This is Proposition 8.8 in Folland's Real Analysis:
If $p$ and $q$ are conjugate exponents, $f \in L^P$, and $g \in L^q$, then $f*g(x)$ exists for every $x$, $f*g$ is bounded and uniformly continuous, and $\|f*g\|_u \leq \|f\|_p\|g\|_q$. If $1 < p < \infty$, then $f*g \in C_0(\mathbb R^n)$.
I have a question about the proof of the estimate $$ \|f*g\|_u \leq \|f\|_p\|g\|_q . $$ It says that this estimate follows immediately from Holder's inequality, but it seems to me that Holder would give a bound for $\|f*g\|_1$, not $\|f*g\|_u$. Am I missing something?
Just use the definition: $$f*g(x)=\int_{\mathbb{R}} f(y)g(x-y)\,dy$$ since $f\in L^p$ and $g\in L^q$ (so also $\bigl(y\mapsto g(x-y)\bigr)\in L^q$, with the the same norm as $g$), and Hölder gives you the uniform estimate.