I would like to know what a common method is for proving the following:
Let $\Omega \subseteq \mathbb{R}^{N}$, some natural number $N$. Suppose $f \in L^p(\Omega) \cap L^q(\Omega)$ for some $1 \leq p < q \leq \infty$. Then $f \in L^r(\Omega)$ for all $p < r < q$.
I feel like you could show this with just the generalised Holder inequality, but not sure. Any help appreciated.
Case 1: $q<\infty$. From $$|f(x)|^r \leq |f(x)|^q, \qquad x \in \{|f|>1\}$$ and $$|f(x)|^r \leq |f(x)|^p, \qquad x \in \{|f| \leq 1\},$$ it follows that \begin{align*}\int |f(x)|^r \, dx &= \int_{\{|f| \leq 1\}} |f(x)|^r \, dx + \int_{\{|f| > 1\}} |f(x)|^r \, dx \\ &\leq \int |f(x)|^q \,dx + \int |f(x)|^p \, dx < \infty.\end{align*}
Case 2: $q=\infty$. Then
$$\int |f(x)|^r \, dx = \int |f(x)|^{p+(r-p)} \, dx \leq \|f\|_{\infty}^{r-p} \int |f(x)|^p \, dx < \infty.$$
