论文标题
$ \ mathbb {r}^{4} $和应用程序中某些单数度量的傅立叶变换的衰减估计值
A decay estimate for the Fourier transform of certain singular measures in $\mathbb{R}^{4}$ and applications
论文作者
论文摘要
我们考虑，对于一类函数$φ：\ mathbb {r}^{2} \ setMinus \ {{{\ bf 0} \} \ to \ to \ mathbb {r}^{2}^{2} $ $ \ mathbb {r}^{4} $由\ [μ（e）= \ int_ {u}χ_{ t^{α_2} s）：c <s <d，\，0 <t <1 \} $。本文的目的是给出$ \hatμ$的衰减估计，对于$φ$的非椭圆点点是$ \ bar {u} \ setMinus \ {{\ bf 0} \} $的曲线。从此估计中，我们获得了通常的傅立叶变换为$φ_{u}的图表：u \ to \ mathbb {r}^{2} $的限制定理。我们还提供$ l^{p} $ - 改善卷积操作员的属性$t_μf =μ\ ast f $。
We consider, for a class of functions $φ: \mathbb{R}^{2} \setminus \{ {\bf 0} \} \to \mathbb{R}^{2}$ satisfying a nonisotropic homogeneity condition, the Fourier transform $\hatμ$ of the Borel measure on $\mathbb{R}^{4}$ defined by \[ μ(E) = \int_{U} χ_{E}(x, φ(x)) \, dx \] where $E$ is a Borel set of $\mathbb{R}^{4}$ and $U = \{ (t^{α_1}, t^{α_2}s) : c < s < d, \, 0 < t < 1 \}$. The aim of this article is to give a decay estimate for $\hatμ$, for the case where the set of nonelliptic points of $φ$ is a curve in $\bar{U} \setminus \{ {\bf 0} \}$. From this estimate we obtain a restriction theorem for the usual Fourier transform to the graph of $φ_{U} : U \to \mathbb{R}^{2}$. We also give $L^{p}$-improving properties for the convolution operator $T_μ f = μ\ast f$.
