Math 554
Homework 4
Due October 30, 2008, before class
Problem I Prove that W1,∞(Rn),→C0,1(Rn). (Hint: use mollifiers but be
careful because C∞(Rn) is NOT dense in W1,∞(Rn).)
Problem II Let U⊂Rnbe bounded and 0 < ν < λ ≤1.Show that
C0,λ(¯
U) is compactly imbedded in C(¯
U) and C0,ν (¯
U). Also show that
for m∈N, Cm,λ (¯
U) is compactly imbedded in Cm(¯
U) and Cm,ν (¯
U).
Problem III Let U⊂Rnbe measurable and Sa precompact set in Lp(U).
Show that for any > 0 :
(i) there exists a compact G⊆Usuch that
ZU\G
|f(x)|pdx < p∀f∈S;
(ii) there exists δ > 0 such that for all h∈Rn,|h|< δ and all f∈S
we have ZU
|˜
f(x+h)−˜
f(x)|pdx < p,
where ˜
fis the extension of fby zero outside U.
Problem IV Let Xbe a Banach space, U⊆Rnbe measurable, and 1 ≤
q1< q2.Assume that X ,→Lq2(U) and Xis compactly imbedded in
Lq1(U).Show that Xis compactly imbedded in Lq(U) for all q1≤q <
q2.
