Arkiv för Matematik

  • Ark. Mat.
  • Volume 33, Number 1 (1995), 81-115.

Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers

Vladimir G Maz'ya and Igor E Verbitsky

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Some new characterizations of the class of positive measures γ on Rn such that H ${}_{p}^{l}$ ∉Lp(γ) are given where H ${}_{p}^{l}$ (1< p<∞ 0< l∞) is the space of Bessel potentials This imbed ding as well as the corresponding trace inequality $||J_l u||_{L_p (\gamma )} \leqslant C||u||_{L_p } $ for Bessel potentials Jl=(1-Δ)-1/2 is shown to be equivalent to one of the following conditions

  1. Jl(Jlγ)pCJ a e
  2. Ml(Mlγ)p’CM a e
  3. For all compact subsets E of Rn
$\int_E {(J_{l\gamma } )^p dx} \leqslant C{\text{ }}cap (E H_p^l )$ where 1/p+1/p'=1Ml is the fractional maximal operator and cap (H ${}_{p}^{l}$ ) is the Bessel capacity In particular it is shown that the trace inequality for a positive measure \gg holds if and only if it holds for the measure (Jl\gg)p'dx Similar results are proved for the Riesz potentials Ilγ=|x|l-n* γ

These results are used to get a complete characterization of the positive measures on Rn giving rise to bounded pointwise multipliers M(H ${}_{p}^{m}$ →H ${}_{p}^{−l}$ ) Some applications to elliptic partial differential equations are considered including coercive estimates for solutions of the Poisson equation and existence of positive solutions for certain linear and semi linear equations

Article information

Ark. Mat., Volume 33, Number 1 (1995), 81-115.

Received: 15 April 1993
First available in Project Euclid: 31 January 2017

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1995 © Institut Mittag Leffler


Maz'ya, Vladimir G; Verbitsky, Igor E. Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers. Ark. Mat. 33 (1995), no. 1, 81--115. doi:10.1007/BF02559606.

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