Is it true that
$$
||f\*g||\_p \le ||\,|f|^\* \* |g|^\*||\_p\quad ?
$$
where $|f|^\*$ and $|g|^\*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ and $g$ this is true?



Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?
Yes, this follows from the [Riesz rearrangement inequality](http://en.wikipedia.org/wiki/Riesz_rearrangement_inequality):
$$
\int\_{\mathbb R^n} h(x)(f\*g)(x)\, dx \le \int\_{\mathbb R^n} h^\*(x) (f^\* \*g^\*)(x)\, dx
$$
(We can assume that all functions are $\ge 0$.) Since $\|h^\*\|\_q=\|h\|\_q$, this shows that
$$
\|f\*g\|\_p = \sup\_{\|h\|\_q=1} \int |h(f\*g)| \le \|f^\* \*g^\*\|\_p ,
$$
as desired (with $1/p+1/q=1$).


