Prove that $L^p(\Omega) \subset L^q (\Omega)$ whenever $\Omega$ is a bounded open set and $1 ≤ q ≤ p ≤ +\infty$.

Question

Let $\Omega$ be a bounded open set and $1 ≤ q ≤ p ≤ +\infty$. Prove that $L^p(\Omega) \subset L^q (\Omega)$. Where $L^p(\Omega)$ and $L^q(\Omega)$ are the spaces of functions $f$ for which $|f|^p$ and $|f|^q$ is integrable.

In trying to solve this, many books refer me to Holder's inequality. However, I have failed to connect it to this problem.

Write $|f|^{q}$ as $(1) (|f|^{q})$ and apply Holder with conjugate indices $\frac p q$ and $\frac p {p-q}$. — Kavi Rama Murthy, Sep 29 '21 at 23:32

score 2 · Accepted Answer · answered Sep 30 '21 at 15:46

2

Suppose $(X,\mu)$ is a measure space with $\mu(X)<\infty.$ Assume $0<q<p<\infty$ (the case $p=\infty$ is very simple). Let $f\ge 0,$ with $f\in L^p(X).$ Set $A=\{f\le 1\}, B = \{f> 1\}.$ Then

$$\int_X f^q\,d\mu = \int_A f^q\,d\mu + \int_B f^q\,d\mu \le \mu(A) + \int_B f^p\,d\mu <\infty.$$

answered Sep 30 '21 at 15:46

zhw.

105,693

Why the downvote? This is easiest solution to this problem that I know of. – zhw. Sep 30 '21 at 15:55

Prove that $L^p(\Omega) \subset L^q (\Omega)$ whenever $\Omega$ is a bounded open set and $1 ≤ q ≤ p ≤ +\infty$.

1 Answers1