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On the $$L^p$$ norm of spectral clusters for compact manifolds with boundary. (English) Zbl 1189.58017
Obtaining bounds, for example, the $$L^p$$-norm, on eigenfunctions of large eigenvalues of Riemannian manifolds is an important problem. Similarly, obtaining $$L^p$$-norm on spectral clusters of elliptic operators is also an important problem. In this paper, the authors obtain $$L^p$$-bounds on spectral clusters of an elliptic, second order differential operator with vanishing zero-order term on a compact 2-dimensional manifold $$M$$ with boundary.
Specifically, let $$P$$ be an elliptic, second-order differential operator on $$M$$, self-adjoint with respect to a density $$d\mu$$, and with vanishing zero-order term, so that in local coordinates the operator can be written as $(Pf)(x)=\varrho(x)^{-1}\sum_{j,k=1}^{n} \partial_j(\varrho(x)g^{jk}(x)\partial_k f(x)),\qquad d\mu=\varrho(x)dx,$ where the $$g^{jk}$$’s are positive, so that the Dirichlet eigenvalues of $$P$$ can be written as $$\{-\lambda_j^2\}_{j=0}^\infty$$.
Let $$\chi_\lambda$$ be the projection of $$L^2(d\mu)$$ onto the subspace spanned by Dirichlet eigenfunctions for which $$\lambda_j\in[\lambda,\lambda+1]$$.
If $$M$$ is of dimension 2 and the boundary has a point of strict convexity with respect to the metric $$g$$, it was known that the following bounds cannot be improved: $||\chi_\lambda f||_{L^q(M)}\leq C\lambda^{(2/3)(1/2-1/q)}|| f||_{L^2(M)},\quad 2\leq q\leq 8$
$||\chi_\lambda f||_{L^q(M)}\leq C\lambda^{2(1/2-1/q)-1/2}|| f||_{L^2(M)}, \quad 8\leq q\leq\infty.$
In the paper the authors show that the above estimates hold on any 2-dimensional compact manifold with boundary, for $$P$$ as above and either Dirichlet or Neumann conditions assumed.

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 35J15 Second-order elliptic equations 35P15 Estimates of eigenvalues in context of PDEs 47F05 General theory of partial differential operators 47G10 Integral operators
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##### References:
 [1] Calderón, A. P.: Commutators of singular integral operators. Proc. Nat. Acad. Sci. U.S.A., 53, 1092–1099 (1965) · Zbl 0151.16901 [2] Coifman, R., Meyer, Y.: Commutateurs d’intégrales singulières et opérateurs multi-linéaires. Ann. Inst. Fourier (Grenoble), 28(3), xi, 177–202 (1978) · Zbl 0368.47031 [3] Córdoba, A., Fefferman, C.: Wave packets and Fourier integral operators. Comm. Partial Differential Equations, 3, 979–1005 (1978) · Zbl 0389.35046 [4] DeLeeuw, K. Unpublished, (1967) [5] Grieser, D.: L p Bounds for Eigenfunctions and Spectral Projections of the Laplacian near Concave Boundaries. Ph.D. Thesis, UCLA, Los Angeles, CA (1992) [6] Grieser, D,: Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary. Comm. Partial Differential Equations, 27, 1283–1299 (2002) · Zbl 1034.35085 [7] Koch, H., Tataru, D.: Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure Appl. Math., 58, 217–284 (2005) · Zbl 1078.35143 [8] Mockenhaupt, G., Seeger, A., Sogge, C. D.: Local smoothing of Fourier integral operators and Carleson–Sjölin estimates. J. Amer. Math. Soc., 6, 65–130 (1993) · Zbl 0776.58037 [9] Rayleigh, J. W.: The problem of the whispering gallery. Philos. Mag., 20, 1001–1004 · JFM 41.0911.02 [10] Rayleigh, J. W., Strutt, B.: The Theory of Sound. Dover, New York, NY (1945) · Zbl 0061.45904 [11] Smith, H. F., Spectral cluster estimates for C 1,1 metrics. Amer. J. Math., 128, 1069–1103 (2006) · Zbl 1284.35149 [12] Smith, H. F.: Sharp L 2 q bounds on spectral projrctorsd for low regularity metrics. Math. Res. Lett., 13, 967–974 (2006) · Zbl 1287.35008 [13] Smith, H. F., Sogge, C. D.: On the critical semilinear wave equation outside convex obstacles. J. Amer. Math. Soc., 8, 879–916 (1995) · Zbl 0860.35081 [14] Sogge, C. D.: Concerning the L p norm of spectral clusters for second-order elliptic operators on compact manifolds. J. Funct. Anal., 77, 123–138 (1988) · Zbl 0641.46011 [15] Sogge, C. D.: Fourier Integrals in Classical Analysis. Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge (1993) · Zbl 0783.35001 [16] Sogge, C. D.: Eigenfunction and Bochner Riesz estimates on manifolds with boundary. Math. Res. Lett., 9, 205–216 (2002) · Zbl 1017.58016 [17] Stein, E. M.: Unpublished (1967) [18] Tataru, D.: Strichartz estimates for operators with nonsmooth coefficients and the non-linear wave equation. Amer. J. Math., 122, 349–376 (2000) · Zbl 1057.11506 [19] Tataru, D.: Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III. J. Amer. Math. Soc., 15, 419–442 (2002) · Zbl 0990.35027 [20] Taylor, M. E.: Pseudodifferential Operators and Nonlinear PDE. Progress in Mathematics, 100. Birkhäuser, Boston, MA (1991) · Zbl 0746.35062
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