Oscillatory integrals I(λ) generally have the form
and become interesting at large (and real) λ. Usually it's assumed that the phase and amplitude are real-valued. The principle of stationary phase says that while the gradient of the phase is non-vanishing, the integral rapidly decreases in λ and the dependency becomes negligible, and the main contribution of I(λ) is reduced to the points, where the gradient of the phase vanishes. For |∇Φ(x)| ≥ c > 0 for all x in the support of φ, then for every N ≥ 0,
For N = 1, c can be assumed to be 3. Check first the case for a static phase for some point x, and a non-degenerate critical point. For a phase of x²
The curvature form or second fundamental form of a surface M is
restricted to tangent vectors to M. It is independent of normalized defining functions.
where the eigenvalues λ are the "principal curvatures" of M at a critical point, and their product forms the total curvature / Gauss curvature of M. Next check the induced Lebesgue measure dσ on M. For any continuous function on M with compact support,
A measure dμ is a surface-carried measure on M with smooth density, if dμ = φdσ, and φ is a C[∞] function of compact support.
For hypersurfaces M with surface-carried measures dμ, define a map A
A is the averaging operator. In Riesz diagrams, A forms a closed triangle in the (1/p, 1/q) plane with (0,0) (1,1), (d/(d+1), 1/(d+1)). For a hypersurface M with at least m non-vanishing principle curvatures, one can substitute as follows: k = m/2, p = (m+2)/(m+1), q = m+2. A version of the Riesz interpolation theorem in which the operator is allowed to vary, which requires an analytic family of operators. These can be defined on the strip S = {a ≤ Re(s) ≤ b} by a linear map T(s): ℝᵈ → ℝᵈ that are locally integrable. For two simple functions f, g, define the function Φ that is continuous, and bounded and analytical in S, and through which one can define the 1-dimensional Fourier transformation I.
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