Convolution operator in $L^p$

Question

This is exercise 10.16 from Measure and Integral by Wheeden and Zygmund, the major part is easy, but what exactly is the "generalization"? I'm not sure what to prove.

Answer 1

First, you had the assumption $$ \|Tf\|_p \leq M\|f\|_p \quad \forall f $$ and then you were able to prove that $$ \|Tf\|_{p'} \leq M \|f\|_{p'} \quad \forall f $$ with $p'$ being the conjugate exponent.

Now for the generalization you have the assumption $$ \|Tf\|_q \leq M\|f\|_p \quad \forall f $$ with $p\neq q$. The case $p=q$ was already the first part, and hence the word "generalization". The way i understand it, you should now prove a result of the type $$ \|Tf\|_r \leq \tilde M\|f\|_s \quad \forall f $$ with suitable $r,s$ and $\tilde M$ (An important part of the problem is to figure out which values for $r,s$ should be chosen).