The Oscillatory Hyper-Hilbert Transform Associated with Plane Curves

Li, Junfeng ; Yu, Haixia

Canadian mathematical bulletin, Tome 61 (2018) no. 4, pp. 802-811

Résumé

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 $\frac{2\beta }{2\beta -3\alpha }\,<\,p\,<\,\frac{2\beta }{3\alpha }$ .

Détail

DOI : 10.4153/CMB-2017-064-9

Mots-clés : 42B20, 42B35, oscillatory hyper-Hilbert transform, oscillatory integral

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