$p \leqslant q \leqslant r$. If $f \in L^p$ and $f \in L^r$ then $ f \in L^q$? [duplicate]

Possible Duplicate:
Proving an interpolation inequality

Let $f \in L^p$ and $f \in L^r$ where $1 \leqslant p \leqslant r$ . Then can we say that $f \in L^q$ if $p \leqslant q \leqslant r$? ($f : \mathbb R^n \to \mathbb R$)

Solution 1:

Yes: we write $q=ap+(1-a)r$ where $0<a<1$ (if $a\in \{0,1\}$ it's clear). We apply Hölder's inequality to the exponent $\frac 1a>1$ (it's conjugate is $\frac 1{1-a}$). We get $$\int_{\Bbb R^n}|f|^qdx=\int_{\Bbb R^n}(|f|^p)^a(|f|^r)^{1-a}dx\leq \left(\int_{\Bbb R^n}|f|^p\right)^a\left(\int_{\Bbb R^n}|f|^r\right)^{1-a},$$ which is finite. We also have the inequality $$\lVert f\rVert_{L^q}\leq \lVert f\rVert_{L^p}^{\frac aq}\cdot \lVert f\rVert_{L^r}^{\frac{1-a}q}.$$

Solution 2:

HINT Split the $L^q$ integral over the sets $A$ and $A^C$, where $A = \{x \in \mathbb{R}^n: f(x) \leq 1\}$ argue out why both are finite making use of the fact that $f \in L^p, L^r$.