In the first part we consider restriction theorems for hypersurfaces Γ in Rn, with the affine curvature KΓ 1/(n+1) introduced as a mitigating factor. Sjölin, [19], showed that there is a universal restriction theorem for all convex curves in R2. We show that in dimensions greater than two there is no analogous universal restriction theorem for hypersurfaces with non-negative curvature.
In the second part we discuss decay estimates for the Fourier transform of the density KΓ 1/2 supported on the surface and investigate the relationship between restriction and decay in this setting. It is well-known that restriction theorems follow from appropriate decay estimates; one would like to know whether restriction and decay are, in fact, equivalent. We show that this is not the case in two dimensions. We also go some way towards a classification of those curves/surfaces for which decay holds by giving some sufficient conditions and some necessary conditions for decay.
@article{urn:eudml:doc:41458,
title = {Restriction and decay for flat hypersurfaces.},
journal = {Publicacions Matem\`atiques},
volume = {46},
year = {2002},
pages = {405-434},
mrnumber = {MR1934361},
zbl = {1043.42007},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41458}
}
Carbery, Anthony; Ziesler, Sarah. Restriction and decay for flat hypersurfaces.. Publicacions Matemàtiques, Tome 46 (2002) pp. 405-434. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41458/