MATH 307 Final, Spring 2003
American University of Beirut · Topics in Analysis · 1 page
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caent MATVERSIT
LIBWARY
OF BEIRUT
Math 307-Final Exam (Spring 03) B. Shayya
Problem 1
(20 pts) Let N be a positive integer. We showed in class that the estimate
|f(z)|
. wid.
IF Il z2(25(0,20)) S [Bi (0, om! T 1+ Ra)"
holds for all f € L'(R") + L?(R") with Supp f C B(0,1/R), where Cy is a constant that depends
only on N and n. Is it possible to obtain the stronger estimate
Cy
BOA (0, R)| /a(0,2R)
lf llzo(Beo,r |f (a) |dx?
Problem 2
(20 pts) Let E be a compact subset of R" and let 2 < p < co. Suppose there is a measure p € P(E)
with fi € L?(R"). Prove that dim £ > 2n/p.
Problem 3
(20 pts) Prove that to every interval J C R and number B > 0 there is a function € CS°(R) with
lislic= <1 and 16 (@)|dg > B.
Problem 4
(20 pts) Let 1 > €, > € >--++ be a decreasing sequence of positive numbers and let & be a Cantor
set with dissection ratios en}: start with the interval {0, 1], remove the middle €; proportion, then
remove the middle €, proportion of each of the resulting intervals and so forth. Prove that
dim E = —°8?
log <=:
where € = limp_yoo €-
Problem 5
(20 pts) Let
Za €
E={xeER = €, = Oor 1}
Prove that dim F = 0.
Problem 6
(20 pts) Let E be a Lebesgue measurable subset of R”. True or False:
(i) dim B=n => |E|>0?
(ii) dim £=0 => E is countable?