A decay estimate for the Fourier transform of certain singular measures in $\mathbb{R}^{4}$ and applications

TL;DR

This paper establishes decay estimates for the Fourier transform of certain singular measures in four-dimensional space, leading to restriction theorems and $L^p$-improving properties for associated convolution operators.

Contribution

It provides new decay estimates for Fourier transforms of measures supported on graphs defined by functions with nonisotropic homogeneity, extending restriction theory and convolution analysis.

Findings

Derived decay estimates for Fourier transforms of specific singular measures.

Established restriction theorems for Fourier transforms on graph surfaces.

Proved $L^p$-improving properties for convolution operators associated with these measures.

Abstract

We consider, for a class of functions $\varphi : \mathbb{R}^{2} \setminus \{ {\bf 0} \} \to \mathbb{R}^{2}$ satisfying a nonisotropic homogeneity condition, the Fourier transform $\hat{\mu}$ of the Borel measure on $\mathbb{R}^{4}$ defined by \[ \mu(E) = \int_{U} \chi_{E}(x, \varphi(x)) \, dx \] where $E$ is a Borel set of $\mathbb{R}^{4}$ and $U = \{ (t^{\alpha_1}, t^{\alpha_2}s) : c < s < d, \, 0 < t < 1 \}$. The aim of this article is to give a decay estimate for $\hat{\mu}$, for the case where the set of nonelliptic points of $\varphi$ 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 $\varphi_{U} : U \to \mathbb{R}^{2}$. We also give $L^{p}$-improving properties for the convolution operator $T_{\mu} f = \mu \ast f$.