PROBLEM 3 (Young's inequality)

In Part (b), you are asked to apply the Riesz-Thorin 
interpolation theorem (Theorem 6.27 in Folland). In 
the case of R^n, with Lebesgue measure, the theorem 
works as follows:

ASSUMPTION: Given two pairs of exponents p0, p1 and 
q0, q1 in [1, infinity].  Suppose T defines a 
bounded linear transformation 
    from L^{p0} to L^{q0} (with norm M0), 
and also 
    from L^{p1} to L^{q1} (with norm M1). 
(See Note 1 below).  

INTERPOLATION: Given p  between p0 and p1, we determine 
t in[0,1] so that 
    1/p = (1-t)/p0 + t/p1 (linearly interpolate the reciprocals). 
Define q by 
    1/q=(1-t)/q0 + t/q1   (i.e., likewise interpolate for 1/q),
and set 
    M= (M0)^{1-t) (M1)^t  (the corresponding geometric mean).
(See Note 2)

CONCLUSION: T defines a bounded linear operator 
    T: L^p -> L^q (with norm less or equal to M).

Problem 3b asks to apply this to the convolution operator.
At the endpoints (p0,q0) and (p1, q1), use Holder's inequality.

***

NOTE 1: Technically, we are dealing with two different operators
    T0: L^{p0} -> L^{q0} 
    T1: L^{p1} -> L^{q1}
that agree on the intersection of L^{p0} and L^{p1}. 
Folland views T as an operator on the larger space 
L^{p0} + L^{p1}, which contains both L^{p0} and L^{p1}.

NOTE 2: It's helpful to sketch, in the unit square, the line 
segment with endpoints (1/p0, 1/q0) and (1/p1, 1/q1).

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PROBLEM 4 (differentiating the L^p-norm): 

(b) Typo: The "Re" (real part) in front of the 
integral was missing when I first posted the problem

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PROBLEM 6 (difference of L^1-norms)

You may assume that the measure space is sigma-finite
(in order to apply Fubini's theorem, Theorem 2.37 of Folland).

(This assumption is actually not necessary, because
     S_0 = {x: f(x)+g(x)>0} 
is the union of the countably many sets
     S_{1/n} = {x: f(x)+g(x)>1/n},   n=1,2, ... ,
each of which has finite measure)