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Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers. (English) Zbl 0834.31006
The trace inequality for Bessel potentials is considered and the connection with applications in partial differential equations is studied.
Authors’ abstract: Some new characterizations of the class of positive measures \(\gamma\) on \(\mathbb R^n\) such that \(H_p^l\subset L_p(\gamma)\) are given where \(H_p^l\) \((1<p<\infty, \, 0<l<\infty)\) is the space of Bessel potentials This imbedding as well as the corresponding trace inequality \[ \| J_lu \|_{L_p(\gamma)}\leq C\,\| u\|_{L_p} \] for Bessel potentials \(J_l =(1-\Delta)^{-l/2}\) is shown to be equivalent to one of the following conditions
(a)\(\quad J_l (J_l\gamma)^p \leq C\,J_l\gamma\) a. e.
(b)\(\quad M_l (M_l\gamma)^{p'} \leq C\,M_l\gamma\) a. e.
(c)\(\quad\) For all compact subsets \(E\) of \(\mathbb R^n\) \[ \int_E(J_l\gamma)^p\,dx\leq C \text{cap}(E\,H_l^p) \] where \(1/p+1/p'=1\), \(M_l\) is the fractional maximal operator and \(\text{cap}(H_p^l)\) is the Bessel capacity. In particular it is shown that the trace inequality for a positive measure \(\gamma\) holds if and only if it holds for the measure \((J_l\gamma)^{p'}\, dx\). Similar results are proved for the Riesz potentials \(I_l\gamma=|x|^{l-n}*\gamma\).
These results are used to get a complete characterization of the positive measures on \(\mathbb R^n\) giving rise to bounded pointwise multipliers \(M(H_p^m \to H_p^{-l})\). 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 semilinear equations.

31C15 Potentials and capacities on other spaces
26A33 Fractional derivatives and integrals
Full Text: DOI
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