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.
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F. Bernicot, D. Frey and S. Petermichl, Sharpweighted normestimates beyond Calderón-Zygmund theory, Anal. PDE 9 (2016), 1079–1113.
S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), 253–272.
C. P. Calderón, Lacunary spherical means, Illinois J. Math. 23 (1979), 476–484.
L. Cladek and B. Krause, Improved endpoint bounds for the lacunary spherical maximal operator, ArXiv:1703.01508[math.CA].
L. Cladek and Y. Ou, Sparse domination of Hilbert transforms along curves, Math. Res. Lett. 25 (2018), 415–436.
R. R. Coifman and G. Weiss, Book Review: Littlewood-Paley and multiplier theory, Bull. Amer. Math. Soc. 84 (1978), 242–250.
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.
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.
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.
A. Culiuc, R. Kesler and M. T. Lacey, Sparse bounds for the discrete cubic Hilbert transform, Anal. PDE 12 (2019), 1259–1272.
F. C. de França Silva and P. Zorin-Kranich, Sparse domination of sharp variational truncations, ArXiv:1604.05506[math.CA].
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.
J. Duoandikoetxea and L. Vega, Spherical means and weighted inequalities, J. London Math. Soc. (2) 53 (1996), 343–353.
R. L. Jones, A. Seeger and J. Wright, Strong variational and jump inequalities in harmonic analysis, Trans. Amer.Math. Soc. 360 (2008), 6711–6742.
B. Krause, M. Lacey and M. Wierdl, On convergence of oscillatory ergodic Hilbert transforms, Indiana Univ. Math. J. 68 (2019), 641–662.
B. Krause and M. T. Lacey, A weak type inequality for maximal monomial oscillatory Hilbert transforms, ArXiv:1609.01564[math.CA].
B. Krause and M. T. Lacey, Sparse bounds for maximally truncated oscillatory singular integrals, ArXiv:1701.05249[math.CA].
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.
M. T. Lacey, An elementary proof of the A2 bound, Israel J. Math. 217 (2017), 181–195.
M. T. Lacey and D. Mena, The sparse T1 theorem, Houston J. Math. 43 (2016), 111–127.
M. T. Lacey and S. Spencer, Scott, Sparse bounds for oscillatory and random singular integrals, ArXiv:1609.06364[math.CA].
S. Lee, Endpoint estimates for the circular maximal function, Proc. Amer.Math. Soc. 131 (2003), 1433–1442.
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.
K. Li, Two weight inequalities for bilinear forms, Collect. Math. 68 (2017), 129–144.
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.
W. Littman, L p - L q-estimates for singular integral operators arising from hyperbolic equations, (1973), 479–481.
R. Manna, Weighted inequalities for spherical maximal operator, Proc. Japan Acad. Ser. AMath. Sci. 91 (2015), 135–140.
K. Moen, Sharp weighted bounds without testing or extrapolation, Arch.Math. (Basel) 99 (2012), 457–466.
R. Oberlin, Sparse bounds for a prototypical singular Radon transform, Canad. Math. Bull. 62 (2019), 405–415.
W. Schlag, A generalization of Bourgain’s circular maximal theorem, J. Amer. Math. Soc. 10 (1997), 103–122.
W. Schlag and C. D. Sogge, Local smoothing estimates related to the circular maximal theorem, Math. Res. Lett. 4 (1997), 1–15.
A. Seeger, T. Tao and J. Wright, Endpoint mapping properties of spherical maximal operators, J. Inst. Math. Jussieu 2 (2003), 109–144.
A. Seeger, T. Tao and J. Wright, Singular maximal functions and Radon transforms near L 1, 126 (2004), 607–647.
E. M. Stein, Maximal functions. I. Spherical means, 73 1976, 2174–2175.
R. S. Strichartz, Convolutions with kernels having singularities on a sphere, 148 (1970), 461–471.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by National Science Foundation grant DMS-1600693, and by Australian Research Council grant DP160100153. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 Semester.
Rights and permissions
About this article
Cite this article
Lacey, M.T. Sparse bounds for spherical maximal functions. JAMA 139, 613–635 (2019). https://doi.org/10.1007/s11854-019-0070-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-019-0070-2