Math 554
Homework 2
Due September 25, 2008, before class
Problem I Preliminaries:
1. Show that the sequence {an}has a limit if and only if lim inf(an) =
lim sup(an).
2. Prove the following variant of Minkowski Inequality: If f , g are
measurable functions on Rnthen:
essup|f+g| ≤ essup|f|+ essup|g|
Problem II Weak derivatives
1. Show that the piecewise C1functions which are also continuous
have weak derivative which coincide a.e. with the classical deriv-
ative. What happens if the continuity hypothesis is removed?
2. (suggested) Find a function defined on Rwith classical derivative
almost everywhere and the derivative locally integrable such that
the function has no weak derivative.
3. Give an example of a continuous function that has classical deriv-
atives almost everywhere but has no weak derivative.
4. Let fbe locally integrable on R. Show that F(x) = Rx
0f(s)ds
has a weak derivative equal to falmost everywhere. (suggested)
Show that Fhas classical derivative almost everywhere.
5. If fis locally integrable and for all φ∈C∞
c(Rn)
ZRn
fφdx = 0
then show that f= 0 a.e.
Problem III Sobolev spaces. Here Uis an open set in Rn.
