Some Properties of a Kind of Singular Integral Operator with Weierstrass Function Kernel ()
1. Introduction
The properties of singular integral operator with Cauchy or Hilbert kernel on simple closed smooth curve or open arc have been elaborately discussed in [1-3]. Based on these, for the boundary curve is a closed curve or an open arc, the authors discussed the singular integral operators and corresponding equation with Cauchy kernel or Hilbert kernel in [1-3]. In recent years, many authors discussed the numerical solution of a class of systems of Cauchy singular integral equations with constant coefficients, Numerical methods for nonlinear singular Volterra integral equations in [4-6].
In this paper, we consider a kind of singular integral operator with Weierstrass function kernel on a simple closed smooth curve in a fundamental period parallelogram. Our goal is to develop the Bertrand poincaré formula for changing order of the corresponding integration, and some important properties of the above singular integral operator.
2. Preliminaries
Definition 1 Suppose that 
are complex constants with
, and P denotes the fundamental period parallelogram with vertices
. Then the function
is called the Weierstrass 
-function, where
denotes the sum of all
, except for
.
Definition 2 Suppose that 
is a smooth closed curve in the counterclockwise direction, lying entirely in the fundamental period parallelogram P, with 
and the origin lying in the domain 
enclosed by
. The following operator
(1)
is called the singular integral operator with 
-function kernel on
, where 
is the unknown function, and
are the given functions.
Letting
, then (1) becomes
(2)
Since 
is uniformly convergent in any closed bounded region lying entirely in P,
for any
, where 
is some positive finite constant. By noting that
, we obtain
, where 
is some positive finite constant. Write
then (1) can be rewritten in the form
, (3)
where 
is a Fredholm operator and 
is called the characteristic operator of
. Now the index of 
is defined as
, where
and for definiteness we assume that
, namely we assume that 
is an operator of normal type.
Now the associated operator of (1) takes the form
(4)
or
(4)′
and so that the associated operator of 
becomes
In addition, if we write
then (4) can be rewritten as
(5)
where
(
,
is some finite constant).
So 
is a Fredholm operator, and then the characteristic operator of 
operator becomes
(6)
Therefore, we concluded that 
usually can not be established, that is
.
For convenience, we write
where the fixed nonzero point 
and the origin lie in
. It is not difficult to get the following results.
Lemma 1 Suppose that
, and with the same 
as mentioned before, then a) 
b) (Poincare-Bertrand formula)
3. Some Properties of Operator K
1) If
, then
.
Proof Through calculation and estimation, we have
(7)
for any
, where 
and 
are all finite constant. While for any
, we have
(8)
where 
is some finite constant. Substituting (8) into (7), we obtain
(9)
Similarly we know that
Consequently, we have
.
2) If 
are singular integral operator, then 
is also a singular integral operator. That is, if
then
, (10)
where the sum of the former two terms in the right hand of Equation (10) are the characteristic operator, and the remainder in that is a Fredholm operator.
Proof By definition, we deduce that
where
By virtue of Lemma 1 (b), 
can be rewritten in the form
Consequently, (10) is established.
Now we write
where
, 
,
,
.
By [1], we know that 
is a Fredholm integral. For
, we know from
that 
is continuous about the variable
, and so that 
is also a Fredholm integral. By nothing that 
have the same form, we only need to discuss either one of them. Here we consider the integral
. Write
then 
is analytic in P and so that
. Consequently, we read from
that 
and so that 
is continuous on
, therefore 
is also a Fredholm integral.
So far, we conclude that 
is a singular integral operator.
3) Let
, where 
denotes the indices of
, then
.
Proof From 2), we know
and
so
.
In addition, we can see from
and
that when 
are normal, 
is also normal.
4)
.
5) If 
is a singular integral operator, and 
is a Fredholm integral operator of the first kind, then 
and 
are also Fredholm integral operators of the first kind.
6) If the indies of 
and 
are 
and 
respectively , then
.
7)
.
Through careful calculation, we may obtain 4) - 7).
8) Generally speaking,
can not be established for
.
Proof By definition and calculation, we have
. (11)
Whereas
. (12)
Let
then by Lemma 1(a), we have
, (13)
Substituting (13) into (12), we see that
. (14)
Therefore, 
cannot be established.