True. Multiplication by a <Schwartz function> is a bounded map $H^s(\mathbb R^n)\to H^s(\mathbb R^n)$ for every real $s$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 327 / 2 / b / i / Solution

A Schwartz function is an element of the <Schwartz space>: it is a <smooth function> and every derivative decreases faster than every inverse polynomial. Equivalently, $x^\alpha D^\beta f(x)$ defines a <bounded function> for every pair of <multi-indices> $\alpha,\beta$.

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