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LinaresandPonce

LinaresandPonce - Introduction to Non-linear dispersive...

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Introduction to Non-linear dispersive wave equations: Linares and Ponce Sudarshan Balakrishnan August 8, 2011 Chapter 1 Exercise 1.1 Let n 1 and f ( x ) = e - 2 π | x | . Show that: ˆ f ( ξ ) = Γ[( n + 1) / 2] π ( n +1) / 2 (1 + | ξ | 2 ) ( n +1) / 2 Proof In order to prove this statement, we have that by example 1.5 Z -∞ cos( ax ) dx x 2 + b 2 = π b e - ab . (1) For a = β and b = 1, we notice that (1) becomes: e - β = 2 π Z 0 cos( βx ) 1 + x 2 dx. (2) Combining (2) with 1 1 + x 2 = Z 0 e - (1+ x ) 2 ρ dρ, (3) we get, e - β = Z 0 e - ρ ρ e - β 2 4 ρ dρ. (4) The fourier transform of f is defined by, 1
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ˆ f ( ξ ) = Z 0 f ( x ) e - 2 πixξ . (5) Substituting f ( x ) in (5) we get: ˆ f ( ξ ) = Z 0 e - 2 π | x | e - 2 πixξ dx = Z 0 e - 2 π | x |- 2 πixξ dx. (6) The idea is to use the identity in (6) to obtain the desired result. But first, the Gamma function is defined by, Γ( n ) = Z 0 y n - 1 e x dx. (7) In (6), using identity (4) we get: Z 0 Z 0 ρ - 1 2 e - ρ e (2 π | x | +2 πixξ ) dxdρ (8) Integrating (8) by parts we get: Z 0 Z 0 ρ - 1 2 e - ρ e (2 π | x | +2 πixξ ) dxdρ = Γ[( n + 1) / 2] π ( n +1) / 2 (1 + | ξ | 2 ) ( n +1) / 2 (9) The claim follows. Exercise 1.3 Prove Young’s inequality: Let f L p ( R n ), 1 p ≤ ∞ , and g L 1 ( R n ). Then f * g L p ( R n ) with: k f * g k ≤ k f k p k g k 1 . Proof I will try to prove a stronger version of Young’s inequality, the exercise follows almost immediately. For functions, f L p ( R n ) and g L 1 ( R n ) let us prove for p, q, r [1 , ] k f * g k r ≤ k f k p k g k q . (10) In order to see this, we need to use Holder’s Inequality . Holder’s inequality states that for p, q [1 , ] such that 1 q + 1 p = 1 we have: 2
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k fg k 1 ≤ k f k p k g k q . (11) Using (11), we have : | f * g ( x ) | = | Z f ( x - y ) g ( y ) dt | ≤ Z | f ( x - y ) | 1 - t f ( x - y ) | t | g ( y ) | 1 - l | g ( y ) | l dy (12) Z | f ( x - y ) (1 - t ) r g ( y ) (1 - l ) r dy 1 r Z | g ( y ) | lk 2 1 k 2 Z | f ( x - y ) | tk 1 1 k 1 .
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