## Tohoku Mathematical Journal

### Kakeya-Nikodym averages and $L^p$-norms of eigenfunctions

Christopher D. Sogge

#### Abstract

We provide a necessary and sufficient condition that $L^p$-norms, $2<p<6$, of eigenfunctions of the square root of minus the Laplacian on two-dimensional compact boundaryless Riemannian manifolds $M$ are small compared to a natural power of the eigenvalue $\lambda$. The condition that ensures this is that their $L^2$-norms over $O(\lambda^{-1/2})$ neighborhoods of arbitrary unit geodesics are small when $\lambda$ is large (which is not the case for the highest weight spherical harmonics on $S^2$ for instance). The proof exploits Gauss' lemma and the fact that the bilinear oscillatory integrals in Hörmander's proof of the Carleson-Sjölin theorem become better and better behaved away from the diagonal. Our results are related to a recent work of Bourgain who showed that $L^2$-averages over geodesics of eigenfunctions are small compared to a natural power of the eigenvalue $\lambda$ provided that the $L^4(M)$ norms are similarly small. Our results imply that QUE cannot hold on a compact boundaryless Riemannian manifold $(M,g)$ of dimension two if $L^p$-norms are saturated for a given $2<p<6$. We also show that eigenfunctions cannot have a maximal rate of $L^2$-mass concentrating along unit portions of geodesics that are not smoothly closed.

#### Article information

Source
Tohoku Math. J. (2), Volume 63, Number 4 (2011), 519-538.

Dates
First available in Project Euclid: 6 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1325886279

Digital Object Identifier
doi:10.2748/tmj/1325886279

Mathematical Reviews number (MathSciNet)
MR2872954

Zentralblatt MATH identifier
1234.35156

#### Citation

Sogge, Christopher D. Kakeya-Nikodym averages and $L^p$-norms of eigenfunctions. Tohoku Math. J. (2) 63 (2011), no. 4, 519--538. doi:10.2748/tmj/1325886279. https://projecteuclid.org/euclid.tmj/1325886279

#### References

• V. M. Babič and V. F. Lazutkin, The eigenfunctions which are concentrated near a closed geodesic, (Russian) Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 9 (1967), 15–25.
• J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), 147–187.
• J. Bourgain, Some new estimates on oscillatory integrals, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), 83–12, Princeton Math. Ser. 42, Princeton Univ. Press, Princeton, NJ, 1995.
• J. Bourgain, $L\sp p$-estimates for oscillatory integrals in several variables, Geom. Funct. Anal. 1 (1991), 321–374.
• J. Bourgain, Geodesic restrictions and $L^p$-estimates for eigenfunctions of Riemannian surfaces, Linear and complex analysis, 27–35, Amer. Math. Soc. Tranl. Ser. 2, 226, Amer. Math. Soc., Providence, RI, 2009.
• N. Burq, P. Gérard and N. Tzvetkov, Restriction of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J. 138 (2007), 445–486.
• L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Sudia Math. 44 (1972), 287–299.
• Y. Colin de Verdière, Semi-classical measures and entropy [after Nalini Anantharaman and Stéphane Nonnenmacher], (English summary) Séminaire Bourbaki. Vol. 2006/2007, Astrisque No. 317 (2008), Exp. No. 978, ix, 393–414.
• A. Córdoba, A note on Bochner-Riesz operators, Duke Math. J. 46 (1979), 505–511.
• J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39–79.
• C. Fefferman, A note on spherical summation operators, Israel J. Math. 15 (1973), 44–52.
• A. Greenleaf and A. Seeger, Fourier integrals with fold singularities, J. Reine Angew. Math. 455 (1994), 35–56.
• D. Grieser, $L^p$ bounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries, Ph. D. Thesis, University of California, Los Angeles, 1992.
• L. Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), 79–183.
• L. Hörmander, Oscillatory integrals and multipliers on $FL^p$, Ark. Math. II (1973), 1–11.
• V. Ivrii, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, (Russian) Funktsional. Anal. i Prilozhen. 14 (1980), 25–34.
• W. P. Minicozzi and C. D. Sogge, Negative results for Nikodym maximal functions and related oscillatory integrals in curved space, Math. Res. Lett. 4 (1997), 221–237.
• G. Mockenhaupt, A. Seeger and C. D. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), 65–130.
• J. V. Ralston, On the construction of quasimodes associated with stable periodic orbits, Comm. Math. Phys. 51 (1976), 219–242.
• A. Reznikov, Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory..
• Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys 161 (1994), 195–213.
• A. Schnirelman, Ergodic properties of eigenfunctions, Usp. Math. Nauk. 29 (1974), 181–182.
• H. Smith and C. D. Sogge, On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc. 8 (1995), 879–916.
• H. Smith and C. D. Sogge, On the $L\sp p$ norm of spectral clusters for compact manifolds with boundary, Acta Math. 198 (2007), 107–153.
• C. D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), 43–65.
• C. D. Sogge, Concerning the $L^p$ norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), 123–138.
• C. D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Math., Cambridge Univ. Press, Cambridge, 1993.
• C. D. Sogge, Concerning Nikodym-type sets in $3$-dimensional curved spaces, J. Amer. Math. Soc. 12 (1999), 1–31.
• C. D. Sogge, J. Toth and S. Zelditch, About the blowup of quasimodes on Riemannian manifolds, to appear, J. Geom. Anal.
• C. D. Sogge and S. Zelditch, Riemannian manifolds with maximal eigenfunction growth, Duke Math. J. 114 (2002), 387–437.
• E. M. Stein, Oscillatory integrals in Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984), 307–355, Ann. of Math. Stud.,112, Princeton Univ. Press, Princeton, NJ, 1986.
• D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1998), 185–206.
• J. Toth, $L^2$-restriction bounds for eigenfunctions along curves in the quantum completely integrable case, Comm. Math. Phys. 288 (2009), 379–401.
• J. Toth and S. Zelditch, $L^p$ norms of eigenfunctions in the completely integrable case, Ann. Henri Poincaré 4 (2003), 343–368.
• S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919–941.
• S. Zelditch, Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series, J. Funct. Anal. 97 (1991), 1–49.
• A. Zygmund, On Fourier coefficients and transforms of two variables, Studia Math. 50 (1974), 189–201.