Prove that $f \ast g$ is continuous and bounded if $f\in L^1(R^n)$ and $g\in L^\propto (R^n)$

Question

My Engliah is no so good and it is my first time to use this website, so I apologize for it if I didnot make myself clearly:)

Answer 1

Bounded is straightforward: $$|f \ast g(x)| \le \int_{\mathbf R^n} |f(y)| |g(x-y)| \, dy \le ||g||_\infty \int_{\mathbf R^n} |f(y)| \, dy = ||f||_1 ||g||_\infty$$

As for continuity, observe that $$|f\ast g(x) - f\ast g(z)| = \left| \int_{\mathbf R^n} \left( f(y-x) - f(y-z) \right) g(y) \, dy \right| \le ||g||_\infty \int_{\mathbf R^n} |f(y-x) - f(y-z)| \, dy.$$

The fact that the right-hand side tends to zero as $|x-z| \to 0$ is a standard result in $L^1$ theory.