Can we say anything about $fg*h$?

Question

Can we say anything about $fg*h$, for example... is it true that $$(fg)*h=f(g*h)?$$ seems that not, because not necessaryly $$f(x)\int_{\mathbb{R}^n} g(x-y)h(y)dy = \int_{\mathbb{R}^n} (fg)(x-y)h(y)dy.$$ But can we anything of the type hölder (for example) $$\|f*(gh)\|_1\leq \|f*g\|_p\|h\|_q$$ such that $\frac{1}{p}+\frac{1}{q}=1$? Thanks!

Answer 1

Don't know if this qualifies as an answer, but what you can say is that $$\|(fg)*h\|_1\leq\min\{\|f\|_\infty\cdot\|g*h\|_1,\|g\|_\infty\cdot\|f*h\|_1\} $$

because $$\|(fg)*h\|_1=\int|f(x-y)g(x-y)h(y)|dy\leq\|f\|_\infty\int|g(x-y)h(y)|dy=\|f\|_\infty\|g*h\|_1 $$ and likewise one shows that $\|(fg)*h\|_1\leq\|g\|_\infty\|f*h\|_1$.

Also, the $\|\cdot\|_1$ norm is submultiplicative with respect to convolution, so you can also deduce that $\|(fg)*h\|_1\leq\|fg\|_1\cdot\|h\|_1$ and then use Holder's inequality to deduce that $\|(fg)*h\|_1\leq\|f\|_p\|g\|_q\|h\|_1$ for all $p,q$ with $1/p+1/q=1$.