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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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    You guys are right. I have edited the question.2011-11-02

1 Answers 1

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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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    Oh yes, you are right!2011-11-03