Strong capacity-estimates for “fractional” norms

It is proved that for all fractional $l$ the integral $\int_0^\infty(p,l)-\operatorname{cap}(M_t)\,dt^p$ is majorized by the $p$-th power norm of the function $u$ in the space $Z_p^l(R^n)$ (here $M_t=\{x:|u(x)|\geqslant t\}$ and $(p,l)-\operatorname{cap}(e)$ is the $(p,l)$-capacity of the compactum $e\subset R^n$). Similar results are obtained for the spaces $W_p^l(R^n)$ and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in $Z_q(d,\mu)$, where $\mu$ is a nonnegative measure in $R^n$. One considers specially the case $p=1$.