Equality condition for convolution's $L^p$ norm.

Question

Suppose that $1< p< \infty$, $f\in L^1(R)$, and $g\in L^p(R)$ and that $\|f*g\|_p=\|f\|_1\|g\|_p$. Show that then either $f=0$ a.e. or $g=0$ a.e. I have solved for $g=0$ a.e. if $\|f\|_1>0$ using the equality condition for Hölder ; $\alpha f^p=\beta g^q$ for some $\alpha, \beta \ge 0$. Is this still useful for supposing $\|g\|_p>0$ and deriving $f=0$ a.e?

Answer 1

I am 6 years late but I might as well answer. What you have shown is enough. You have shown that if $||f||_1 > 0$ then $g = 0$ a.e.

Now assume $||g||_p > 0$. This means "$g=0$ a.e" doesn't hold. So the contrapositive of what you have shown means $||f||_1 = 0$, so $f = 0$ a.e.

Answer 2

In general, if you want to prove a statement of the form $A\implies (B\ \text{or}\ C)$, it is logically equivalent to show $(A\ \text{and}\ (\text{not}\ B))\implies C$. The other answer essentially shows how this works in this particular case, but also you could convince yourself of this by making a truth table with columns

• $A$,
• $B$,
• $C$,
• $A\implies (B\ \text{or}\ C)$,
• $(A\ \text{and}\ (\text{not}\ B))\implies C$,

and then checking that no matter what values of $0$ or $1$ ($\text{False}$ or $\text{True}$) you assign to the first three statements $A, B, C$, both of the last two statements will agree.