Sobolev estimate with a scaled cutoff

Statement

Let $\phi\in C_c^\infty(\mathbb R^3)$ , $0\le\phi\le1$ , and $\phi_R(x)=\phi(x/R)$ . For an integer $m\ge1$ and $u\in L^2\cap H^1_{\mathrm{loc}}$ ,

\|\phi_R^m u\|_6\le C_{m,\phi}\bigl(\|\phi_R^m\nabla u\|_2+R^{-1}\|u\|_2\bigr).

Here $H^1_{\mathrm{loc}}$ denotes local Sobolev membership. No global gradient bound is assumed.

Product rule

Apply the three-dimensional Sobolev inequality to the compactly supported $H^1$ function $\phi_R^m u$ . Its gradient is $\phi_R^m\nabla u+m\phi_R^{m-1}(\nabla\phi_R)u$ , and the second term is bounded in $L^2$ by $C R^{-1}\|u\|_2$ .