Local square summability of convolutions

Let $\mathscr P(D)$ and $\mathscr R(D)$ be two convolution operators in $\mathbf R^n$ and $K$ be an arbitrary compact subset of $\mathbf R^n$, having interior points. Necessary and sufficient conditions are given for the boundedness and compactness of the ball $\{u:\|\mathscr P(D)u\|_{L^2(\mathbf R^n)}\leqslant1\}$ in the metric of $\|\mathscr R(D)u\|_{L^2(K}$.