PROBLEM 2 (the Scale of Lp-spaces): 
(Note that I had originally posted a different problem.)

In Part (a), the the monotonicity statement is that 

* the intersection of Lp and Lq increases with p
      (and decreases with q). 
  Thus the intersection of L1 and L-infinity is contained in every Lp.

* the sum Lp + Lq decreases with p (and increases with q).
  In particular, L1 + L-infinity contains all Lp.

Here, "small" and "large" is with respect to inclusion.

Part (d): L1 intersection L-infinity cannot be dense
in L-infinity unless X has finite measure.

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PROBLEM 4 (additive => linear?)

Part (b):  The hint yields relatively directly that
e^{if} is smooth. But e^{if} determines f only up to
a multiple of 2pi. To see that f itself is smooth
you may need to revisit your argument from Part (a).

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PROBLEM 5 (Lp-bound on the maximal operator):

Several typos have been corrected:

* In the Maximal Theorem, ||f||_1 must appear on the right
  hand side of the inequality.

A few lines below (between Parts (a) and (b)):

* p is in (1,infinity);
* the formula for ||h_a||_1 is an identity (not an inequality)