Multiparameter singular integrals and maximal operators along flat surfaces
Rev. Mat. Iberoamericana, Tome 24 (2008) no. 2, p. 1047-1073 / Harvested from Project Euclid
We study double Hilbert transforms and maximal functions along surfaces of the form $(t_1,t_2,\gamma_1(t_1)\gamma_2(t_2))$. The $L^p(\mathbb{R}^3)$ boundedness of the maximal operator is obtained if each $\gamma_i$ is a convex increasing and $\gamma_i(0)=0$. The double Hilbert transform is bounded in $L^p(\mathbb{R}^3)$ if both $\gamma_i$'s above are extended as even functions. If $\gamma_1$ is odd, then we need an additional comparability condition on $\gamma_2$. This result is extended to higher dimensions and the general hyper-surfaces of the form $(t_1,\dots,t_{n},\Gamma(t_1,\dots,t_{n}))$ on $\mathbb{R}^{n+1}$.
Publié le : 2008-04-15
Classification:  singular Radon transform,  multiple Hilbert transform,  flat surface,  42B20,  42B25
@article{1228834304,
     author = {Cho
,  
Yong-Kum and Hong
,  
Sunggeum and Kim
,  
Joonil and Yang
,  
Chan Woo},
     title = {Multiparameter singular integrals and maximal operators along flat surfaces},
     journal = {Rev. Mat. Iberoamericana},
     volume = {24},
     number = {2},
     year = {2008},
     pages = { 1047-1073},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1228834304}
}
Cho
,  
Yong-Kum; Hong
,  
Sunggeum; Kim
,  
Joonil; Yang
,  
Chan Woo. Multiparameter singular integrals and maximal operators along flat surfaces. Rev. Mat. Iberoamericana, Tome 24 (2008) no. 2, pp.  1047-1073. http://gdmltest.u-ga.fr/item/1228834304/