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Discrete Radon transforms and applications to ergodic theory. (English) Zbl 1139.42002

The paper deals with \(L_p\) estimates for discrete operators in certain non-translation invariant settings and their applications to ergodic theorems for certain families of non-commuting operators. After a short description of the type of operators analyzed, the paper contains a systematic study of analogues of the operators in discrete settings which are not translation invariant. A special class of non-translation invariant operators, called “quasi-translation” invariant Radon transforms is considered. Three theorems are given and proved. In Section 2 the first two theorems are reduced to lemmas defined on a group called \(G_0^\#\) on polynomial mappings proved later in the paper. In Section 3 four lemmas on oscillatory integrals on \(L_2\) are proved. Section 4 is on maximal Radon transforms, its main result being two lemmas representing a qualitative \(L_2\) estimate and a restricted \(L^p\) estimate used to prove the lemma in Section 2. The third theorem (ergodic) is proved in Section 5. Section 6 contains the proof of another lemma on singular Radon transform. Section 7, self-contained is on real variable theory on the group \(G_0^\#\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
11L40 Estimates on character sums
44A12 Radon transform
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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[1] Arkhipov, G. I.; Oskolkov, K. I., A special trigonometric series and its applications, Mat. Sb., 134, 176, 147-157 (1987) · Zbl 0665.42003
[2] Bergelson, V.; Leibman, A., A nilpotent Roth theorem, Invent. Math., 147, 429-470 (2002) · Zbl 1042.37001 · doi:10.1007/s002220100179
[3] Bourgain, J., On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math., 61, 39-72 (1988) · Zbl 0642.28010
[4] Bourgain, J., On the pointwise ergodic theorem on L^p for arithmetic sets, Israel J. Math., 61, 73-84 (1988) · Zbl 0642.28011
[5] Bourgain, J., Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math., 69, 5-45 (1989) · Zbl 0705.28008
[6] Christ, M.; Nagel, A.; Stein, E. M.; Wainger, S., Singular and maximal Radon transforms: analysis and geometry, Ann. of Math., 150, 489-577 (1999) · Zbl 0960.44001 · doi:10.2307/121088
[7] García-Cuerva, J.; Rubio de Francia, J. L., Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, 116 (1985), Amsterdam: North-Holland, Amsterdam · Zbl 0578.46046
[8] Ionescu, A. D.; Wainger, S., L^p boundedness of discrete singular Radon transforms, J. Amer. Math. Soc., 19, 357-383 (2006) · Zbl 1158.42007 · doi:10.1090/S0894-0347-05-00508-4
[9] Leibman, A., Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math., 146, 303-315 (2005) · Zbl 1080.37002
[10] Magyar, A., Stein, E. M. & Wainger, S., Discrete analogues of integral operators: maximal functions on the discrete Heisenberg group with applications to ergodic theory. To appear in J. Anal. Math. · Zbl 1149.22009
[11] Ricci, F.; Stein, E. M., Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals, J. Funct. Anal., 73, 179-194 (1987) · Zbl 0622.42010 · doi:10.1016/0022-1236(87)90064-4
[12] Ricci, F.; Stein, E. M., Harmonic analysis on nilpotent groups and singular integrals. II. Singular kernels supported on submanifolds, J. Funct. Anal., 78, 56-84 (1988) · Zbl 0645.42019 · doi:10.1016/0022-1236(88)90132-2
[13] Rubio de Francia, J. L., A Littlewood-Paley inequality for arbitrary intervals, Rev. Mat. Iberoamericana, 1, 1-14 (1985) · Zbl 0611.42005
[14] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, 43 (1993), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 0821.42001
[15] Stein, E. M.; Wainger, S., Discrete analogues of singular Radon transforms, Bull. Amer. Math. Soc., 23, 537-544 (1990) · Zbl 0718.42015 · doi:10.1090/S0273-0979-1990-15973-7
[16] Stein, E. M.; Wainger, S., Discrete analogues in harmonic analysis. I. l^2 estimates for singular Radon transforms, Amer. J. Math., 121, 1291-1336 (1999) · Zbl 0945.42009 · doi:10.1353/ajm.1999.0046
[17] Stein, E. M.; Wainger, S., Two discrete fractional integral operators revisited, J. Anal. Math., 87, 451-479 (2002) · Zbl 1029.39016 · doi:10.1007/BF02868485
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