Any important inequalities with convolutions?

[jostpuur]

I am interested to know who to find upper bounds for derivatives of convolutions. If I know something of $f$ and $g$, are there any major results about what kind of numbers $C_{f,g}$ exist such that

$$|D_x((f *g)(x))| \leq C_{f,g} ?$$

 

[jostpuur]

Truth is that I didn't think about that much before posting. I thought that it would probably be better to ask about old results, before trying to come up with own ones.

But now, after very short thinking, I already came up with one very natural result. It seems that the following is true:

$$|D_x(f*g)(x)| \leq \|f'\|_{\infty} \|g\|_1$$

It could be that this is what I was after. If somebody has something else to add, I'm still all ears.

 

[jostpuur]

Suppose $\psi$ is a function, which is mostly smooth, but has a little spike somewhere so that $\psi'$ jumps badly. Also suppose that the spike is so small that the values of $\psi$ don't jump very much. Only the derivative jumps. And suppose that $\varphi$ is some typical convolution kernel, which is approximately a delta function, but still so wide that it makes the spike in $\psi$ almost vanish.

It should be possible to prove that $D_x(\varphi *\psi)(x)$ is almost the same as $\psi'(x)$ with exception of the $x$ that is close to the little spike. Close to the spike $\psi'(x)$ jumps, but $D_x(\varphi *\psi)(x)$ behaves as if the spike did not exist.

Approximation

$$|D_x(\varphi *\psi)(x)| \leq \|\varphi'\|_{\infty} \|\psi\|_1$$

is useless because $\|\varphi'\|_{\infty}$ is very large, and

$$|D_x(\varphi *\psi)(x)| \leq \|\psi'\|_{\infty} \|\varphi\|_1$$

is useless too because $\|\psi'\|_{\infty}$ is very large because of the spike.

So there must be some other upper bound for $|D_x(f *g)(x)|$, better than the one I mentioned in the #2 post.