2 JOONIL KIM
in this paper, we are not concerned with this coefficient dependence. We rather
search for a condition on Λ so that
for all PΛ, sup
ξ∈Rd
|I(PΛ,ξ,r)| CPΛ ∞. (1.2)
(3) The dependence on the domain [−rj,rj] is observed for the case Λ = {(2, 2),
(3, 3)},
sup
ξ∈R ,0r1,r2 1
r2
−r2
r1
−r1
eiξ(t1t2+t1t2)
2 2 3 3
dt1
t1
dt2
t2
∞,
sup
ξ∈R ,0r1,r2
r2
−r2
r1
−r1
eiξ(t1t2+t1t2)
2 2 3 3
dt1
t1
dt2
t2
= ∞.
In the former integral, a monomial t1t2
2 2
dominating t1t2
3 3
with small t1,t2, makes
the vanishing property
dti
ti
= 0 effective. But in the latter integral, a monomial
t1t2,
3 3
dominating t1t2
2 2
with large t1,t2, weakens the cancellation effect of the integral
dti
ti
. Knowing this dependence on whether rj is taken from a finite interval (0, 1)
or an infinite interval (0, ∞), we set up our problem by first fixing the range of r
according to S Nn = {1, · · · , n}:
r I(S) =
n
j=1
Ij (1.3)
where Ij = (0, 1) for j S and Ij = (0, ∞) for j Nn \ S. Instead of (1.2), we
shall find the necessary and sufficient condition on Λ and S that
for all PΛ, sup
ξ∈Rd, r∈I(S)
|I(PΛ,ξ,r)| CPΛ ∞. (1.4)
For each Schwartz function f on Rd and a vector polynomial PΛ, the
multiple Hilbert transform of f associated to is defined to be
(
Hr

f
)
(x) = p.v.
n
j=1
[−rj,rj ]
f
(
x PΛ(t)
)
dt1
t1
· · ·
dtn
tn
.
Here rj = 1 with j S corresponds to a local Hilbert transform, and rj = with
j Nn \S corresponds to a global Hilbert transform. Since I(PΛ,ξ,r) is the Fourier
multiplier of the Hilbert transform Hr , the boundedness (1.4) is equivalent to that
for all PΛ, sup
r∈I(S)
Hr

Lp(Rd)→Lp(Rd)
CPΛ where p = 2. (1.5)
In this paper, we show (1.4) and (1.5) with 1 p for all n and d when
S Nn. To determine the conditions that ensure that (1.4) and (1.5) hold, we
study the concept of faces and their dual faces (cones) of the Newton Polyhedron
associated with Λ and S Nn. It is noteworthy in advance that the necessary
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