Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Nonisotropic dilations and the method of rotations with weight

Author: Shuichi Sato
Journal: Proc. Amer. Math. Soc. 140 (2012), 2791-2801
MSC (2010): Primary 42B20, 42B25
Posted: December 19, 2011
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider maximal functions $Mf(x,\theta )$ , singular integrals
$Hf(x,\theta )$ , and maximal singular integrals $H_*f(x,\theta )$ on $\mathbb{R}^n\times S^{n-1}$ associated with homogeneous curves, for functions on $\mathbb{R}^n$ . We prove certain weighted mixed norm estimates for them. These results are applied to the theory of singular integrals with variable kernels via the method of rotations of Calderón-Zygmund.

References

1. Neal Bez, Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms, J. Funct. Anal. 255 (2008), no. 12, 3281–3302. MR 2469023 (2010a:42061), http://dx.doi.org/10.1016/j.jfa.2008.07.026
2. A.-P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution, Advances in Math. 16 (1975), 1–64. MR 0417687 (54 #5736)
3. A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309. MR 0084633 (18,894a)
4. A.-P. Calderón and A. Zygmund, On singular integrals with variable kernels, Applicable Anal. 7 (1977/78), no. 3, 221–238. MR 0511145 (58 #23379)
5. Lung-Kee Chen, The singular integrals related to the Calderón-Zygmund method of rotations, Appl. Anal. 30 (1988), no. 4, 319–329. MR 967578 (90b:42030), http://dx.doi.org/10.1080/00036818808839810
6. Lung-Kee Chen, The maximal operators related to the Calderón-Zygmund method of rotations, Illinois J. Math. 33 (1989), no. 2, 268–279. MR 987823 (90e:42034)
7. Michael Christ, Estimates for the 𝑘-plane transform, Indiana Univ. Math. J. 33 (1984), no. 6, 891–910. MR 763948 (86k:44004), http://dx.doi.org/10.1512/iumj.1984.33.33048
8. Michael Christ, Javier Duoandikoetxea, and José L. Rubio de Francia, Maximal operators related to the Radon transform and the Calderón-Zygmund method of rotations, Duke Math. J. 53 (1986), no. 1, 189–209. MR 835805 (88d:42032), http://dx.doi.org/10.1215/S0012-7094-86-05313-5
9. Michael Cowling and Giancarlo Mauceri, Inequalities for some maximal functions. I, Trans. Amer. Math. Soc. 287 (1985), no. 2, 431–455. MR 768718 (86a:42023), http://dx.doi.org/10.1090/S0002-9947-1985-0768718-5
10. 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 (87f:42046), http://dx.doi.org/10.1007/BF01388746
11. E. B. Fabes and N. M. Rivière, Singular integrals with mixed homogeneity, Studia Math. 27 (1966), 19–38. MR 0209787 (35 #683)
12. E. B. Fabes and N. M. Rivière, Symbolic calculus of kernels with mixed homogeneity, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 106–127. MR 0236773 (38 #5067)
13. C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 0284802 (44 #2026)
14. R. Fefferman, On an operator arising in the Calderón-Zygmund method of rotations and the Bramble-Hilbert lemma, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 12, Phys. Sci., 3877–3878. MR 706542 (85g:42019), http://dx.doi.org/10.1073/pnas.80.12.3877
15. Philip T. Gressman, 𝐿^{𝑝}-improving properties of X-ray like transforms, Math. Res. Lett. 13 (2006), no. 5-6, 787–803. MR 2280775 (2007i:47056)
16. Philip T. Gressman, Sharp 𝐿^{𝑝}-𝐿^{𝑞} estimates for generalized 𝑘-plane transforms, Adv. Math. 214 (2007), no. 1, 344–365. MR 2348034 (2008m:47047), http://dx.doi.org/10.1016/j.aim.2007.01.018
17. Alexander Nagel, Néstor M. Rivière, and Stephen Wainger, On Hilbert transforms along curves. II, Amer. J. Math. 98 (1976), no. 2, 395–403. MR 0450900 (56 #9191b)
18. N. M. Rivière, Singular integrals and multiplier operators, Ark. Mat. 9 (1971), 243–278. MR 0440268 (55 #13146)
19. José L. Rubio de Francia, Francisco J. Ruiz, and José L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. in Math. 62 (1986), no. 1, 7–48. MR 859252 (88f:42035), http://dx.doi.org/10.1016/0001-8708(86)90086-1
20. Shuichi Sato, Maximal functions associated with curves and the Calderón-Zygmund method of rotations, Trans. Amer. Math. Soc. 293 (1986), no. 2, 799–806. MR 816326 (87f:42051), http://dx.doi.org/10.1090/S0002-9947-1986-0816326-0
21. Shuichi Sato, Estimates for singular integrals along surfaces of revolution, J. Aust. Math. Soc. 86 (2009), no. 3, 413–430. MR 2529333 (2010h:42023), http://dx.doi.org/10.1017/S1446788708000773
22. 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 (80k:42023), http://dx.doi.org/10.1090/S0002-9904-1978-14554-6

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|