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Additional info for Acta Numerica 1996: Volume 5

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Satisﬁes n r = CR r for any r ∈ [1, ∞], where C = φ r . Also, by the chain rule ∇ψ (58) 1 = R φ 1. We know that f = ψ ∗ f . In the case of ﬁrst derivatives, 1. therefore follows from (57) and (24). The general case of 1. then follows by induction. , let r satisfy 1q = 1p − r1 . Apply Young’s inequality obtaining f ψ∗f q ψ r f p = ≤ q n Rr f = R p n( p1 − 1q ) f p. We now extend the Lp → Lq bound to ellipsoids instead of balls, using change of variables. An ellipsoid in Rn is a set of the form (59) E = {x ∈ Rn : j |(x − a) · ej |2 ≤ 1} rj2 for some a ∈ Rn (called the center of E), some choice of orthonormal basis {ej } (the axes) and some choice of positive numbers rj (the axis lengths).

J 1Actually the order is exactly 2j but we have no need to know that. 6. THE STATIONARY PHASE METHOD 41 (ii)Assume that ∇φ(p) = 0, and Hφ (p) is invertible. Then, for a supported in a small neighborhood of p, n dk πiλφ(p) (e I(λ)) ≤ Ck λ−( 2 +k) . k dλ Proof. 1 after diﬀerentiating under the integral sign as in the proof below. For (ii) we need the following. Claim. Let {φi }M i=1 be real valued smooth functions and assume that φi (p) = 0, ∇φi (p) = 0. Let Φ = ΠM i=1 φi . Then all partial derivatives of Φ of order less than 2M also vanish at p.

All the basic formulas for the L1 Fourier transform extend to the L1 +L2 Fourier transform by approximation arguments. This was done above in the case of the duality relation. Let us note in particular that the transformation formulas in Chapter 1 extend to L1 + L2 . For example, in the case of (5), one has (36) f ◦ T = |det (T )|−1 fˆ ◦ T −t if f ∈ L1 + L2 . Since we already know this when f ∈ L1 , it suﬃces to prove it when f ∈ L2 . Choose {fk } ⊂ L1 ∩ L2 , fk → f in L2 . Composition with T is continuous on L2 , as is Fourier transform, so we can write down (36) for the {fk } and pass to the limit.