Strong capacity-estimates for ?fractional? norms

Abstract

It is proved that for all fractionall the integral $$\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p$$ is majorized by the P-th power norm of the functionu in the space ℒ p l (Rn) (here Mt={x∶¦u(x)¦⩾t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e⊂Rn). Similar results are obtained for the spaces W p l (Rn) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ℒq(dμ), whereμ is a nonnegative measure in Rn. One considers specially the case p=1.