Weighted Fourier capacities and Frostman restriction estimates

We study a weighted Fourier capacity attached to compact sets, $c_{p,k}(K)=\inf \{\mathrm{\parallel }k\widehat{u}{\mathrm{\parallel }}_{L^p({\mathbb{R}}^d)}:u\in C_0^{\mathrm{\infty }}({\mathbb{R}}^d),u\ge 1\text{ on }K\},$ where $k$ is a positive tempered weight. The paper develops complementary upper and lower estimates for this quantity. The upper bounds are obtained from Frostman data, Fourier restriction estimates, and Bessel-potential majorants, which together provide admissible smooth majorants of the compact set. The lower bounds follow from a dual testing argument against positive measures supported on $K$, yielding estimates in terms of the weighted $L^{p^{\mathrm{\prime }}}$ size of their Fourier transforms. For sets of positive Lebesgue measure, testing with ${\chi }_K\mathrm{d}x$ connects the capacity with the weighted integrability of $\widehat{{\chi }_K}$. A Lebedev-type Sobolev criterion then gives non-triviality ranges governed by the upper Minkowski dimension of $\mathrm{\partial }K$ and the growth of the weight. We also record consequences for Ahlfors-David regular sets and positive-measure Cantor-type examples.