Comparison of $L^p$ and $L^q$ norms to establish the inclusion between corresponding spaces

Question

We can deduce that; for any $x \in \ell^p,$ the space of $p$-summable real sequences ($p \geq 1$), $$\lVert x \lVert_q \leq \lVert x \lVert_p,~p \leq q < \infty,$$ by just letting $e=\frac{x}{\lVert x \lVert_p},~x \neq 0$, to get $\lVert e \lVert_q \leq 1$. This inequality essentially gives $ \ell^p \subseteq \ell^q$.

But the inclusion in the case of spaces, $$L^p(E)=\{f:E \to \Bbb R~:~\int_E|f(t)|^p d \mu < \infty\},$$ where $E$ is a non empty measurable set in the Lebesgue measure space $(X, \mathcal M, \mu)$, should be $$L^q(E) \subseteq L^p(E),~~p \leq q < \infty.$$ What is the nature of the comparion between the norms $\lVert \cdot \lVert_p $ and $\lVert \cdot \lVert_q $ for $f \in L^q(E)$ to establish the above inclusion? How to justify the comparison?

Answer 1

The inequalities and inclusions depend highly on the measure $\mu$ considered.

If $\mu$ is a counting measure, the functions $p \mapsto ||x||_p$ decrease (in a wide sense) and the space $\ell^p = L^p(\mu)$ increases with $p$.

If $\mu$ is a probability measure, the functions $p \mapsto ||x||_p$ increase (in a wide sense) and the space $L^p(\mu)$ decreases when $p$ increases.

If $\mu$ is finite, by applying the previous result to $\mu(E)^{-1}\mu$, one can derive inequalities with constants depending on $p$, $q$ and $\mu(E)$, so the space $L^p(\mu)$ decreases when $p$ increases.

When $\mu$ is infinite, no inclusion hold in general. For example, the function $x \mapsto \sin(x)/x$ is in $L^2(\mathbf{R})$ but not in $L^1(\mathbf{R})$. the function $x \mapsto |x|^{-1/2}\mathbf{1}_{0<|x|<1}$ in $L^1(\mathbf{R})$ but not in $L^2(\mathbf{R})$.