2015/12/07 by Pablo Ramacher, Ramacher, Pablo
Mathematics · #14E15 #35P20 #42B70 #57S15 #57S17 #58J40 #58J50 #58K55 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Analysis and Transform Methods #Nonlinear Partial Differential Equations #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1512.02193
openalex publication_date 2015/12/07 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Let M be a compact boundaryless Riemannian manifold, carrying an effective\nand isometric action of a compact Lie group G, and P0 an invariant\nelliptic classical pseudodifferential operator on M. Using Fourier integral\noperator techniques, we prove a local Weyl law with remainder estimate for the\nequivariant (or reduced) spectral function of P0 for each isotpyic component\nin the Peter-Weyl decomposition of L2(M), generalizing work of\nAvacumovi vc, Levitan, and H "ormander. From this we deduce a generalized\nKuznecov sum formula for periods of G-orbits, and recover the local Weyl law\nfor orbifolds shown by Stanhope and Uribe. Relying on recent results on\nsingular equivariant asymptotics of oscillatory integrals, we further\ncharacterize the caustic behaviour of the reduced spectral function near\nsingular orbits, which allows us to give corresponding point-wise bounds for\nclusters of eigenfunctions in specific isotypic components. In case that G\nacts on M without singular orbits, we are able to deduce hybrid Lp-bounds\nfor 2 \≤ p \≤ \∞ in the eigenvalue and isotypic aspect that improve\non the classical estimates of Seeger and Sogge for generic eigenfunctions. Our\nresults are sharp in the eigenvalue aspect, but not in the isotypic aspect, and\nreduce to the classical ones in the case G= e .\n