The relation of the Homogeneous Sobolev norm and general Sobolev norm

Question

I'm wondering if the inequality $$ \left\| F\right\|_{\dot H^k(\mathbb R^n)} \le C\left\| f\right\|_{L^\infty(\mathbb R^n)} \left\| f\right\|_{\dot H^k(\mathbb R^n)} $$ holds for $k\in[0,10]$ then $$ \left\| F\right\|_{H^k(\mathbb R^n)} \le C\left\| f\right\|_{L^\infty(\mathbb R^n)} \left\| f\right\|_{H^k(\mathbb R^n)} $$ holds for $k\in[0,10]$, where $\dot H^k$ denotes the homogeneous Sobolev norm and $H^k$ denotes the Sobolev norm of order $k$.