍曲底分形测量的傅立叶衰变
Fourier decay of fractal measures on hyperboloids
论文作者
论文摘要
令$μ$为$α$维度的概率度量。我们证明了傅立叶变换$ \wideHatμ$的双曲平均值的衰减率的新上限和下限。更准确地说，如果$ \ mathbb {h} $是$ \ mathbb {r}^d $中的截断双曲线抛物面r^{ - β} $$对于所有$ r> 1 $。我们对$β$的估计取决于$ \ mathbb {h} $的正主曲率和负主曲率的最小值；如果此数字尽可能大，我们的估计值在所有维度上都很清晰。
Let $μ$ be an $α$-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform $\widehatμ$. More precisely, if $\mathbb{H}$ is a truncated hyperbolic paraboloid in $\mathbb{R}^d$ we study the optimal $β$ for which $$\int_{\mathbb{H}} |\hatμ(Rξ)|^2 \, d σ(ξ)\leq C(α, μ) R^{-β}$$ for all $R > 1$. Our estimates for $β$ 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.
