Injections, Embeddings and the Trace Theorem

We have showed previously that the Fourier transform is an isometric isomorphism on L2 = CcR ∩ L2Rn. In particular, we have a ∀u ∈ L2 ||u||0 = ||û||0 Parseval relation b ∀u,v ∈ L2, u,v0 = û, v̂0 Plancherel relation The results (a) and (b) hold for every test function, and since the test functions are dense in L2Rn, they extend to L2Rn by continuity. Then the Fourier transform can be extended to L2Rn by continuity as well. In addition, i if u,Dαu ∈ L2 then TFDu = izûz ∈ L2 ii if u,xαux ∈ L2 then TFxux = iDzûz ∈ L2. i.e., TFDu = ∫ Rn Dαuxe−ix⋅zdx̂ = −1 |α | ∫ Rn uxDxe dx̂