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On the maximal Riesz-transforms along surfaces


Author: Lung-Kee Chen
Journal: Proc. Amer. Math. Soc. 103 (1988), 487-496
MSC: Primary 42B25
DOI: https://doi.org/10.1090/S0002-9939-1988-0943072-2
MathSciNet review: 943072
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Abstract: Let $ b(t)$ be an arbitrary bounded radial function. For $ x = ({x_1},{x_2}),t = ({t_1},{t_2})$ in $ {R^2},\left\vert t \right\vert = {({t_1} + {t_2})^{1/2}}$, we establish that the following maximal Riesz-transforms along the surfaces $ ({t_1},{t_2},\vert t{\vert^a}),a > 0$:

$\displaystyle {T^*}f(x) = \mathop {\sup }\limits_{\varepsilon > 0} \left\vert {... ...ert^a})b(t)\left. {\frac{{{t_1}}}{{\vert t{\vert^3}}}dt} \right\vert} } \right.$

are bounded in $ {L^p}({R^3})$ for all $ 1 < p < \infty $. The $ n$-dimensional result can be found at the end of this paper.

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DOI: https://doi.org/10.1090/S0002-9939-1988-0943072-2
Article copyright: © Copyright 1988 American Mathematical Society

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