I am working on the following problem:
Suppose $f \in L^{p}(X)$ for some $0 < p < \infty$ and the space $X$ is such that each set of positive measure has measure $\geq m$ for some $m > 0$. Show that $f \in L^{q}(X)$ for $p < q \leq \infty$ with $\|f\|_{L^{q}} \leq m^{1/q - 1/p}\|f\|_{L^{p}}$.
I think one could first do it for the characteristic functions, then simple functions, and then use the Dominated/Monotone Convergence Theorem, but is there a way to do it with Holder's Inequality (or other inequalities)? Using simple functions doesn't seem to illuminate when the case of equality holds.
Let $\mu$ be the measure on $X$. For any $\lambda>0$, the measure of the set $E_\lambda= \{x\in X:|f(x)|\ge \lambda\}$ can be estimated as follows: $$\|f\|_{L^p}^p=\int_X |f|^p \ge \int_{E_\lambda} |f|^p \ge \lambda^p \mu(E_\lambda) \tag{1}$$ When $\lambda>m^{-1/p}\|f\|_{L^p}$, the estimate (1) yields $\mu(E_\lambda)<m$, hence $\mu(E_\lambda)=0$. Conclusion: $$|f|\le m^{-1/p}\|f\|_{L^p}\qquad \text{a.e. }\tag{2}$$ From (2) the desired inequality follows easily: $$\int_X |f|^q \le \left(m^{-1/p}\|f\|_{L^p}\right)^{q-p}\int_X |f|^p$$ hence $$\|f\|_{L^q} \le m^{(p-q)/(pq)}\|f\|_{L^p}^{1-p/q+p/q} = m^{1/q-1/p}\|f\|_{L^p}$$