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 PΛ the

family of all vector polynomials PΛ of the following form:

PΛ = PΛ : 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 ∈ 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 PΛ 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