$\|\,f\|_{h} \leq \|\,f\|_{p}^{\frac{p}{h}} + \|\,f\|_{q}^{\frac{q}{h}},\,$ for every $\,h\in(\,p,q)$.

Question

Let $\,f \in L^{p}(E) \cap L^{q}(E),$ with $p<q$. How to prove that $f \in L^{h}(E)$ for every $h \in (p,q)$ and the following interpolation inequality: $$\,\|\,f\|_{h} \leq \|\,f\|_{p}^{\frac{p}{h}} + \|\,f\|_{q}^{\frac{q}{h}}.$$

Thanks.

Answer 1

Note that for every $t\ge 0$, we have that $$ t^h\le t^p+t^q. $$ To see this consider the possibilities $t\le 1$ and $t>1$.

Hence $$ \lvert f(x)\rvert \le \lvert f(x)\rvert^{p/h}+\lvert f(x)\rvert^{q/h}=g_1(x)+g_2(x)=g(x), $$ and thus $$ \|f\|_h\le \|g\|_h\le \|g_1\|_h+\|g_2\|_h=\left(\int_E g_1^h\right)^{1/h}+\left(\int_E g_2^h\right)^{1/h}=\left(\int_E \lvert f\rvert^p\right)^{1/h}+\left(\int_E \lvert f\rvert^q\right)^{1/h} \\ =\|f\|_p^{p/h}+\|f\|_q^{q/h} $$ Note. I have assumed that $h\ge 1$, in order to apply Minkowski's inequality.