Defect measures of eigenfunctions with maximal L ∞ growth
Annales de l'Institut Fourier, to appear, 42 p.

We characterize the defect measures of sequences of Laplace eigenfunctions with maximal L ∞ growth. As a consequence, we obtain new proofs of results on the geometry of manifolds with maximal eigenfunction growth obtained by Sogge–Toth–Zelditch, and generalize those of Sogge–Zelditch to the smooth setting. We also obtain explicit geometric dependence on the constant in Hörmander’s L ∞ bound for high energy eigenfunctions, improving on estimates of Donnelly.

Nous caractérisons les mesures de défauts de séquences de fonctions propres de Laplace avec croissance L ∞ maximale. En conséquence, nous obtenons des nouvelles preuves de résultats sur la géométrie des variétés avec une croissance des fonctions propres maximale obtenus par Sogge–Toth–Zelditch, et nous généralisons ceux de Sogge–Zelditch au cas lisse. Nous obtenons également une dépendance géométrique explicite de la constante de Hörmander L ∞ liée aux functions propres de haute énergie, améliorant les estimations de Donnelly.

Received : 2017-08-31
Revised : 2018-02-13
Accepted : 2018-07-12
Classification:  35P20,  58J50
Keywords: eigenfunctions, defect measures, sup-norms 
@unpublished{AIF_0__0_0_A7_0,
     author = {Galkowski, Jeffrey},
     title = {Defect measures of eigenfunctions with maximal $L^\infty $ growth},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}

Defect measures of eigenfunctions with maximal $L^\infty $ growth. Annales de l'Institut Fourier, to appear, 42 p.

[1] Avakumović, Vojislav G. Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z., Tome 65 (1956), pp. 327-344 | Article | MR 0080862

[2] Bérard, Pierre H. On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., Tome 155 (1977) no. 3, pp. 249-276 | Article | MR 0455055

[3] Blair, David E. Riemannian geometry of contact and symplectic manifolds, Birkhäuser, Progress in Mathematics, Tome 203 (2010), xvi+343 pages | Article | MR 2682326

[4] Blair, Matthew D.; Sogge, Christopher D. Refined and microlocal Kakeya–Nikodym bounds for eigenfunctions in two dimensions, Anal. PDE, Tome 8 (2015) no. 3, pp. 747-764

[5] Blair, Matthew D.; Sogge, Christopher D. Refined and Microlocal Kakeya–Nikodym Bounds of Eigenfunctions in Higher Dimensions, Commun. Math. Phys., Tome 356 (2017) no. 2, pp. 501-533

[6] Brin, Michael; Stuck, Garrett Introduction to dynamical systems, Cambridge University Press (2002), xii+240 pages | Article | MR 1963683

[7] Donnelly, Harold Bounds for eigenfunctions of the Laplacian on compact Riemannian manifolds, J. Funct. Anal., Tome 187 (2001) no. 1, pp. 247-261 | Article | MR 1867351

[8] Dyatlov, Semyon; Zworski, Maciej Mathematical theory of scattering resonances (2017) (http://math.berkeley.edu/~zworski/res.pdf )

[9] Galkowski, Jeffrey; Toth, John A. Eigenfunction scarring and improvements in L ∞ bounds, Anal. PDE, Tome 11 (2017) no. 3, pp. 801-812 | Zbl 1386.35307

[10] Heinonen, Juha Lectures on analysis on metric spaces, Springer, Universitext (2001), x+140 pages | Article | MR 1800917

[11] Hörmander, Lars The spectral function of an elliptic operator, Acta Math., Tome 121 (1968), pp. 193-218 | Article | MR 0609014

[12] Iwaniec, Henryk; Sarnak, Peter L ∞ norms of eigenfunctions of arithmetic surfaces, Ann. Math., Tome 141 (1995) no. 2, pp. 301-320 | Article | MR 1324136 | Zbl 0833.11019

[13] Koch, Herbert; Tataru, Daniel; Zworski, Maciej Semiclassical L p estimates, Ann. Henri Poincaré, Tome 8 (2007) no. 5, pp. 885-916 | Article | MR 2342881

[14] Levitan, Boris M. On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order, Izv. Akad. Nauk SSSR, Ser. Mat., Tome 16 (1952), pp. 325-352 | MR 0058067

[15] Safarov, Yuri G. Asymptotic of the spectral function of a positive elliptic operator without the nontrap condition, Funct. Anal. Appl., Tome 22 (1988) no. 3, pp. 213-223 | Article | Zbl 0679.35074

[16] Sogge, Christopher D. Concerning the L p norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal., Tome 77 (1988) no. 1, pp. 123-138

[17] Sogge, Christopher D. Fourier integrals in classical analysis, Cambridge University Press, Cambridge Tracts in Mathematics, Tome 105 (1993), x+237 pages | Article | MR 1205579

[18] Sogge, Christopher D. Kakeya–Nikodym averages and L p -norms of eigenfunctions, Tôhoku Math. J., Tome 63 (2011) no. 4, pp. 519-538 | Zbl 1234.35156

[19] Sogge, Christopher D.; Toth, John A.; Zelditch, Steve About the blowup of quasimodes on Riemannian manifolds, J. Geom. Anal., Tome 21 (2011) no. 1, pp. 150-173 | Article | MR 2755680

[20] Sogge, Christopher D.; Zelditch, Steve Riemannian manifolds with maximal eigenfunction growth, Duke Math. J., Tome 114 (2002) no. 3, pp. 387-437 | Article | MR 1924569

[21] Sogge, Christopher D.; Zelditch, Steve Focal points and sup-norms of eigenfunctions, Rev. Mat. Iberoam., Tome 32 (2016) no. 3, pp. 971-994 | Article | MR 3556057

[22] Sogge, Christopher D.; Zelditch, Steve Focal points and sup-norms of eigenfunctions II: the two-dimensional case, Rev. Mat. Iberoam., Tome 32 (2016) no. 3, pp. 995-999 | Article | MR 3556058

[23] Toth, John A.; Zelditch, Steve Riemannian manifolds with uniformly bounded eigenfunctions, Duke Math. J., Tome 111 (2002) no. 1, pp. 97-132 | Article | MR 1876442

[24] Toth, John A.; Zelditch, Steve Norms of modes and quasi-modes revisited, Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), American Mathematical Society (Contemporary Mathematics) Tome 320 (2003), pp. 435-458 | Article | MR 1979955

[25] Zworski, Maciej Semiclassical analysis, American Mathematical Society, Graduate Studies in Mathematics, Tome 138 (2012), xii+431 pages | Article | MR 2952218