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] Beltran D, Guo S, Hickman J, Seeger A. Sharp L^p bounds for the helical maximal function. Amer. J. Math., 2025, 147(1): 149-234

[3] Beltran D, Oberlin R, Roncal L, Seeger A, Stovall B. Variation bounds for spherical averages. Math. Ann., 2022, 382(1–2): 459-512

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

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

[6] Guo S, Roos J, Yung P. Sharp variation-norm estimates for oscillatory integrals related to Carleson’s theorem. Anal. PDE, 2020, 13(5): 1457-1500

[7] Guth L, Wang H, Zhang R. A sharp square function estimate for the cone in ℝ³. Ann. of Math. (2), 2020, 192(2): 551-581

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

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

[10] Jones R, Seeger A, Wright J. Strong variational and jump inequalities in harmonic analysis. Trans. Amer. Math. Soc., 2008, 360(12): 6711-6742

[11] Ko H, Lee S, Oh S. 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] Lee S. 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] Mockenhaupt G, Seeger A, Sogge CD. Wave front sets, local smoothing and Bourgain’s circular maximal theorem. Ann. of Math. (2), 1992, 136(1): 207-218

[16] Oberlin R, Seeger A, Tao T, Thiele C, Wright J. A variation norm Carleson theorem. J. Eur. Math. Soc. (JEMS), 2012, 14(2): 421-464

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

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

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

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

[21] Stein EMHarmonic Analysis—Real-variable Methods, Orthogonality, and Oscillatory Integrals, 1993Princeton, NJPrinceton University Press43

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

[23] Triebel HTheory of Function Spaces, 1983BaselBirkhäuser Verlag 78

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Peking University