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October  2022, 21(10): 3283-3307. doi: 10.3934/cpaa.2022100

Circular average relative to fractal measures

Department of Mathematical Sciences and RIM, Seoul National University, Seoul 08826, Republic of Korea

*Corresponding author

Received  October 2021 Revised  May 2022 Published  October 2022 Early access  June 2022

Fund Project: This work was supported by the NRF (Republic of Korea) grants No. 2017R1C1B2002959 (Ham), No. 2022R1I1A1A01055527 (Ko), and No. 2022R1A4A1018904 (Lee)

We prove new $ L^p $–$ L^q $ estimates for averages over dilates of the circle with respect to fractal measures, which unify different types of maximal estimates for the circular average. Our results are consequences of $ L^p $–$ L^q $ smoothing estimates for the wave operator relative to fractal measures. We also discuss similar results concerning the spherical averages.

Citation: Seheon Ham, Hyerim Ko, Sanghyuk Lee. Circular average relative to fractal measures. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3283-3307. doi: 10.3934/cpaa.2022100
References:
[1]

D. BeltranR. OberlinL. RoncalA. Seeger and B. Stovall, Variation bounds for spherical averages, Math. Ann., 382 (2022), 459-512.  doi: 10.1007/s00208-021-02218-2.

[2]

J. BennettA. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math., 196 (2006), 261-302.  doi: 10.1007/s11511-006-0006-4.

[3]

A. S. Besicovitch and R. Rado, A plane set of measure zero containing circumferences of every radius, J. London Math. Soc., 43 (1968), 717-719.  doi: 10.1112/jlms/s1-43.1.717.

[4]

J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Anal. Math., 47 (1986), 69-85.  doi: 10.1007/BF02792533.

[5]

J. Bourgain and C. Demeter, The proof of the $l^2$ decoupling conjecture, Ann. Math., 182 (2015), 351-389.  doi: 10.4007/annals.2015.182.1.9.

[6]

J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal., 21 (2011), 1239-1295.  doi: 10.1007/s00039-011-0140-9.

[7]

C.-H. ChoS. Ham and S. Lee, Fractal Strichartz estimate for the wave equation, Nonlinear Anal., 150 (2017), 61-75.  doi: 10.1016/j.na.2016.11.006.

[8]

L. GuthH. Wang and and R. Zhang, A sharp square function estimate for the cone in $\mathbb R^{3}$, Ann. Math., 192 (2020), 551-581.  doi: 10.4007/annals.2020.192.2.6.

[9]

S. Ham, H. Ko and S. Lee, Dimension of divergence set of the wave equation, Nonlinear Anal., 215 (2022), 112631, 10 pp. doi: 10.1016/j. na. 2021.112631.

[10]

T. L. J. Harris, Improved decay of conical averages of the Fourier transform, Proc. Amer. Math. Soc., 147 (2019), 4781-4796.  doi: 10.1090/proc/14747.

[11]

A. IosevichB. KrauseE. SawyerK. Taylor and I. Uriarte-Tuero, Maximal operators: scales, curvature and the fractal dimension, Anal. Math., 45 (2019), 63-86.  doi: 10.1007/s10476-018-0307-9.

[12]

J. R. Kinney, A thin set of circles, Amer. Math. Monthly, 75 (1968), 1077-1081.  doi: 10.2307/2315733.

[13]

L. Kolasa and T. Wolff, On some variants of the Kakeya problem, Pacific J. Math., 190 (1999), 111-154.  doi: 10.2140/pjm.1999.190.111.

[14]

S. Lee, Endpoint estimates for the circular maximal function, Proc. Amer. Math. Soc., 131 (2003), 1433-1442.  doi: 10.1090/S0002-9939-02-06781-3.

[15]

S. Lee, Square function estimates for the Bochner-Riesz means, Anal. PDE, 11 (2018), 1535-1586.  doi: 10.2140/apde.2018.11.1535.

[16]

S. Lee and A. Vargas, On the cone multiplier in $\mathbb R^3$, J. Funct. Anal., 263 (2012), 925-940.  doi: 10.1016/j.jfa.2012.05.010.

[17] P. Mattila, Fourier Analysis and Hausdorff Dimension, Cambridge University Press, Cambridge, United Kingdom, 2015.  doi: 10.1017/CBO9781316227619.
[18]

T. Mitsis, On a problem related to sphere and circle packing, J. London Math. Soc., 60 (1999), 501-516.  doi: 10.1112/S0024610799007838.

[19]

G. MockenhauptA. Seeger and C. D. Sogge, Wave front sets, local smoothing and Bourgain's circular maximal theorem, Ann. Math., 136 (1992), 207-218.  doi: 10.2307/2946549.

[20]

D. Oberlin, Packing spheres and fractal Strichartz estimates in ${\mathbb R}^d$ for $d\ge3$, Proc. Amer. Math. Soc., 134 (2006), 3201-3209.  doi: 10.1090/S0002-9939-06-08356-0.

[21]

D. Oberlin and R. Oberlin, Spherical means and pinned distance sets, Commun. Korean Math. Soc., 30 (2015), 23-34. 

[22]

G. Polya and G. Szegö, Problems and Theorems in Analysis, Die Grundlehren der mathematischen Wissenschaften, Band 216, Springer-Verlag, New York-Heidelberg, 1976.

[23]

J. Roos and A. Seeger, Spherical maximal functions and fractal dimensions of dilation sets, To appear in Amer. J. Math., arXiv: 2004.00984.

[24]

W. Schlag, A generalization of Bourgain's circular maximal theorem, J. Amer. Math. Soc., 10 (1997), 103-122.  doi: 10.1090/S0894-0347-97-00217-8.

[25]

W. Schlag and C. D. Sogge, Local smoothing estimates related to the circular maximal theorem, Math. Res. Let., 4 (1997), 1-15.  doi: 10.4310/MRL.1997.v4.n1.a1.

[26]

E. M. Stein, Maximal functions: spherical means, Proc. Nat. Acad. Sci. USA, 73 (1976), 2174-2175.  doi: 10.1073/pnas.73.7.2174.

[27]

T. Wolff, A Kakeya type problem for circles, Amer. J. Math., 119 (1997), 985-1026. 

[28]

T. Wolff, Local smoothing estimates on $L^p$ for large $p$, Geom. Funct. Anal., 10 (2000), 1237-1288.  doi: 10.1007/PL00001652.

[29]

J. Zahl, On the Wolff circular maximal function, Illinois J. Math., 56 (2012), 1281-1295. 

show all references

References:
[1]

D. BeltranR. OberlinL. RoncalA. Seeger and B. Stovall, Variation bounds for spherical averages, Math. Ann., 382 (2022), 459-512.  doi: 10.1007/s00208-021-02218-2.

[2]

J. BennettA. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math., 196 (2006), 261-302.  doi: 10.1007/s11511-006-0006-4.

[3]

A. S. Besicovitch and R. Rado, A plane set of measure zero containing circumferences of every radius, J. London Math. Soc., 43 (1968), 717-719.  doi: 10.1112/jlms/s1-43.1.717.

[4]

J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Anal. Math., 47 (1986), 69-85.  doi: 10.1007/BF02792533.

[5]

J. Bourgain and C. Demeter, The proof of the $l^2$ decoupling conjecture, Ann. Math., 182 (2015), 351-389.  doi: 10.4007/annals.2015.182.1.9.

[6]

J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal., 21 (2011), 1239-1295.  doi: 10.1007/s00039-011-0140-9.

[7]

C.-H. ChoS. Ham and S. Lee, Fractal Strichartz estimate for the wave equation, Nonlinear Anal., 150 (2017), 61-75.  doi: 10.1016/j.na.2016.11.006.

[8]

L. GuthH. Wang and and R. Zhang, A sharp square function estimate for the cone in $\mathbb R^{3}$, Ann. Math., 192 (2020), 551-581.  doi: 10.4007/annals.2020.192.2.6.

[9]

S. Ham, H. Ko and S. Lee, Dimension of divergence set of the wave equation, Nonlinear Anal., 215 (2022), 112631, 10 pp. doi: 10.1016/j. na. 2021.112631.

[10]

T. L. J. Harris, Improved decay of conical averages of the Fourier transform, Proc. Amer. Math. Soc., 147 (2019), 4781-4796.  doi: 10.1090/proc/14747.

[11]

A. IosevichB. KrauseE. SawyerK. Taylor and I. Uriarte-Tuero, Maximal operators: scales, curvature and the fractal dimension, Anal. Math., 45 (2019), 63-86.  doi: 10.1007/s10476-018-0307-9.

[12]

J. R. Kinney, A thin set of circles, Amer. Math. Monthly, 75 (1968), 1077-1081.  doi: 10.2307/2315733.

[13]

L. Kolasa and T. Wolff, On some variants of the Kakeya problem, Pacific J. Math., 190 (1999), 111-154.  doi: 10.2140/pjm.1999.190.111.

[14]

S. Lee, Endpoint estimates for the circular maximal function, Proc. Amer. Math. Soc., 131 (2003), 1433-1442.  doi: 10.1090/S0002-9939-02-06781-3.

[15]

S. Lee, Square function estimates for the Bochner-Riesz means, Anal. PDE, 11 (2018), 1535-1586.  doi: 10.2140/apde.2018.11.1535.

[16]

S. Lee and A. Vargas, On the cone multiplier in $\mathbb R^3$, J. Funct. Anal., 263 (2012), 925-940.  doi: 10.1016/j.jfa.2012.05.010.

[17] P. Mattila, Fourier Analysis and Hausdorff Dimension, Cambridge University Press, Cambridge, United Kingdom, 2015.  doi: 10.1017/CBO9781316227619.
[18]

T. Mitsis, On a problem related to sphere and circle packing, J. London Math. Soc., 60 (1999), 501-516.  doi: 10.1112/S0024610799007838.

[19]

G. MockenhauptA. Seeger and C. D. Sogge, Wave front sets, local smoothing and Bourgain's circular maximal theorem, Ann. Math., 136 (1992), 207-218.  doi: 10.2307/2946549.

[20]

D. Oberlin, Packing spheres and fractal Strichartz estimates in ${\mathbb R}^d$ for $d\ge3$, Proc. Amer. Math. Soc., 134 (2006), 3201-3209.  doi: 10.1090/S0002-9939-06-08356-0.

[21]

D. Oberlin and R. Oberlin, Spherical means and pinned distance sets, Commun. Korean Math. Soc., 30 (2015), 23-34. 

[22]

G. Polya and G. Szegö, Problems and Theorems in Analysis, Die Grundlehren der mathematischen Wissenschaften, Band 216, Springer-Verlag, New York-Heidelberg, 1976.

[23]

J. Roos and A. Seeger, Spherical maximal functions and fractal dimensions of dilation sets, To appear in Amer. J. Math., arXiv: 2004.00984.

[24]

W. Schlag, A generalization of Bourgain's circular maximal theorem, J. Amer. Math. Soc., 10 (1997), 103-122.  doi: 10.1090/S0894-0347-97-00217-8.

[25]

W. Schlag and C. D. Sogge, Local smoothing estimates related to the circular maximal theorem, Math. Res. Let., 4 (1997), 1-15.  doi: 10.4310/MRL.1997.v4.n1.a1.

[26]

E. M. Stein, Maximal functions: spherical means, Proc. Nat. Acad. Sci. USA, 73 (1976), 2174-2175.  doi: 10.1073/pnas.73.7.2174.

[27]

T. Wolff, A Kakeya type problem for circles, Amer. J. Math., 119 (1997), 985-1026. 

[28]

T. Wolff, Local smoothing estimates on $L^p$ for large $p$, Geom. Funct. Anal., 10 (2000), 1237-1288.  doi: 10.1007/PL00001652.

[29]

J. Zahl, On the Wolff circular maximal function, Illinois J. Math., 56 (2012), 1281-1295. 

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