Tempered distributions and Sobolev spaces

UPMC Basic functional analysis

Master 1, MM05E 2011-2012

Tempered distributions and Sobolev spaces

1) Topology of the Schwartz space.We recall that a functionf ∈ S(R^N)if for any multi-indexesαandβ∈N^N,

sup

x∈R^N

|x^αD^βf(x)|<+∞.

For each n∈Nandf ∈ S(R^N), define

pn(f) := sup

α,β∈N^N,|α|≤n,|β|≤n

sup

x∈R^N

|x^αD^βf(x)|.

For allf andg∈ S(R^N), let

d(f, g) :=X

n∈N

1 2ⁿ

p_n(f−g) 1 +pn(f−g).

Show thatdis a distance onS(R^N)whose induced topology defines a complete metric space.

2) Compute the Fourier transform of the following tempered distributions : a)δ0.

b)1.

c)δ₀⁰, δ⁰⁰₀, . . . , δ^(k)₀ . d)δa.

e)x7→e^−2iπax.

3) Letµ∈ M(R^N)be a bounded Radon measure inR^N. Show that its Fourier transform is given by the function

Fµ(ξ) = Z

R^N

e^−2iπξ·xdµ(x) for allξ∈R^N.

4) Cauchy principal value. For allx∈R^∗, definef(x) := ln|x|.

a) Show thatf ∈L¹_loc(R).

b) Show that the mapping

T :ϕ∈ S(R)7→

Z

R

ϕ(x) ln|x|dx∈R

defines a tempered distribution.

c) Show that the derivative ofT is theprincipal valueof1/x, which is defined by the tempered distribution

pv 1

x

:ϕ∈ S(R)7→lim

ε→0

Z

{|x|>ε}

ϕ(x) x dx.

5) Hilbert transform.For allu∈ S(R), we denote byHu∈ S(R)theHilbert transformofudefined by

Hu:= 1 πu∗pv

1 x

.

a) Show that for everyx∈R,

(Hu)(x) = lim

ε→0

1 π

Z

{|x−y|>ε}

u(y) x−ydy.

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b) Show that its Fourier transform is given by

Hu(ξ) =d −isign(ξ)ˆu(ξ) for allξ∈R. Deduce that for everyu∈ S(R),

kHuk_L2(R)=kuk_L2(R)

c) Deduce thatH extends to an isometry fromL²(R)toL²(R), and thatH²=−I, whereIis the identity mapping overL²(R).

6) Letχ∈ C^∞_c (R)be such that χ= 1 in a neighborhood of0.

a) Show that any functionϕ∈ S(R)admits the following decomposition

ϕ(x) =ϕ(0)χ(x) +xψ(x) ∀x∈R,

for someψ∈ S(R).

b) SolvexT = 0inS⁰(R).

c) SolvexT = 1in S⁰(R).

7) We recall that a tempered distributionu∈ S⁰(R^N)belongs to theSobolev spaceH^s(R^N)(withs∈R) if and only ifuˆ∈L²_loc(R^N)and

Z

R^N

(1 +|ξ|²)^s|ˆu(ξ)|²dξ <∞.

Show thatδ₀∈H^s(R^N)if and only ifs <−N/2.

8) The aim of this exercise is to show that for each s≥0, (H^s(R^N))⁰ can be identified to H^−s(R^N). More precisely, for each T ∈(H^s(R^N))⁰ there exists a uniqueu∈H^−s(R^N)such that

hT, vi_(Hs(R^N))⁰,H^s(R^N)= Z

R^N

ˆ

uˆv dξ for everyv∈H^s(R^N),

and

kTk(H^s(R^N))⁰ =kukH^−s(R^N). – Show that ifu∈H^−s(R^N)andv∈H^s(R^N), thenuˆˆv∈L¹(R^N).

– Deduce that the mappingv7→R

R^Nuˆˆv dξ defines an element of(H^s(R^N))⁰.

– Show that the Fourier transform F is an isometrical isomorphism from H^s(R^N)to L²(R^N, µ), where µis the absolutely continuous Radon measure(1 +|ξ|²)^sL^N.

– LetT ∈(H^s(R^N))⁰, and define T˜:=T ◦ F⁻¹. Deduce thatT˜ is a linear continuous map onL²(R^N, µ).

– Conclude.

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