By Alex Iosevich

ISBN-10: 3034806248

ISBN-13: 9783034806244

ISBN-10: 3034806256

ISBN-13: 9783034806251

The Plancherel formulation says that the *L*^2 norm of the functionality is the same as the *L*^2 norm of its Fourier rework. this means that not less than on typical, the Fourier remodel of an *L*^2 functionality decays at infinity. This e-book is devoted to the examine of the speed of this decay less than a number of assumptions and conditions, a ways past the unique *L*^2 surroundings. Analytic and geometric homes of the underlying capabilities have interaction in a unbroken symbiosis which underlines the big variety impacts and functions of the strategies lower than consideration.

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**Example text**

Without loss of generality we may assume that in a neighborhood of the point x ¯ the surface S is given by the equation xd = g(x1 , . . , xd−1 ). Thus x = (x1 , . . , xd−1 ) may be regarded as local coordinates on S. The statement about the Legendre transform follows from [159] and [213]. 23) holds along this curve. We may choose this curve to be γ = {θ : θi = θ¯i , i = 1, . . , d}, θ0 is a parameter on γ. Since the dual of S is S, the hyperplane tangent to S at the point θ¯ is given by the equation d x ¯i θi + θ0 = 0, i=1 in which x¯ is the point on ∂D deﬁned above.

2. Indeed, −1 e−2πi(tξ1 +γ(t)ξ2 ) ψ(t) dt = e−2πiξ2 (tξ1 ξ2 +γ(t)) ψ(t) dt. 2 implies that −1 e−2πiξ2 (tξ1 ξ2 +γ(t)) ψ(t) dt ≤ C|ξ2 | − 12 . Since we are in the regime where |ξ1 | ≤ C|ξ2 |, the right-hand side above is −1 bounded by C|ξ| 2 and the proof is complete. , calculations with Hardy’s L-functions (see [91]) by Miyachi in [145]. Let us give a couple of versions of the Stationary Phase Method theorems in several dimensions. The ﬁrst one can be found in [70]. 4. 5) × | det S (v0 )|− 2 ϕ(v0 ) + O R−1 1 k−1 +R 1−d 2 e aj R−j + O R iRS(v0 ) 1−d 2 −k , j=1 0 where v0 = (v10 , v20 , .

N N N m+2≤dist(k,∂(N S))≤CN u1 um λ; , . . 44) 1 2 . We may estimate from above each term in the last sum by the maximal value of λ at the points with integer coordinates. This yields m Δk 0≤dist(k,∂(N S)) ≤m+2 N u1 um λ; , . . , N N 2 ≤C 0≤dist(k,∂(N S)) ≤m+2 2 d−1 2 m+2 N . , [204]. Therefore, the last sum is not greater than CN −2 d−1 2 1 ≤ C1 N d−1−2 d−1 2 = O(1). 44). Use m times the mean value theorem for the directional derivative. The sum being estimated is transformed to the form ∂ mλ N −2m m+2≤dist(k,∂(N S))≤CN k N + σ1 uN1 + · · · + σm uNm ∂u1 · · · ∂um 2 .

### Decay of the Fourier Transform: Analytic and Geometric Aspects by Alex Iosevich

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