Let us consider only Lebesgue measure on $\mathbb{R}^n$.
The norms are defined by
$$ \| f \|_p := \left(\int_{\mathbb{R}^n} |f|^p dx \right)^{1/p} \qquad \text{and} \qquad \|f\|_\infty := \inf\{C \ge 0 : |f(x)|\le C \text{ for almost every $x$}\}. $$
and define Schwartz space $\mathcal{S}$, for every $\alpha, \beta \in \mathbb{N}^n$,
$$f\in\mathcal{S} \quad \iff \quad \sup_{x\in\mathbb{R}^n} |x^\alpha D^\beta f(x)|<\infty \; \text{for every}\; f \in C^{\infty}(\mathbb{R}^n).$$
My question is here.
For every $1\le p < \infty$, is the following inequality holds:
$$ \|f\|_p \le \|f\|_\infty$$
for every $f \in \mathcal{S}$ ?
Best Answer
Not true. Take $n=1,p=1,f(x)=e^{-cx^{2}}$ where $c>0$. The inequality gives $\int_{\mathbb R} e^{-cx^{2}} dx \leq 1$. Letting $c \to 0$ and using Fatou's Lemma we get $\infty=\int_{\mathbb R} 1 dx\leq 1$, a contradiction.