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

<br /> |D_x((f *g)(x))| \leq C_{f,g} ?<br />

 

[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:

<br /> |D_x(f*g)(x)| \leq \|f&#039;\|_{\infty} \|g\|_1<br />

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&#039; 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&#039;(x) with exception of the x that is close to the little spike. Close to the spike \psi&#039;(x) jumps, but D_x(\varphi *\psi)(x) behaves as if the spike did not exist.

Approximation

<br /> |D_x(\varphi *\psi)(x)| \leq \|\varphi&#039;\|_{\infty} \|\psi\|_1<br />

is useless because \|\varphi&#039;\|_{\infty} is very large, and

<br /> |D_x(\varphi *\psi)(x)| \leq \|\psi&#039;\|_{\infty} \|\varphi\|_1<br />

is useless too because \|\psi&#039;\|_{\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.