CHAPTER 1
Introduction
Let Z+ denote the set of all nonnegative integers and let Λν Z+ n be the finite
set of multi-indices for each ν = 1, · · · , d. Given Λ = (Λ1, · · · , Λd), we set the
family of all vector polynomials of the following form:
= : PΛ(t) =
m ∈Λ1
cm
1 tm,
· · · ,
m ∈Λd
cm
d tm
with t
Rn
(1.1)
where cm’s
ν
are nonzero real numbers. Given PΛ, ξ = (ξ1, · · · , ξd)
Rd
and
r = (r1, · · · , rn) R+,
n
we define a multi-parameter oscillatory singular integral:
I(PΛ,ξ,r) = p.v.
[−rj,rj ]
ei ξ,PΛ(t)
dt1
t1
· · ·
dtn
tn
where the principal value integral is defined by
lim
→0
{
j
|tj |rj }
ei ξ,PΛ(t)
dt1
t1
· · ·
dtn
tn
where = (
1
, · · · ,
n
) with
j
0. The existence of this limit follows by the
Taylor expansion of t ei ξ,PΛ(t) and the cancelation property dtν /tν = 0 with
ν = 1, · · · , n.
We see that whether supξ
|I(PΛ,ξ,r)| is finite or not depends on
(1) Sets Λν of exponents of monomials in PΛ(t).
(2) Coefficients of polynomial
PΛ(t).
(3) Domain of integral [−rj,rj].
(1) The dependence on set Λν of exponents is observed in the following simple cases:
sup
ξ∈R
|I(PΛ,ξ, (1, 1))| =





supξ∈R
1
−1
1
−1
sin(ξt1t2)
dt1
t1
dt2
t2
= if Λ = {(1, 1)}
supξ∈R
1
−1
1
−1
sin(ξt1t2)
2
dt1
t1
dt2
t2
= 0 if Λ = {(2, 1)}
(2) The dependence on coefficients of polynomials first appeared in [12], later in
[1] and [13]. Let PΛ(t) = t1t2
1 3
t1t2
3 1
and QΛ(t) = t1t2
1 3
+ t1t2.
3 1
Then these have the
same exponent set Λ, but supξ I(PΛ,ξ,r) and supξ I(QΛ,ξ,r) = ∞. However,
1
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