Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516 513
embedded respectively in bmoz( ¯Ω) (functions whose extension by 0 to R n is in BMO(R n )) and
the second in the larger space bmor( ¯Ω) (restrictions of functions in BMO(R n ) to Ω) (see [8] for
the definitions).
Acknowledgments
The author thanks Haïm Brezis who suggested the problem, and who encouraged and discussed
the progress of this work. He also thanks Thierry De Pauw for discussions, and acknowledges
the hospitality of the Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie in
Paris, at which this work was initiated.
Appendix A. Density of compactly supported forms
A.1. The closure of closed k-forms
This appendix is devoted to the study of dense sets in the space L 1 (R n ; Λ k R n ) of summable
closed forms.
Definition A.1. Let 1 p0, the s-dimensional Hausdorff
measure of Σ vanishes [9].
In a similar way, one can prove
Lemma A.4. There exists a sequence (ζm)m in D(Rn ) such that 0 ζm 1, ζm → 1 almost
everywhere and
|∇ζm| n dx → 0
as m →∞.
R n