Gagliardo-Nirenberg by liamei12345

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									ASSIGNMENT PROBLEM – GAGLARDO-NIRENBERG


  The Sobolev spaces H s (Rn ) for s > n/2 are subspaces of C0 (Rn ),
                                                                 0
                                         n
the bounded continuous functions on R with supremum norm. In fact
they are also subalgebras, i.e. are closed under pointwise products:
                   H s (Rn ) · H s (Rn ) ⊂ H s (Rn ), s > n/2
and indeed form a C ∞ -algebra, meaning that if f is a smooth function
on Rk then f (u1 , . . . , uk ) ∈ H s (Rn ) if all the ui ∈ H s (Rn ). This result
follows from the Gagliardo-Nirenberg estimates.
   You maybe should start with the case s ∈ N when the estimates are
more straightforward.




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