The question is:
(Young's inequality) Consider standard Lebesgue measure on $\mathbb{R}^n$. Prove $$ \| f * g\|_r \leq \|f\|_p\|g\|_q, f \in L^p(\mathbb{R}^3), g \in L^q(\mathbb{R}^3)$$ where $f*g$ is convolution, $(f * g)(x) =\int_{R^3} f(x - y)g(y)dy.$
(Remark) If we consider $K(x,y) := f(x * y)$, then we can apply Theorem 6.36 (in folland). But in this case $K$ is $L^\infty L^p$ instead of $L^\infty {L_w}^p$ and we have much sharper operator bound in fact.
I don't understand what is mean "But in this case $K$ is $L^\infty L^p$ instead of $L^\infty {L_w}^p$ and we have much sharper operator bound in fact." part in remark. Can I know the meaning of $K$ in $L^\infty {L_w}^p$??