By the discrete Young inequality, we know that $\|a_n * b_n\|_q \leq \|a_n\|_p \|b_n\|_r$ when $1/q+1= 1/p+1/r$.
My question is that if there exists $C>0$, such that $\|a_n * b_n\|_q \leq C \|a_n\|_p $ is true for any $a_n \in \ell^p$, can we have $q\geq p$ and $b_n\in \ell ^r$?
Any idea will be helpful. Thanks a lot.
This is false.
Take $(a_n)\in\ell^{p}$ and $(b_n)\in\ell^{q}$ where $p,q\geq 1$, then by Youngs inequality $$\|(a_n)*(b_n)\|_{\ell^r}\leq \|(a_n)\|_{\ell^p}\|(b_n)\|_{\ell^q}$$ provided $$\dfrac{1}{r}+1=\dfrac{1}{p}+\dfrac{1}{q}$$ and then $$\|(a_n)*(b_n)\|_{\ell^s}\leq\|(a_n)\|_{\ell^p}\|(b_n)\|_{\ell^q}$$ whenever $s>r$ (see How do you show monotonicity of the $\ell^p$ norms?), even though $$\frac1s +1< \frac1p+\frac1q$$