I'm doing an exercise asking us to show that if for $ 1\leq p\leq 2$, the inequality for convolution $||f*g||_p\leq ||f||_1 ||g||_p$ holds, then it holds for $p\geq 2$.
The exercise suggests using Theorem 15.9 [Bass] which is the following:
For $1<p<\infty$, and $1/p+1/q=1$, suppose $f\in L^p$. Then $ ||f||_p =\operatorname{sup} \{ \int fgd\mu:||g||_q\leq 1\}.$
Suppose $p\geq 2$, take $h= f*g/(||f||_1||g||_q)$. Then $||h||_q\leq 1$ and $\int (f*g )h d\mu\leq ||f*g||_p$. If using Minkovski's inequality to get $\int (f*g )h d\mu\leq ||f*g||_p||f*g||_q/(||f||_1||g||_q)$, I'm thinking if there is a function $h$ that makes the equality holds so that I can conclude the result. I'm not sure if this is the right approach. Can someone suggest how to solve this problem please?
The following is the full description.
Hint: If $f,g,h \in C_c(\Bbb R^n),$ then by Fubini's theorem we have $$ \int_{\Bbb R^n} (f \ast g)(x) h(x) \,\mathrm{d} x = \int_{\Bbb R^n} \int_{\Bbb R^n} f(y) g(x-y) h(x) \,\mathrm{d} y \,\mathrm{d} x = \int_{\Bbb R^n} f(y) (g \ast h)(y) \,\mathrm{d}y. $$ Using this you can establish the inequality $$ \left|\int_{\Bbb R^n}(f \ast g)(x) h(x) \,\mathrm{d} x \right| \leq \lVert f \rVert_p \lVert g \rVert_1 \lVert h \rVert_{p'}. $$ Can you take it form here?