What is the relationship between the Schwartz's space and the sets of functions of moderate decrease?

Question

There are two notions in the book by Stein and Shakarchi. The Schwartz's space on $\mathbb{R}$ consists of the set of all indefinitely differentiable functions $f$ so that $f$ and all its derivatives $f^{'}, f^{''},\ldots,f^{(\ell)},\ldots,$ are rapidly decreasing in the sense that $$\sup_{x\in \mathbb{R}}|x|^k|f^{(\ell)}(x)|<\infty$$ for every $k,\ell\ge 0.$ A function $f$ defined on $\mathbb{R}$ is said to be of moderate decrease if $f$ is continuous and there exists a constant $A>0$ such that $$|f(x)|\le \frac{A}{1+x^2}, x\in \mathbb{R}.$$

What is the relationship between the two spaces? Is the Schwartz's space contained in the set of functions of moderate decrease? Can we find a function which is of moderate decrease but not in Schwartz space?

Answer 1

If $f$ is in the Schwarz space then $|f(x)|$ and $x^{2}f(x)$ are bounded. So $(1+x^{2}) f(x)$ is bounded which means $f$ is moderately decreasing.

$f(x)=\frac 1 {1+x^{2}}$ is moderately decreasing but it is not in the Schwarz space becasue $x^{3}f(x)$ is not bounded.