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 lp (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 lp (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.
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 161–168, 1977.
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Maz'ya, V.G. Strong capacity-estimates for “fractional” norms. J Math Sci 23, 1997–2003 (1983). https://doi.org/10.1007/BF01093280
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DOI: https://doi.org/10.1007/BF01093280