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Sparse bounds for spherical maximal functions

  • Michael T. LaceyEmail author
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Abstract

We consider the averages of a function f on ℝn over spheres of radius 0 < r < ∞ given by \({A_r}f(x) = \int_{{\mathbb{S}^{n - 1}}} {f(x - ry)d\sigma (y)} \), where σ is the normalized rotation invariant measure on 𝕊n−1. We prove a sharp range of sparse bounds for two maximal functions, the first the lacunary spherical maximal function, and the second the full maximal function.
$${M_{1ac}}f = \mathop {\sup }\limits_{j \in \mathbb{Z}} {A_{{2^j}}}f,\;\;\;{M_{\text{full}}}f = \mathop {\sup }\limits_{r > 0} {A_r}f.$$

The sparse bounds are very precise variants of the known Lp bounds for these maximal functions. They are derived from known Lp-improving estimates for the localized versions of these maximal functions, and the indices in our sparse bound are sharp. We derive novel weighted inequalities for weights in the intersection of certain Muckenhoupt and reverse Hölder classes.

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References

  1. [1]
    F. Bernicot, D. Frey and S. Petermichl, Sharpweighted normestimates beyond Calderón-Zygmund theory, Anal. PDE 9 (2016), 1079–1113.MathSciNetCrossRefGoogle Scholar
  2. [2]
    S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), 253–272.MathSciNetCrossRefGoogle Scholar
  3. [3]
    C. P. Calderón, Lacunary spherical means, Illinois J. Math. 23 (1979), 476–484.MathSciNetCrossRefGoogle Scholar
  4. [4]
    L. Cladek and B. Krause, Improved endpoint bounds for the lacunary spherical maximal operator, ArXiv:1703.01508[math.CA].Google Scholar
  5. [5]
    L. Cladek and Y. Ou, Sparse domination of Hilbert transforms along curves, Math. Res. Lett. 25 (2018), 415–436.MathSciNetCrossRefGoogle Scholar
  6. [6]
    R. R. Coifman and G. Weiss, Book Review: Littlewood-Paley and multiplier theory, Bull. Amer. Math. Soc. 84 (1978), 242–250.MathSciNetCrossRefGoogle Scholar
  7. [7]
    J. M. Conde-Alonso, A. Culiuc, F. Di Plinio and Y. Ou, A sparse domination principle for rough singular integrals, Anal. PDE 10 (2017), 1255–1284.MathSciNetCrossRefGoogle Scholar
  8. [8]
    M. Cowling, J. Garcí a Cuerva and H. Gunawan, Weighted estimates for fractional maximal functions related to spherical means, Bull. Austral. Math. Soc. 66 (2002), 75–90.MathSciNetCrossRefGoogle Scholar
  9. [9]
    A. Culiuc, F. Di Plinio and Y. Ou, Domination of multilinear singular integrals by positive sparse forms, J. Lond. Math. Soc. (2)98 (2018), 369–392.MathSciNetCrossRefGoogle Scholar
  10. [10]
    A. Culiuc, R. Kesler and M. T. Lacey, Sparse bounds for the discrete cubic Hilbert transform, Anal. PDE 12 (2019), 1259–1272.MathSciNetCrossRefGoogle Scholar
  11. [11]
    F. C. de França Silva and P. Zorin-Kranich, Sparse domination of sharp variational truncations, ArXiv:1604.05506[math.CA].Google Scholar
  12. [12]
    F. Di Plinio, Y. Q. Do and G. N. Uraltsev, Positive sparse domination of variational Carleson operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), 1443–1458.MathSciNetzbMATHGoogle Scholar
  13. [13]
    J. Duoandikoetxea and L. Vega, Spherical means and weighted inequalities, J. London Math. Soc. (2) 53 (1996), 343–353.MathSciNetCrossRefGoogle Scholar
  14. [14]
    R. L. Jones, A. Seeger and J. Wright, Strong variational and jump inequalities in harmonic analysis, Trans. Amer.Math. Soc. 360 (2008), 6711–6742.MathSciNetCrossRefGoogle Scholar
  15. [15]
    B. Krause, M. Lacey and M. Wierdl, On convergence of oscillatory ergodic Hilbert transforms, Indiana Univ. Math. J. 68 (2019), 641–662.MathSciNetCrossRefGoogle Scholar
  16. [16]
    B. Krause and M. T. Lacey, A weak type inequality for maximal monomial oscillatory Hilbert transforms, ArXiv:1609.01564[math.CA].Google Scholar
  17. [17]
    B. Krause and M. T. Lacey, Sparse bounds for maximally truncated oscillatory singular integrals, ArXiv:1701.05249[math.CA].Google Scholar
  18. [18]
    B. Krause and M. T. Lacey, Sparse bounds for random discrete Carleson theorems, in 50 Years with Hardy Spaces, Birkhäuser, Cham, 2018, pp. 317–332.CrossRefGoogle Scholar
  19. [19]
    M. T. Lacey, An elementary proof of the A2 bound, Israel J. Math. 217 (2017), 181–195.MathSciNetCrossRefGoogle Scholar
  20. [20]
    M. T. Lacey and D. Mena, The sparse T1 theorem, Houston J. Math. 43 (2016), 111–127.MathSciNetzbMATHGoogle Scholar
  21. [21]
    M. T. Lacey and S. Spencer, Scott, Sparse bounds for oscillatory and random singular integrals, ArXiv:1609.06364[math.CA].Google Scholar
  22. [22]
    S. Lee, Endpoint estimates for the circular maximal function, Proc. Amer.Math. Soc. 131 (2003), 1433–1442.MathSciNetCrossRefGoogle Scholar
  23. [23]
    A. K. Lerner, S. Ombrosi and I. P. Rivera-Ríos, On pointwise and weighted estimates for commutators of Calderón-Zygmund operators, Adv. Math. 319 (2017), 153–181.MathSciNetCrossRefGoogle Scholar
  24. [24]
    K. Li, Two weight inequalities for bilinear forms, Collect. Math. 68 (2017), 129–144.MathSciNetCrossRefGoogle Scholar
  25. [25]
    K. Li, C. Pérez, I. P. Rivera-Ríos and L. Roncal, Weighted norm inequalities for rough singular integral operators, J. Geom. Anal. 29 (2019), 2526–2564.MathSciNetCrossRefGoogle Scholar
  26. [26]
    W. Littman, L p - L q-estimates for singular integral operators arising from hyperbolic equations, (1973), 479–481.zbMATHGoogle Scholar
  27. [27]
    R. Manna, Weighted inequalities for spherical maximal operator, Proc. Japan Acad. Ser. AMath. Sci. 91 (2015), 135–140.MathSciNetCrossRefGoogle Scholar
  28. [28]
    K. Moen, Sharp weighted bounds without testing or extrapolation, Arch.Math. (Basel) 99 (2012), 457–466.MathSciNetCrossRefGoogle Scholar
  29. [29]
    R. Oberlin, Sparse bounds for a prototypical singular Radon transform, Canad. Math. Bull. 62 (2019), 405–415.MathSciNetCrossRefGoogle Scholar
  30. [30]
    W. Schlag, A generalization of Bourgain’s circular maximal theorem, J. Amer. Math. Soc. 10 (1997), 103–122.MathSciNetCrossRefGoogle Scholar
  31. [31]
    W. Schlag and C. D. Sogge, Local smoothing estimates related to the circular maximal theorem, Math. Res. Lett. 4 (1997), 1–15.MathSciNetCrossRefGoogle Scholar
  32. [32]
    A. Seeger, T. Tao and J. Wright, Endpoint mapping properties of spherical maximal operators, J. Inst. Math. Jussieu 2 (2003), 109–144.MathSciNetCrossRefGoogle Scholar
  33. [33]
    A. Seeger, T. Tao and J. Wright, Singular maximal functions and Radon transforms near L 1, 126 (2004), 607–647.Google Scholar
  34. [34]
    E. M. Stein, Maximal functions. I. Spherical means, 73 1976, 2174–2175.Google Scholar
  35. [35]
    R. S. Strichartz, Convolutions with kernels having singularities on a sphere, 148 (1970), 461–471.Google Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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