• Previous Article
    Analysis of COVID-19 epidemic transmission trend based on a time-delayed dynamic model
  • CPAA Home
  • This Issue
  • Next Article
    Large time behavior of the solutions with spreading fronts in the Allen-Cahn equations on $ \mathbb R^n $
doi: 10.3934/cpaa.2022100
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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, 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. 

[1]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235

[2]

Émilie Chouzenoux, Henri Gérard, Jean-Christophe Pesquet. General risk measures for robust machine learning. Foundations of Data Science, 2019, 1 (3) : 249-269. doi: 10.3934/fods.2019011

[3]

Arno Berger, Roland Zweimüller. Invariant measures for general induced maps and towers. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3885-3901. doi: 10.3934/dcds.2013.33.3885

[4]

Raz Kupferman, Asaf Shachar. On strain measures and the geodesic distance to $SO_n$ in the general linear group. Journal of Geometric Mechanics, 2016, 8 (4) : 437-460. doi: 10.3934/jgm.2016015

[5]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control and Related Fields, 2021, 11 (4) : 829-855. doi: 10.3934/mcrf.2020048

[6]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

[7]

David Rojas, Pedro J. Torres. Bifurcation of relative equilibria generated by a circular vortex path in a circular domain. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 749-760. doi: 10.3934/dcdsb.2019265

[8]

Martin Bauer, Thomas Fidler, Markus Grasmair. Local uniqueness of the circular integral invariant. Inverse Problems and Imaging, 2013, 7 (1) : 107-122. doi: 10.3934/ipi.2013.7.107

[9]

Hengrui Luo, Alice Patania, Jisu Kim, Mikael Vejdemo-Johansson. Generalized penalty for circular coordinate representation. Foundations of Data Science, 2021, 3 (4) : 729-767. doi: 10.3934/fods.2021024

[10]

R. Enkhbat , N. Tungalag, A. S. Strekalovsky. Pseudoconvexity properties of average cost functions. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 233-236. doi: 10.3934/naco.2015.5.233

[11]

Robert S. Strichartz. Average error for spectral asymptotics on surfaces. Communications on Pure and Applied Analysis, 2016, 15 (1) : 9-39. doi: 10.3934/cpaa.2016.15.9

[12]

Sujay Jayakar, Robert S. Strichartz. Average number of lattice points in a disk. Communications on Pure and Applied Analysis, 2016, 15 (1) : 1-8. doi: 10.3934/cpaa.2016.15.1

[13]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012

[14]

Matthew Foreman, Benjamin Weiss. From odometers to circular systems: A global structure theorem. Journal of Modern Dynamics, 2019, 15: 345-423. doi: 10.3934/jmd.2019024

[15]

Yirmeyahu J. Kaminski. Equilibrium locus of the flow on circular networks of cells. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1169-1177. doi: 10.3934/dcdss.2018066

[16]

Igor Chueshov, Tamara Fastovska. On interaction of circular cylindrical shells with a Poiseuille type flow. Evolution Equations and Control Theory, 2016, 5 (4) : 605-629. doi: 10.3934/eect.2016021

[17]

Willard S. Keeran, Patrick D. Leenheer, Sergei S. Pilyugin. Circular and elliptic orbits in a feedback-mediated chemostat. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 779-792. doi: 10.3934/dcdsb.2007.7.779

[18]

A. V. Borisov, I.S. Mamaev, S. M. Ramodanov. Dynamics of two interacting circular cylinders in perfect fluid. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 235-253. doi: 10.3934/dcds.2007.19.235

[19]

A. V. Borisov, I. S. Mamaev, S. M. Ramodanov. Dynamics of a circular cylinder interacting with point vortices. Discrete and Continuous Dynamical Systems - B, 2005, 5 (1) : 35-50. doi: 10.3934/dcdsb.2005.5.35

[20]

Tero Laihonen. Information retrieval and the average number of input clues. Advances in Mathematics of Communications, 2017, 11 (1) : 203-223. doi: 10.3934/amc.2017013

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (84)
  • HTML views (41)
  • Cited by (0)

Other articles
by authors

[Back to Top]