## Illinois Journal of Mathematics

### On the Wolff circular maximal function

Joshua Zahl

#### Abstract

We prove sharp $L^{3}$ bounds for a variable coefficient generalization of the Wolff circular maximal function $M^{\delta}f$. For each fixed radius $r$, $M^{\delta}f(r)$ is the maximal average of $f$ over the $\delta$-neighborhood of a circle of radius $r$ and arbitrary center. In this paper, we consider maximal averages over families of curves satisfying the cinematic curvature condition, which was first introduced by Sogge to generalize the Bourgain circular maximal function. Our proof manages to avoid a key technical lemma in Wolff’s original argument, and thus our arguments also yield a shorter proof of the boundedness of the (conventional) Wolff circular maximal function. At the heart of the proof is an induction argument that employs an efficient partitioning of $\mathbb{R}^{3}$ into cells using the discrete polynomial ham sandwich theorem.

#### Article information

Source
Illinois J. Math., Volume 56, Number 4 (2012), 1281-1295.

Dates
First available in Project Euclid: 6 May 2014

https://projecteuclid.org/euclid.ijm/1399395832

Digital Object Identifier
doi:10.1215/ijm/1399395832

Mathematical Reviews number (MathSciNet)
MR3231483

Zentralblatt MATH identifier
1301.42038

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory

#### Citation

Zahl, Joshua. On the Wolff circular maximal function. Illinois J. Math. 56 (2012), no. 4, 1281--1295. doi:10.1215/ijm/1399395832. https://projecteuclid.org/euclid.ijm/1399395832

#### References

• T. Bagby, L. Bos and N. Levenberg, Multivariate Simultaneous Approximation, Constr. Approx. 18 (2002), no. 4, 569–577.
• S. Barone and S. Basu, Refined bounds on the number of connected components of sign conditions on a variety, Discrete Comput. Geom. 47 (2012), no. 3, 577–597.
• S. Basu, R. Pollack and M. Roy, Algorithms in real algebraic geometry, Springer, Berlin, 2006.
• J. Bochnak, M. Coste and M. Roy, Real algebraic geometry, Springer, Berlin, 1998.
• J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. 21 (2011), no. 6, 1239–1295.
• J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Anal. Math. 47 (1986), 69–85.
• G. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decompostion, Automata theory and formal languages (Second GI Conf., Kaiserslautern, 1975), Lect. Notes Comput. Sci., vol. 33, Springer, Berlin, 1975, pp. 134–183.
• L. Guth, The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture, Acta Math. 205 (2010), no. 2, 263–286.
• L. Guth and N. Katz, On the Erdos distinct distance problem in the plane, 2011, available at \arxivurlarXiv:1011.4105v3.
• L. Kolasa and T. Wolff, On some variants of the Kakeya problem, Pacific J. Math. 190 (1999), no. 1, 111–154.
• T. Kővari, V. Sós and P. Turan, On a problem of K. Zarankiewicz, Colloquium Math. 3 (1954), 50–57.
• J. Marstrand, Packing circles in the plane, Proc. London Math. Soc. 55 (1987), no. 3, 37–58.
• W. Schlag, On continuum incidence problems related to harmonic analysis, J. Funct. Anal. 201 (2003), no. 2, 480–521.
• C. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991), 349–376.
• J. Solymosi and T. Tao, An incidence theorem in higher dimensions, Discrete Comput. Geom. 48 (2012), no. 2, 255–280.
• T. Wolff, A Kakeya-type problem for circles, Amer. J. Math. 119 (1997), no. 5, 985–1026.
• T. Wolff, Recent work connected with the Kakeya problem, Prospects in Mathematics (H. Rossi, ed.), AMS, Providence, RI, 1999.
• T. Wolff, Local smoothing type estimates on $L^p$ for large $p$, Geom. Funct. Anal. 10 (2000), no. 5, 1237–1288.
• J. Zahl, $L^3$ estimates for an algebraic variable coefficient Wolff circular maximal function, Rev. Mat. Iberoam. 28 (2012), no. 4, 1061–1090.