The Riesz-Thorin interpolation theorem gives us the following:
Let $I=(a,b)\subset \mathbb{R}$ where $I$ can be infinite. if we have $0<p_0<p_1<\infty$, and $p_0<p<p_1$ then for some parameter $t\in (0,1)$, and $f\in L^{p_0}(I)\cap L^{p_1}(I)$, we obtain the inequality $||f||_{L^p_t}\leq ||f||_{p_0}^{1-t}||f||_{p_1}^{t}$
I am aware of the Holder's inequality proof, but what about the following situation:
Question 1:
If we have $\int_{I}|f(x)|^{p_0}$ and $\int_{I}|f(x)|^{p_1}$, I want to show that we also have $\int_{I}|f(x)|^p$. I am sure there is some clever substitution trick, but I am just not seeing it.
Question 2:
Similarly, I believe we can also derive the inequality $(\int_{I}|f|^p)^{1/p}\leq (\int_{I}|f|^{p_0})^{\frac{1-t}{p_0}}(\int_{I}|f|^{p_1})^{\frac{t}{p_1}}$ similarly, and again, I am just not seeing the substitution trick.