For n[ges ]2, let S(Rn)
be the space of Schwartz class functions. The dual space of
S(Rn) is denoted by
S′(Rn) and is called the tempered
distributions.
For f∈S(Rn),
define fˆ(w)=∫Rne−iw·xf(x)dx
and for
T∈S′(Rn), define
Tˆ∈S′(Rn)
by the formula
〈Tˆ, f〉=〈T, fˆ〉
for all
f∈S(Rn). For
1[les ]p[les ]∞, let
FLp(Rn)=
{T∈S′(Rn)[ratio ]
Tˆ∈Lp(Rn)}.
Let β be a multi-index of nonnegative integers,
β=(β1, β2, …, βn−1)
with [mid ]β[mid ]=
β1+β2+…+βn−1.
For a
nonnegative integer k and bounded Borel measures σβ
on a
compact subset K of Rn,
[mid ]β[mid ][les ]k, we define
T∈S′(Rn) by the
formula
formula here
If T is given by (1), then we know that supp(T)⊂K
and the order of T is at
most k. In particular, when k=0, we say that T
is a bounded measure on K. Denote
EK(Rn)=
{T∈S′(Rn)[ratio ]T
is a bounded measure on K} and
DK(Rn)=
{T∈S′(Rn)[ratio ]
supp(T)∪K}. If in addition we assume some connectness
property
of K, then one can show ([5, th. 2·3·10])
that if T∈DK(Rn),
then T does have the representation (1) for some k[ges ]0.
The
question is when we can say that k=0. From
the Hausdorff–Young inequality and the proof of theorem
7·6·6 in [5], one sees that
if the interior of K (in Rn) is not
empty, then
DK(Rn)
∩FLp(Rn)
⊂EK(Rn) if
and
only if 1[les ]p[les ]2. Much remains to be understood if
the interior of K is empty.
In this paper, we seek to prove a similar result for compact subsets
of hypersurfaces
of Rn, motivated from the differential
equations with constant coefficients.