PROBLEM SET 5, 18.155
DUE MIDNIGHT FRIDAY 14 OCTOBER, 2011
(1) Show that if s ∈ R and u ∈ S 0 (Rn ) satisfies
|u(φ)| ≤ CkφkH s ∀ φ ∈ S(Rn )
R
then u ∈ H −s (Rn ); here kφk2H s = hξi2s |φ̂(ξ)|2 .
(2) Show that the Dirac delta ‘function’, δ(φ) = φ(0) for all φ ∈
n
S(Rn ) is an element of H − 2 − (Rn ) for any > 0.
(3) Suppose m ∈ R and u ∈ H m (Rn ). Show that if Djk u ∈ H m (Rn )
for fixed k ∈ N and all j = 1, . . . , n then u ∈ H m+k (Rn ).
(4) Derive the ‘Hölder-type’ characterization of the spaces H s (Rn )
for 0 < s < 1, that u ∈ L2 (Rn ) and
Z
|u(x) − u(y)|2
dxdy < ∞.
|x − y|n+2s
R2n
Hint: Look it up! Or proceed formally by changing variable to
x = y + t; then use Parseval’s identity to rewrite the y integral
in terms of the Fourier transform arriving at a constant multiple
of
Z
|û(η)|2 |eiη·t − 1|2 |t|−n−2s dtdη.
R2n
Now, think about the t integral, note that this is integrable
using the observation that |t|s is integrable near zero if s > −n
(remember Taylor’s formula at t = 0 for exp(iη · t)) and near
infinity if s < −n. The result is a multiple of |η| (try scaling t
by |η|) and so combined with the L2 norm bounds the H s norm
above and below. Now go back and make sure it is all rigorous
– maybe use the density of S(Rn ) in H s (Rn ).
(5) Show that integration in the last variable and ‘extension as
constant in the last variable’ are well-defined maps
In : S(Rn ) −→ S(Rn−1 ),
Z
In φ(x1 , . . . , xn−1 ) =
φ(x1 , . . . , xn−1 , dxn )dxn ,
En : S 0 (Rn−1 ) −→ S 0 (Rn ), En (u)(φ) = u(In (φ))
1
2
PROBLEM SET 5, 18.155 DUE MIDNIGHT FRIDAY 14 OCTOBER, 2011
and that these fit into short exact sequences
0
S(Rn )
Dn
S 0 (Rn−1 )
En
/
0
/
/
/
S(Rn )
In
S 0 (Rn )
Dn
/
/
S(Rn−1 )
/
S 0 (Rn )
/
0
0.