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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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