On the maximal Riesz-transforms along surfaces
HTML articles powered by AMS MathViewer
- by Lung-Kee Chen PDF
- Proc. Amer. Math. Soc. 103 (1988), 487-496 Request permission
Abstract:
Let $b(t)$ be an arbitrary bounded radial function. For $x = ({x_1},{x_2}),t = ({t_1},{t_2})$ in ${R^2},\left | t \right | = {({t_1} + {t_2})^{1/2}}$, we establish that the following maximal Riesz-transforms along the surfaces $({t_1},{t_2},|t{|^a}),a > 0$: \[ {T^*}f(x) = \sup \limits _{\varepsilon > 0} \left | {\int _{|t| > \varepsilon } {f({x_1} - {t_1},{x_2} - {t_2},{x_3} - |t{|^a})b(t)\left . {\frac {{{t_1}}}{{|t{|^3}}}dt} \right |} } \right .\] 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.References
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
- A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309. MR 84633, DOI 10.2307/2372517
- Lung-Kee Chen, On a singular integral, Studia Math. 85 (1986), no. 1, 61–72 (1987). MR 879417, DOI 10.4064/sm-85-1-61-72
- Javier Duoandikoetxea and José L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), no. 3, 541–561. MR 837527, DOI 10.1007/BF01388746
- C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 284802, DOI 10.2307/2373450
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Elias M. Stein and Stephen Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295. MR 508453, DOI 10.1090/S0002-9904-1978-14554-6
Additional Information
- © Copyright 1988 American Mathematical Society
- 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