# Third (delayed) Problem set for 18.155 Due October 17 in class or 2-174.

These are Problems 24 - 30 of the notes on the web.

Problem 1   Prove the continuity in the mean of functions on that

 as

You will probably have to go back to first principles to do this. Show that it is enough to assume has compact support. Then show it is enough to assume that is a simple, and integrable, function. Finally look at the definition of Lebesgue measure and show that if is Borel and has finite Lebesgue measure then

where Lebesgue measure and

Problem 2   Prove Leibniz' formula

for any functions and and . Here and are multiindices, means for each and

I suggest induction!

Problem 3   Prove the generalization of a result from class that , (which means that for all such that in for some implies there are constants for , for some , such that

Hint This is not so easy! I would be happy if you can show that , implies To see this, you can show that

To prove the general case you need something similar -- that given if and for then , in such that in the norm.

Problem 4   If , show that and for all implies . Is the converse true?

Problem 5   Show that every element can be written as a sum

Problem 6   Consider for , the locally integrable function (the Heaviside function),

Show that ; what is the constant ?

Problem 7   For what range of orders is it true that ?