The Boundedness of the Dilated Averages over the Parabola

Abstract

This paper investigates the variation bounds and mixed norm bounds for the dilated averages operator. The established bounds are shown to be sharp, except for certain endpoint cases.

Keywords

dilated averages operator / parabola / variation / mixed norm / 42B20 / 42B25

References

[1] Beltran D., Duncan J., Hickman J., Off-diagonal estimates for the helical maximal function. Proc. Lond. Math. Soc. (3), 2024, 128(4): Paper No. e12594, 36 pp.

[2] BeltranD, GuoS, HickmanJ, SeegerA. Sharp L^p bounds for the helical maximal function. Amer. J. Math., 2025, 147(1): 149-234

[3] BeltranD, OberlinR, RoncalL, SeegerA, StovallB. Variation bounds for spherical averages. Math. Ann., 2022, 382(1–2): 459-512

[4] BourgainJ. Averages in the plane over convex curves and maximal operators. J. Analyse Math., 1986, 47: 69-85

[5] BourgainJ. Pointwise ergodic theorems for arithmetic sets. Inst. Hautes Etudes Sci. Publ. Math., 1989, 69: 5-45

[6] GuoS, RoosJ, YungP. Sharp variation-norm estimates for oscillatory integrals related to Carleson’s theorem. Anal. PDE, 2020, 13(5): 1457-1500

[7] GuthL, WangH, ZhangR. A sharp square function estimate for the cone in ℝ³. Ann. of Math. (2), 2020, 192(2): 551-581

[8] HickmanJ. Uniform L_x^p−L_x,r^q improving for dilated averages over polynomial curves. J. Funct. Anal., 2016, 270(2): 560-608

[9] HörmanderL. Estimates for translation invariant operators in L^p spaces. Acta Math., 1960, 104: 93-140

[10] JonesR, SeegerA, WrightJ. Strong variational and jump inequalities in harmonic analysis. Trans. Amer. Math. Soc., 2008, 360(12): 6711-6742

[11] KoH, LeeS, OhS. Maximal estimates for averages over space curves. Invent. Math., 2022, 228(2): 991-1035

[12] Ko H., Lee S., Oh S., Sharp smoothing properties of averages over curves. Forum Math. Pi, 2023, 11: Paper No. e4, 33 pp.

[13] LeeS. Endpoint estimates for the circular maximal function. Proc. Amer. Math. Soc., 2023, 131(5): 1433-1442

[14] Mirek M., Stein E.M., Zorin-Kranich P., Jump inequalities for translation-invariant operators of Radon type on ℝ^d. Adv. Math., 2020, 365: Paper No. 107065, 57 pp.

[15] MockenhauptG, SeegerA, SoggeCD. Wave front sets, local smoothing and Bourgain’s circular maximal theorem. Ann. of Math. (2), 1992, 136(1): 207-218

[16] OberlinR, SeegerA, TaoT, ThieleC, WrightJ. A variation norm Carleson theorem. J. Eur. Math. Soc. (JEMS), 2012, 14(2): 421-464

[17] SchlagW. A generalization of Bourgains circular maximal theorem. J. Amer. Math. Soc., 1997, 10(1): 103-122

[18] SchlagW, SoggeCD. Local smoothing estimates related to the circular maximal theorem. Math. Res. Lett., 1997, 4(1): 1-15

[19] SoggeCD. Propagation of singularities and maximal functions in the plane. Invent. Math., 1991, 104(2): 349-376

[20] SteinEM. Maximal functions, I. Spherical means. Proc. Nat. Acad. Sci. U.S.A., 1976, 73(7): 2174-2175

[21] SteinEMHarmonic Analysis—Real-variable Methods, Orthogonality, and Oscillatory Integrals, 1993, Princeton, NJ. Princeton University Press. 43

[22] StrichartzRS. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J., 1977, 44(3): 705-714

[23] TriebelHTheory of Function Spaces, 1983, Basel. Birkhäuser Verlag. 78

RIGHTS & PERMISSIONS

Peking University

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.