Elementary Sobolev and propagator bounds

For the spatial Sobolev norm

Fourier inversion and Cauchy–Schwarz give

Indeed, ${\int }_{{\mathbb{R}}^3}(1+|p{|}^2{)}^{-2}{\mathrm{d}}^{3}p<\infty$. Spatially differentiated versions follow by applying the same inequality to the derivative.

For $D_{\mu }=\sqrt{-\Delta +{\mu }^2}$, including $\mu =0$,

At zero frequency the sine quotient is its continuous value $\tau$. Also ${D_{\mu }}^2:H^{s+2}\to H^s$ is bounded, with a constant depending on $\mu$.

If $F$ is spatially supported in a fixed ball, its integer Sobolev norm at each time is bounded by a fixed-volume constant times the maximum of its spatial derivatives of the required order. This also holds for its time derivatives.