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Let $f$ and $g$ be functions on $\mathbb{R}^n$. Let $x_0$ be a given point in the unit ball $B(0,1)$. I am looking for sufficient conditions for the convolution $$ (f \ast g)(x) = \int_{B(0,1)} f(y)g(x-y) dy $$ to be continuous at $x_0$.

I would appreciate simple proofs or references to proofs that conditions given in an answer are sufficient.

In my specific application, $f$ and $g$ are continuous in $B(0,1) \setminus \{0\}$ and $x_0 \neq 0$, but I would be very interested to see conditions for other (more general) situations as well.

I would also be very interested to see conditions for the situation where $B(0,1)$ is replaced by $\mathbb{R}^n$.

Thanks very much!

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maxpower you have to extend the domain of $g$ a bit more in order to compute $g(x-y)$ for any $x\in B(0,1)$. I guess that you have to let $g$ to be defined at least in $B(0,2)$. – Leandro Nov 2 '11 at 3:13
in the OP's application $x_0 = 0$, so $h(x_0) = (f \ast g)(x_0)$ does make sense but asking whether $h$ is continuous at $x_0$ does not since the $h$ is only defined at the single point $x_0$. However, if you let $g$ be define on $B(0, 1 + \epsilon)$ for any $\epsilon > 0$, then $h$ is defined on a neighborhood of $x_0$ so the question makes sense. – user12014 Nov 2 '11 at 4:09
You guys are right. I have edited the question. – maxpower Nov 2 '11 at 4:52

Continuity is not needed in general, but sufficient integrability is helpful: Let $p>1$ and $q$ be conjugate exponents ($1/p + 1/q = 1$) and $f\in L^p(\mathbb{R}^n)$, $g\in L^q(\mathbb{R}^n)$. Then by Hölder's inequality for any $x,h\in\mathbb{R^n}$: $$ |f*g(x+h) - f*g(x)| \leq \|f\|_{L^p} (\int |g(x+h-y) - g(x-y)|^q dy)^{1/q}. $$ The last term vanishes for $h\to 0$ since $L^q$ functions are "continuous in the $q$th mean" (which can be seen by density of the continuous functions with compact support).

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$q\ne\infty$ for sure. – AD. Nov 3 '11 at 13:07
Oh yes, you are right! – Dirk Nov 3 '11 at 13:59

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