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Oscillatory Hyper-Hilbert 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
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Abstract

In this paper, the bounded properties of oscillatory hyper-Hilbert transform along certain plane curves $\gamma(t)$ $$T_{\alpha,\beta}f(x,y)=\int_{0}^1f(x-t,y-\gamma(t))e^{ i t^{-\beta}}\frac{\textrm{d}t}{t^{1+\alpha}}$$ 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 $\beta\gt 3\alpha$, $\frac{2\beta}{2\beta-3\alpha}\lt p\lt \frac{2\beta}{3\alpha}$.
Keywords: oscillatory hyper-Hilbert transform, oscillatory integral oscillatory hyper-Hilbert transform, oscillatory integral
MSC Classifications: 42B20, 42B35 show english descriptions Singular and oscillatory integrals (Calderon-Zygmund, etc.)
Function spaces arising in harmonic analysis
42B20 - Singular and oscillatory integrals (Calderon-Zygmund, etc.)
42B35 - Function spaces arising in harmonic analysis
 

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