By Mitsuo Morimoto
This ebook treats round harmonic enlargement of actual analytic features and hyperfunctions at the sphere. simply because a one-dimensional sphere is a circle, the easiest instance of the speculation is that of Fourier sequence of periodic services. the writer first introduces a method of complicated neighborhoods of the field via the Lie norm. He then reviews holomorphic services and analytic functionals at the advanced sphere. within the one-dimensional case, this corresponds to the learn of holomorphic capabilities and analytic functionals at the annular set within the advanced airplane, counting on the Laurent sequence growth. during this quantity, it really is proven that an identical inspiration nonetheless works in a higher-dimensional sphere. The Fourier-Borel transformation of analytic functionals at the sphere can be tested; the eigenfunction of the Laplacian should be studied during this method.
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Extra info for Analytic Functionals on the Sphere
2. Let f, g ∈ S, and let μ be a (say) compactly supported measure. Then (75) fˆgˆdμ = (ˆ μ ∗ g) · f dx. Proof. Recall that gˆ˜ = gˆ, so that ˆ gˆˆ = gˆ = g 7. THE RESTRICTION PROBLEM 49 by the inversion theorem. Now apply the duality relation and (74), obtaining fˆgˆdμ = f · (ˆ g μ)ˆdx f · (g ∗ μ ˆ)dx = as claimed. 3. Let μ be a ﬁnite positive measure. The following are equivalent for any q and any C. (1) f dμ q ≤ C f 2 , f ∈ L2 (dμ). (2) gˆ L2 (dμ) ≤ C g q , g ∈ S. (3) μ ˆ ∗ f q ≤ C 2 f q , f ∈ S.
Morse Lemma Suppose Ω ⊂ Rn is open, f : Ω → R is C ∞ , p ∈ 2f Ω, ∇f (p) = 0, and suppose that the Hessian matrix Hf (p) = ∂x∂i ∂x (p) is j invertible. Then, for a unique k (= number of positive eigenvalues of Hf ; see 37 38 6. 3 below) there are neighborhoods U and V of 0 and p respectively and a C ∞ diﬀeomorphism G : U → V with G(0) = p and k n f ◦ G(x) = f (p) + x2j j=1 − x2j . j=k+1 We consider now I(λ) ﬁrst when a is supported near a regular point, and then when a is supported near a nondegenerate critical point.
This gives the result, since |ˆ a(ξ)|ξ|2N +2 dξ is bounded in terms of Schwartz space seminorms of a ˆ, and therefore in terms of derivatives of a. Now we consider the case of a general phase function with a nondegenerate critical point. It is clear that this should be reducible to the Gaussian case using the Morse lemma and remark 3. above. 1, since we need to obtain the correct form for the asymptotic expansion. 3. Suppose that φ is smooth, ∇φ(p) = 0 and G is a smooth diﬀeomorphism, G(0) = p.
Analytic Functionals on the Sphere by Mitsuo Morimoto