For an exercise I'm being asked to prove that, given two functions $f,g \in\mathcal{L}^2(\mathbb{R}^d)$, their convolution $ f \star g$ is "almost everywhere defined, and in $\mathcal{L}^\infty$ once we fill in the undefined points. But it looks like to me like $ f \star g $ should be defined everywhere. My "proof" goes by noting that, if $x \in \mathbb{R}^d$, the function given by $t \mapsto g(x-t) $ is also squared integrable, since its square norm is the same one as $g$'s by substitution. So we should have:
$$ f \star g (x) = \int_{\mathbb{R}^d} f(t)g(x-t) \, dt \leq \lVert f \rVert_2 \lVert g \rVert_2 $$
by the Hölder inequality, which also gives us that the function is in $\mathcal{L}^\infty$.
Can anyone tell me why this doesn't work? Or is the convolution truly defined anywhere?
Your teacher probably meant $\int_{\mathbb{R}^d} f(t)g(x-t) \, dt$ as a Riemann Stjejes integral. If you interpret it as a Lebesgue integral :
By Cauchy-Schwarz $$| f \ast g(x)| = |\langle f(.) , g(x-.) \rangle| \le \|f\|_{L^2} \|g\|_{L^2}$$ is well-defined.
Also letting $\varphi_\epsilon(x) = \epsilon e^{-|x|^2/\epsilon^2}$, for every $\epsilon > 0$ we have $$\lim_{h \to 0}\|\varphi_\epsilon \ast (g(.+h)-g(.))\|_{L^2} = \lim_{h \to 0}\|g \ast (\varphi_\epsilon(.+h)-\varphi_\epsilon(.))\|_{L^2}=0$$ so that $$\lim_{h \to 0}\|g(.+h)-g(.)\|_{L^2} = 0$$ hence $$| f \ast g(x+h)- f \ast g(x)| \le \|f\|_{L^2} \|g(.+h)-g(.)\|_{L^2} $$ shows that $f \ast g$ is uniformly continuous