On the Lp norm of spectral clusters for compact manifolds with boundary

By Hart F. Smith, Christopher D. Sogge. 2006

115 citations recorded by the source corpus.

We use microlocal and paradifferential techniques to obtain L8 norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal Lq bounds, in the range 2⩽q⩽∞, for L2 - normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp Lq estimates in higher dimensions for a range of exponents q̅n⩽q⩽∞.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2a826687aafa4057a77a18a0700db949416677f3

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In Mathematical Physics

Citations in the corpus

On eigenfunction restriction estimates and $L^4$-bounds for compact surfaces with nonpositive curvature (2011, background): “Smith and the first author in [15], and it would be interesting to obtain analogues of the results in the present paper for this setting, but this appears to be difficult.”
Injections de Sobolev probabilistes et applications (2011, background): “En ce qui concerne l’estimée ex,h ≤ C0Nh, elle est conséquence dans le cas avec bord, des travaux de Sogge [28] pour les conditions aux limites de Dirichlet, puis Smith-Sogge [26] pour les conditions de Neuman.”
Decay estimates for variable coefficient wave equations in exterior domains (2008, methodology): “eneral geometries. See, e.g., [3]. Certain global-in-time Strichartz estimates are also known in exterior domains, but, except for certain very special cases (see [4], [2], which are closely based on [21]), require that the obstacle be strictly convex. Local-in-time estimates were shown in [19] for ”
Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case (2012, methodology): “ of type Lp tL q x are particularly useful. Moreover, the arguments leading to such estimates turn out to be useful when considering spectral cluster estimates, which are of independent interest (see [18]). On any smooth Riemannian manifold without boundary, the following set of so-called Strichartz”
On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles (2008, background): “Recently, there has been work on proving local Strichartz estimates when X = L r (Ω) for more general exterior domains ([5], [6], [3], [34]), but only partial results for a more restrictive range of exponents than the ones described in Corollary 1.6 have been obtained.”
Strichartz estimates for the wave equation on manifolds with boundary (2008, methodology): “Also, we abbreviate Lp(I ;Lq(U)) by LpLq(I × U).”
Random data Cauchy theory for supercritical wave equations I: local theory (2007, unknown): “projector estimate is proved by Smith and Sogge [ 14, Theorem 7.1 ]”
Global existence for energy critical waves in 3-d domains (2006, background): “The main ingredient in the proof is an L(5) spectral projector estimate, obtained recently by Smith and Sogge [12], combined with a precise study of the boundary value problem.”
Kakeya-Nikodym averages and $L^p$-norms of eigenfunctions (2009, background): “Recently, Smith and the author [24] were able to obtain sharp eigenfunction estimates in this case.”