3 results
The boundedness of the bilinear oscillatory integral along a parabola
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 1 / February 2023
- Published online by Cambridge University Press:
- 03 April 2023, pp. 54-88
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In this paper, the
$L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$ boundedness of the bilinear oscillatory integral along parabolais set up, where β > 1 or β < 0,
\begin{equation*}T_\beta(f, g)(x)=p.v.\int_{{\mathbb R}} f(x-t)g(x-t^{2})\,{\rm e}^{i |t|^{\beta}}\,\frac{{\rm d}t}{t}\end{equation*}
$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and 
$\frac{1}{2}\lt r\lt\infty$, p > 1 and q > 1. The result for the case β < 0 extends the 
$L^\infty\times L^2\to L^2$ boundedness obtained by Fan and Li (D. Fan and X. Li, A bilinear oscillatory integral along parabolas, Positivity 13(2) (2009), 339–366) by confirming an open question raised in it.
The Oscillatory Hyper-Hilbert Transform Associated with Plane Curves
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- Journal:
- Canadian Mathematical Bulletin / Volume 61 / Issue 4 / 01 December 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 802-811
- Print publication:
- 01 December 2018
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In this paper, the bounded properties of oscillatory hyper-Hilbert transformalong certain plane curves
$\gamma \left( t \right)$,
$${{T}_{\alpha ,\beta }}f\left( x,\,y \right)\,=\,\int_{0}^{1}{f\left( x\,-\,t,\,y\,-\,\gamma \left( t \right) \right){{e}^{i{{t}^{-\beta }}}}\frac{\text{d}t}{{{t}^{1}}+\alpha }}$$are studied. For general curves, these operators are bounded in
${{L}^{2}}\left( {{\mathbb{R}}^{2}} \right)$ if 
$\beta \,\ge \,3\alpha $. Their boundedness in ${{L}^{p}}\left( {{\mathbb{R}}^{2}} \right)$ is also obtained, whenever $\beta \,\ge \,3\alpha $ and 
$\frac{2\beta }{2\beta -3\alpha }\,<\,p\,<\,\frac{2\beta }{3\alpha }$.
BOUNDS FOR A CONE-TYPE MULTIPLIER OPERATOR OF NEGATIVE INDEX IN
${ \mathbb{R} }^{3} $
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- Journal:
- Journal of the Australian Mathematical Society / Volume 94 / Issue 3 / June 2013
- Published online by Cambridge University Press:
- 10 April 2013, pp. 321-347
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- June 2013
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In this paper we obtain some sharp
${L}^{p} - {L}^{q} $ estimates and the restricted weak-type endpoint estimates for the multiplier operator of negative order associated with conic surfaces in 
${ \mathbb{R} }^{3} $ which have finite type degeneracy.
