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We consider a class of nonlocal operators associated with an action of a compact Lie group G on a smooth closed manifold. Ellipticity condition and Fredholm property for elliptic operators are obtained. This class of operators is studied using pseudodifferential uniformization, which reduces the problem to a pseudodifferential operator acting in sections of infinite-dimensional bundles.
Sharp estimates are proven for oscillatory integrals with phase functions Φ(x,y), (x,y) ∈ X × Y, under the assumption that the canonical relation projects to T*X and T*Y with fold singularities.
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