Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?

Question

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?

Answer 1

Yes, this follows from the 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 $1/p+1/q=1$).