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

J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516 511
The improvement of Dk(R n ) on BMO(R n ) should thus not be seen as an improvement of the
integrand, but as an improvement on the domains of integration: by the trace Theorem 3.4, the
embedding Theorem 5.3, and the John–Nirenberg inequality, functions in Vk(R n ) are exponentially
integrable on (n − k + 1)-dimensional subspaces.
6. Further problems
6.1. Traces of V1(R n ) on VMO(R n−1 )
By Theorems 3.4 and 5.3, functions in Vk(R n ) have VMO traces on (n − k + 1)-dimensional
spaces. The dimension n − k seems more natural: functions in Vn(R n ) are continuous, and hence
have traces on 0-dimensional spaces, i.e. points. If there was such a trace inequality, one could
define D0(R n ) = BMO(R n ). This notation would be consistent with the mutual injection Theorem
3.1, the extension Theorem 3.2 and the examples of Proposition 4.6. It would then be nice
to have a definition of Dk(R n ) which encompasses the case k = 0. The two-dimensional case
would already solve the problem of traces of Vn−1(R n ) on lines.
6.2. Geometric characterizations
By Propositions 2.10 and 2.11, the spaces V1(R n ) and Vn−1(R n ) can be defined by oscillations
respectively along boundaries of bounded domains and along closed curves. Further
refinements would restrict the set of domains and of curves. The most striking result would be if
the oscillation could be simply evaluated respectively on spheres and on circles.
The spaces Vk(R n ) for 1