Abstract view
Oscillatory HyperHilbert Transform Associated with Plane Curves


Junfeng Li,
Laboratory of Math and Complex systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China
Haixia Yu,
Laboratory of Math and Complex systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China
Abstract
In this paper, the bounded properties of oscillatory hyperHilbert
transform along certain plane curves $\gamma(t)$
$$T_{\alpha,\beta}f(x,y)=\int_{0}^1f(xt,y\gamma(t))e^{ i t^{\beta}}\frac{\textrm{d}t}{t^{1+\alpha}}$$
were studied. For a general curves, these operators are bounded
in ${L^2(\mathbb{R}^{2})}$, if $\beta\geq 3\alpha$. And their
boundedness in $L^p(\mathbb{R}^{2})$
were also obtained, whenever $\beta\gt 3\alpha$, $\frac{2\beta}{2\beta3\alpha}\lt p\lt \frac{2\beta}{3\alpha}$.