ABSTRACT
Given , we discuss the embedding of in . In particular, for we deduce its compactness on all open sets on which it is continuous. We then relate, for all q up the fractional Sobolev conjugate exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in in a suitable weak sense, for every open set . The proofs make use of a non-local Hardy-type inequality in , involving the fractional torsion function as a weight.
Acknowledgements
The author wishes to thank Prof. Lorenzo Brasco for his useful comments on a preliminary version of the present manuscript, as well as for suggesting the problem, in Osaka in May 2017, during the Workshop ‘Geometric Properties for Parabolic and Elliptic PDEs’, the organisers of which are also gratefully acknowledged.
Notes
No potential conflict of interest was reported by the authors.
1 It suffices to apply inequality of [Citation8, Proposition 3.5] with , , , , and .
2 In fact, that estimate implies (Equation5.14(5.14) (5.14) ) with , but a close inspection of its proof at scale 1 reveals that minor arrangements allow for the parameter to appear.