convolution $f * g$ decays at infinity for $f\in L^p(\mathbb R), g\in L^q(\mathbb R)$

Question

Here $p$ and $q$ are conjugate exponents with $1<p<\infty$. This question is from Terence Tao's harmonic analysis exercise. It is easy to prove that $f*g \in L^\infty$ by Minkowski-Young inequality, but how can I prove the decay condition?

Answer 1

Hint: If $f$ and $g$ are in $C_c(\mathbb R$) then $f*g$ actually has compact support so the result is true. Now approximate $f$ and $g$ by functions in $C_c(\mathbb R$) and use the fact that $$|(f*g)(x)-(f*g)(x')|\leq \int |f(x-y) -f(x'-y)| |g(y)| dy$$ $$ \leq 2\|f\|_p \|g\|_q$$ for all $x$ and $x'$.