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$f$ is smooth (derivatives of all orders exist) except at $x_1$,$x_2$,$x_3$ where derivatives exist only upto orders of $k_1$,$k_2$,$k_3$ respectively, where $k_i$ is a natural number. Same is the case with another function $g$ which is smooth except at $t_1$,$t_2$,$t_3$ where derivatives exist only upto orders of $l_1$,$l_2$,$l_3$ respectively. Both $f$,$g$ belong to either $L^1$ or $L^2$ or have compact support. What is the behaviour of the function $h=f*g$

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Assume $f$ and $g$ has compact support, and put $h=f*g$, then there is no problem for the derivatives to be integrable. If $D^{n_1}f,D^{n_2}g\in L^1$ then we have $$D^n h = D^{n_1}f*D^{n_2}g, \qquad \text{where } n_1+n_2=n.$$ Hence we certainly have that $h$ is differentiable of order $n=n_1+n_2$ where $n_1=\min(k_1,k_2,k_3)$ and $n_2= \min(l_1,l_2,l_3)$.

Depending on the discontinuity at $x_i$ the integrability of $D^if$ might be lost ($i=1,2,3$) and the same holds for $g$, so we can not do better than this.

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  • $\begingroup$ Assuming the discontinuities are of a type (jump) such that integrability is not lost, what are the points where $h$ is not smooth and at what orders ? $\endgroup$ – Rajesh Dachiraju Nov 7 '10 at 6:15

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