## Acta Mathematica

### Estimates for maximal functions associated with hypersurfaces in ℝ3 and related problems of harmonic analysis

#### Abstract

We study the boundedness problem for maximal operators $\mathcal{M}$ associated with averages along smooth hypersurfaces S of finite type in 3-dimensional Euclidean space. For p > 2, we prove that if no affine tangent plane to S passes through the origin and S is analytic, then the associated maximal operator is bounded on ${L^p}\left( {{\mathbb{R}^3}} \right)$ if and only if p > h(S), where h(S) denotes the so-called height of the surface S (defined in terms of certain Newton diagrams). For non-analytic S we obtain the same statement with the exception of the exponent p = h(S). Our notion of height h(S) is closely related to A. N. Varchenko’s notion of height h(ϕ) for functions ϕ such that S can be locally represented as the graph of ϕ after a rotation of coordinates.

Several consequences of this result are discussed. In particular we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on S and the Lp-boundedness of the associated maximal operator $\mathcal{M}$, and a conjecture by Iosevich and Sawyer which relates the Lp-boundedness of $\mathcal{M}$ to an integrability condition on S for the distance to tangential hyperplanes, in dimension 3.

In particular, we also give essentially sharp uniform estimates for the Fourier transform of the surface measure on S, thus extending a result by V. N. Karpushkin from the analytic to the smooth setting and implicitly verifying a conjecture by V. I. Arnold in our context. As an immediate application of this, we obtain an ${L^p}\left( {{\mathbb{R}^3}} \right) - {L^2}(S)$ Fourier restriction theorem for S.

#### Note

We acknowledge the support for this work by the Deutsche Forschungsgemeinschaft.

#### Article information

Source
Acta Math., Volume 204, Number 2 (2010), 151-271.

Dates
Revised: 7 September 2009
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485892469

Digital Object Identifier
doi:10.1007/s11511-010-0047-6

Mathematical Reviews number (MathSciNet)
MR2653054

Zentralblatt MATH identifier
1223.42011

Subjects
Primary: 35D05 35D10 35G05

Rights

#### Citation

Ikromov, Isroil A.; Kempe, Michael; Müller, Detlef. Estimates for maximal functions associated with hypersurfaces in ℝ 3 and related problems of harmonic analysis. Acta Math. 204 (2010), no. 2, 151--271. doi:10.1007/s11511-010-0047-6. https://projecteuclid.org/euclid.acta/1485892469

#### References

• A rkhipov, G. I., K aratsuba, A. A. & C hubarikov, V. N., Trigonometric integrals. Izv. Akad. Nauk SSSR Ser. Mat., 43 (1979), 971–1003, 1197 (Russian); English translation in Math. USSR–Izv., 15 (1980), 211–239.
• Arnold, V. I., Remarks on the method of stationary phase and on the Coxeter numbers. Uspekhi Mat. Nauk, 28 (1973), 17–44 (Russian); English translation in Russian Math. Surveys, 28 (1973), 19–48.
• A rnold, V. I., G useĭn-Z ade, S. M. & V archenko, A. N., Singularities of Differentiable Maps. Vol. II. Monographs in Mathematics, 83. Birkhäuser, Boston, MA, 1988.
• B ourgain, J., Averages in the plane over convex curves and maximal operators. J. Anal. Math., 47 (1986), 69–85.
• B runa, J., N agel, A. & W ainger, S., Convex hypersurfaces and Fourier transforms. Ann. of Math., 127 (1988), 333–365.
• C arbery, A., W ainger, S. & W right, J., Singular integrals and the Newton diagram. Collect. Math., Vol. Extra (2006), 171–194.
• C hester, C., F riedman, B. & U rsell, F., An extension of the method of steepest descents. Proc. Cambridge Philos. Soc., 53 (1957), 599–611.
• C owling, M. & M auceri, G., Inequalities for some maximal functions. II. Trans. Amer. Math. Soc., 296 (1986), 341–365.
• — Oscillatory integrals and Fourier transforms of surface carried measures. Trans. Amer. Math. Soc., 304 (1987), 53–68.
• D omar, Y., On the Banach algebra A(G) for smooth sets Γ ⊂ Rn. Comment. Math. Helv., 52 (1977), 357–371.
• D uistermaat, J. J., Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math., 27 (1974), 207–281.
• G reenblatt, M., Newton polygons and local integrability of negative powers of smooth functions in the plane. Trans. Amer. Math. Soc., 358 (2006), 657–670.
• — The asymptotic behavior of degenerate oscillatory integrals in two dimensions. J. Funct. Anal., 257 (2009), 1759–1798.
• — Oscillatory integral decay, sublevel set growth, and the Newton polyhedron. Math. Ann., 346 (2010), 857–895.
• — Resolution of singularities, asymptotic expansions of oscillatory integrals, and related phenomena. Preprint, 2007. arXiv:0709.2496v2 [math.CA].
• G reenleaf, A., Principal curvature and harmonic analysis. Indiana Univ. Math. J., 30 (1981), 519–537.
• H örmander, L., The Analysis of Linear Partial Differential Operators. I. Grundlehren der Mathematischen Wissenschaften, 256. Springer, Berlin–Heidelberg, 1990.
• I kromov, I. A., K empe, M. & M üller, D., Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces. Duke Math. J., 126 (2005), 471–490.
• I kromov, I. A. & M üller, D., On adapted coordinate systems. To appear in Trans. Amer. Math. Soc.
• — Uniform estimates for the Fourier transform of surface carried measures and a sharp Lp-L2 Fourier restriction theorem for hypersurfaces in ${\mathbb{R}^3}$. In preparation.
• I osevich, A., Maximal operators associated to families of flat curves in the plane. Duke Math. J., 76 (1994), 633–644.
• I osevich, A. & S awyer, E., Oscillatory integrals and maximal averages over homogeneous surfaces. Duke Math. J., 82 (1996), 103–141.
• — Maximal averages over surfaces. Adv. Math., 132 (1997), 46–119.
• I osevich, A., S awyer, E. & S eeger, A., On averaging operators associated with convex hypersurfaces of finite type. J. Anal. Math., 79 (1999), 159–187.
• K arpushkin, V. N., A theorem on uniform estimates for oscillatory integrals with a phase depending on two variables. Trudy Sem. Petrovsk., 10 (1984), 150–169, 238 (Russian); English translation in J. Soviet Math., 35 (1986), 2809–2826.
• L evinson, N., Transformation of an analytic function of several variables to a canonical form. Duke Math. J., 28 (1961), 345–353.
• M agyar, Á, On Fourier restriction and the Newton polygon. Proc. Amer. Math. Soc., 137 (2009), 615–625.
• M ockenhaupt, G., S eeger, A. & S ogge, C. D., Wave front sets, local smoothing and Bourgain’s circular maximal theorem. Ann. of Math., 136 (1992), 207–218.
• — Local smoothing of Fourier integral operators and Carleson–Sjölin estimates. J. Amer. Math. Soc., 6 (1993), 65–130.
• N agel, A., S eeger, A. & W ainger, S., Averages over convex hypersurfaces. Amer. J. Math., 115 (1993), 903–927.
• P hong, D. H. & S tein, E. M., The Newton polyhedron and oscillatory integral operators. Acta Math., 179 (1997), 105–152.
• P hong, D. H., S tein, E. M. & S turm, J. A., On the growth and stability of real-analytic functions. Amer. J. Math., 121 (1999), 519–554.
• R andol, B., On the asymptotic behavior of the Fourier transform of the indicator function of a convex set. Trans. Amer. Math. Soc., 139 (1969), 279–285.
• S chulz, H., Convex hypersurfaces of finite type and the asymptotics of their Fourier transforms. Indiana Univ. Math. J., 40 (1991), 1267–1275.
• S ogge, C. D., Fourier Integrals in Classical Analysis. Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993.
• — Maximal operators associated to hypersurfaces with one nonvanishing principal curvature, in Fourier Analysis and Partial Differential Equations (Miraflores de la Sierra, 1992), Stud. Adv. Math., pp. 317–323. CRC, Boca Raton, FL, 1995.
• S ogge, C. D. & S tein, E. M., Averages of functions over hypersurfaces in Rn. Invent. Math., 82 (1985), 543–556.
• S tein, E. M., Maximal functions. I. Spherical means. Proc. Nat. Acad. Sci. U.S.A., 73 (1976), 2174–2175.
• Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 1993.
• S vensson, I., Estimates for the Fourier transform of the characteristic function of a convex set. Ark. Mat., 9 (1971), 11–22.
• V archenko, A. N., Newton polyhedra and estimates of oscillatory integrals. Funktsional. Anal. i Prilozhen., 10 (1976), 13–38 (Russian); English translation in Functional Anal. Appl., 18 (1976), 175–196.
• V asil’ev, B. A., The asymptotic behavior of exponential integrals, the Newton diagram and the classification of minima. Funktsional. Anal. i Prilozhen., 11 (1977), 1–11, 96 (Russian); English translation in Functional Anal. Appl., 11 (1977), 163–172.