Question

Let $g$ be a function in the Schwartz space $\mathscr S (\mathbb R)$. Show that for any $l \ge 0$, we have $\sup_x |x|^l |g(x-y)|\le A_l (1+|y|)^l$ by considering separately the cases $|x|\le 2|y|$ and $|x|\ge 2 |y|$.

The Schwartz space is defined as the set $\mathscr S (\mathbb R)$ of all indefinitely differentiable functions $f:\mathbb R\to \mathbb R$ such that $\sup_{x\in\mathbb R} |x|^k |f^{(l)}(x)|<\infty$ for all $k, l \ge 0$.

(This was used in a proof in Elias Stein's book on Fourier Analysis and is not a homework problem. The book just didn't go through this particular step.)

Answer 1

We have $$ \vert x\vert^l\vert g(x-y)\vert\le {(\vert x-y\vert+\vert y\vert)}^l\vert g(x-y)\vert \le (1+\vert y\vert)^l(1+\vert x-y\vert)^l\vert g(x-y)\vert $$ so that $$ \vert x\vert^l\vert g(x-y)\vert\le (1+\vert y\vert)^l\underbrace{\sup_{z\in\mathbb R} (1+\vert z\vert)^l\vert g(z)\vert}_{A_l} $$ which is the sought inequality, where $A_l$ in a semi-norm of $g$ in the Schwartz space. Note that no differentiability property for $g$ is necessary, we have used only fast decay of the function itself.

Bazin.