Weighted weak type (1,1) estimates for oscillatory singular integrals with dini kernels

Weighted weak type (1,1) estimates for oscillatory singular integrals with dini kernels

著者 Sato Shuichi

著者別表示 佐藤 秀一

journal or

publication title

金沢大学教育学部紀要.自然科学編 = Bulletin of the Faculty of Education, Kanazawa University.

Natural sciences

volume 49

page range 1‑22

year 2000‑02‑10

URL http://hdl.handle.net/2297/26408

WEIGHTED WEAK TYPE ^{(1 1)} ESTIMATES FOR

OSCILLATORY SINGULAR INTEGRALS WITH DINI KERNELS

Shuichi Sato

Abstract.

We consider

-weights and prove weighted weak type (1 1) estimates for oscil- latory singular integrals with kernels satisfying a Dini condition.

1. Introduction

We consider an oscillatory singular integral operator of the form:

T ( f )( x ) = p : v :

R n

e ^iP

^{x y}

K ( x

y ) f ( y ) dy = lim

!0 Z

j

x

y

> e ^iP

^{x y}

K ( x

y ) f ( y ) dy where P is a real-valued polynomial:

(1.1) P ( x y ) =

j

M

N a x y and f

( R ⁿ ) (the Schwartz space).

Let K

C

( R ⁿ

⁰

^{) satisfy}

(1.2)

K ( x )

c

x

ⁿ

K ( x )

c

x

ⁿ

(1.3)

Z

a<

x

<b K ( x ) dx = 0 for all a b with 0 < a < b:

The smallest constant for which (1.2) holds will be denoted by C ( K ). The following results are known.

Theorem A. (Ricci-Stein 5]) Let 1 < p <

. Then, T is bounded on L ^p ( R ⁿ ) with the operator norm bounded by a constant depending only on the total degree of P , C ( K ), p and the dimension n .

Received September 16, 1999.

1991 Mathematics Subject Classi cation . Primary 42B20.

Key words and phrases. Oscillatory singular integrals, rough operators.

1

Theorem B. (Chanillo-Christ 1]) The operator T is bounded from L

( R ⁿ ) to the weak L

( R ⁿ ) space with the operator norm bounded by a constant depending only on the total degree of P , C ( K ) and the dimension n .

Let w be a locally integrable positive function on R ⁿ . We say that w

A

if there is a constant c such that

(1.4) M ( w )( x )

cw ( x ) a.e.

where M denotes the Hardy-Littlewood maximal operator. The smallest constant for which (1.4) holds will be denoted by C

( w ).

It is known that T is bounded from L

_w to L

_w

(the weak L

_w space).

Theorem C. (8]) There exists a constant c depending only on the total degree of P , C ( K ), C

( w ) and the dimension n such that

sup >

w (

x

R ⁿ :

T ( f )( x )

>

)

c

f

L

where w ( E ) =

_E w ( x ) dx and

f

_L

=

f ( x )

w ( x ) dx .

Let K be locally integrable away from the origin. Put, for r

1, 0 < t

1 and R > 0,

! r R ( t ) = sup

j

y

Rt=

0

B

@

R

ⁿ

R

x

R

j

R ⁿ ( K ( x

y )

K ( x ))

^r dx

1

C

A 1

=r

: We say that the kernel K satises the D r -condition if

B r =

0

! r ( t ) dt

t <

^where ! r ( t ) = sup _R>

0

! r R ( t ) C _r = sup _R>

0 0

B

@

R

ⁿ

R

x

R

j

R ⁿ K ( x )

^r dx

1

C

A 1

=r

<

:

By the usual modications we can also dene the D

-condition. In this note we shall prove the following results, which will improve Theorems B and C.

Theorem 1. Let r > 1 and 1 =r +1 =u = 1. Suppose the kernel K satisfy the D _r -condition and (1 : 3), and suppose w ^u

A

. Then, there exists a constant c depending only on the total degree of P , B _r , C _r , C

( w ^u ), r and the dimension n such that

sup >

w (

x

R ⁿ :

T ( f )( x )

>

)

c

f

L

:

Theorem 2. Suppose that K satises the D

-condition and (1 : 3). Then, there exists a constant c depending only on the total degree of P , B

, C

and the dimension n such that

sup >

x

R ^{n :}

^{T ( f )( x )}

^>

^c

^f

L

:

Every kernel satisfying (1.2) satises the D

-condition. If K ( x ) =

x

ⁿ ( x

), x

= x=

x

, and if satises the L ^r -Dini condition on S ⁿ

, then K satises the D r -condition.

These theorems will be proved by a double induction as in 5], 1] and 8]. In this note we shall prove only Theorem 1. Theorem 2 can be proved similarly. Let P be a polynomial of the form in (1.1). We assume that there exists such that

= M and a

= 0 for some . We write

(1.5) P ( x y ) =

j

M x Q ( y )

and dene L = max

deg( Q ) : Q

= 0

= M

. Then 0

L

N . We assume that L

1 and max

M

L

a

= 1. Under this assumption on a polynomial P , we dene

T

( f )( x ) =

Z

j

x

y

>

e ^iP

^{x y}

K ( x

y ) f ( y ) dy:

To prove Theorem 1, we shall use the following result in the induction.

Proposition 1. Let , > 0 and let the kernel K , the weight w and the exponents r , u be as in Theorem 1. Then, there exists a constant c depending only on , , the total degree of P , r and the dimension n such that if C

( w ^u )

, B r , C r

,

sup >

w (

x

R ^{n :}

^T

^{( f )( x )}

^>

⁾

^c

^f

L

:

Let A ( f )( x ) = p : v :K

f ( x ). We need the following result for the rst step of induction for the proof of Theorem 1.

Proposition 2. Let the kernel K , the weight w and the exponents r , u be as in Theorem 1. Let , > 0. There exists a constant c depending only on , , r and the dimension n such that if C

( w ^u )

, B r , C r

, then

sup >

w (

x

R ⁿ :

A ( f )( x )

>

)

c

f

_L

:

Since A is bounded on L

(see 6, pp. 25{26]), if A is as in Proposition 2, we see that A is a singular integral operator considered in 6, p. 13]. Hence the conclusion of Proposition 2 will follow from 6, p. 15, Theorem 1.6].

We shall give the outlines of the proofs of Theorem 1 and Proposition 1 in Sections 2

and 4, respectively. Our proof of Proposition 1 is based on the techniques in Christ 3] for

the proofs of the weak (1 1) estimates for rough operators (see also Christ-Rubio 4] and

Sato 7]). We also use the geometrical argument of Chanillo-Christ 1]. We have to prove

a key estimate (Lemma 8 in

5) in the unweighted case in order to apply the method of

Vargas 9] involving an interpolation with change of measure. To prove Lemma 8, we need

a geometrical result for polynomials (Lemma 6 in

5). We shall prove Lemma 6 in

6 by

using the results appearing in the proof of Chanillo-Christ 1, Lemma 4.1]. Lemmas 6

and 8 have been proved in 8]. We include the proofs and some other parts of 8] almost

verbatim for the sake of completeness.

2. Outline of proof of Theorem 1

To apply the induction argument of 5] we need some preparation. We may assume that M

1 and N

1 otherwise Theorem 1 reduces to Proposition 2.

We write a polynomial in (1.1) as follows:

P ( x y ) =

^M

j

X

j

j x Q ( y ) =:

^M

j

P j ( x y ) : We further decompose P j as follows:

P j ( x y ) =

^N

t

X

j

j

j

t

a x y =:

^N

t

P jt ( x y ) : For j = 1 2 ::: M and k = 0 1 ::: N , dene

(2.1) R jk ( x y ) =

^j

s

P s ( x y ) +

^k

t

P jt ( x y ) : Note that R jN =

^j _s

P s ( j = 1 2 ::: M ).

For j = 1 2 ::: M and k = 0 1 ::: N , we consider the following propositions.

Proposition ^{A ( j k )} . ^{Let , >} 0. There exists a constant c depending only on , , j , N , r and the dimension n such that if C

( w ^u )

, B r , C r

and if R jk is a polynomial of the form in (2 : 1), then

sup >

w (

x

R ^{n :}

^T jk ( f )( x )

>

)

c

f

_L

where

T jk ( f )( x ) = p : v :

R n

e ^iR

^{x y}

K ( x

y ) f ( y ) dy:

Then, Theorem 1 follows from Proposition A ( M N ). We shall prove it by double induction. We rst note that A (1 0) follows from the boundedness of the operator A .

Next, we observe that if M

2 and if A ( j N ) (1

j

M

1) is true, so is A ( j +1 0) since

R j

( x y ) = R jN ( x y ) +

j

j

a

x

and hence

T j

( f )( x )

=

T jN ( f )( x )

. Thus, to complete the induction starting from A (1 0) and arriving at A ( M N ), it is sucient to prove A ( j k + 1) assuming A ( j k ) (0

k < N , 1

j

M ). To achieve this, put R = R j k

, R

= R jk , T j k

= S . We note that

R ( x y ) = R

( x y ) +

j

j

j

k

a x y :

We have only to deal with the case C jk = max

_j

_k

a

= 0. Then, by a suitable dilation we may assume C jk = 1. This can be seen as follows. We rst note that, for a > 0,

S ( f )( ax ) = p : v :

e ^iR

^{ax ay}

K a ( x

y ) f ( ay ) dy

where K _a ( x ) = a ⁿ K ( ax ). Assume the boundedness of S for the case C _jk = 1. Then, choosing a to satisfy a ^j

^k

C jk = 1, and using the dilation invariance of both the class A

and the class of the kernels considered in Theorem 1, we get

w (

x

R ^{n :}

^{S ( f )( x )}

^>

^{) = w} a (

x

R ^{n :}

^{S ( f )( ax )}

^>

⁾

c

f ( ax )

a ⁿ w ( ax ) dx

= c

f

_L

:

We split the kernel K as K = K

+ K

, where K

( x ) = K ( x ) if

x

1 and K

( x ) = K ( x ) if

x

> 1. Assuming C jk = 1, we consider the corresponding splitting S = S

+ S

:

S

( f )( x ) = p : v :

e ^iR

^{x y}

K

( x

y ) f ( y ) dy S

( f )( x ) =

Z

e ^iR

^{x y}

K

( x

y ) f ( y ) dy:

In the next section, we shall prove

(2.2) sup _>

0

w (

x

R ^{n :}

^S

^{( f )( x )}

^>

⁾

^c

^f

L

while by Proposition 1 we have

(2.3) sup _>

0

w (

x

R ⁿ :

S

( f )( x )

>

)

c

f

_L

:

Combining (2.2) and (2.3), we shall complete the proof of A ( j k +1), which will nish the proof of Theorem 1.

3. Estimate for S

In this section, we shall prove, under the assumption made in

2, that if C

( w )

, B r , C r

( , > 0), then S

is bounded from L

_w to L

_w

with the operator norm bounded by a constant depending only on j , N , , , r and n ((2.2)).

First, we shall prove

(3.1) w (

x

B (0 1) :

S

( f )( x )

>

)

c

j

y

<

f ( y )

w ( y ) dy

where B ( x r ) denotes the closed ball with center x and radius r > 0.

Lemma 1. ^{Let w w u (1}

^{u <}

⁾

^A

^{. Let T} be an operator of the form:

T ( f )( x ) = p : v :

R n

K ( x y ) f ( y ) dy = lim

!0 Z

j

x

y

> K ( x y ) f ( y ) dy

for f

( R ^{n ). Let 1 =r + 1 =u} = 1 and consider a non-negative function L on R ⁿ

⁰

satisfying J r <

, where J r = sup _R>

0 0

B

@

R

ⁿ

R

x

R ( R ⁿ L ( x )) ^r dx

1

C

A 1

=r

for r <

and J

can be dened by the usual modication. Suppose the kernel K satises

j

K ( x y )

L ( x

y ). For > 0, put T ( f )( x ) = p : v :

j

x

y

< K ( x y ) f ( y ) dy:

Suppose

sup >

w (

x

R ^{n :}

^{T ( f )( x )}

^>

⁾

^c w

f

_L

: Then sup _>

0

w (

x

R ⁿ :

T ( f )( x )

>

)

c ( c w + J r C

( w ^u )

^=u )

f

L

: Proof. The proof is similar to that of Lemma in 5, p. 187]. We shall prove (3.2) w (

x

B ( h = 4) :

T ( f )( x )

>

)

c ( c w + J r C

( w ^u )

^=u )

j

y

h

<

=

f ( y )

w ( y ) dy uniformly in h

R ⁿ . Integrating both sides of the inequality in (3.2) with respect to h , we get the conclusion of Lemma 1.

Split f into 3 pieces: f = f

+ f

+ f

, where f i

( R ⁿ ),

f i

c

f

( i = 1 2 3) supp( f

)

B ( h = 2), supp( f

)

B ( h 11 = 8)

B ( h 3 = 8), supp( f

)

x :

x

h

5 = 4

. Note that if

x

h

= 4, then T ( f

)( x ) = T ( f

)( x ) since

y

h

= 2 and

x

h

= 4 imply

x

y

< . So by the assumption on T , we have

w (

x

B ( h = 4) :

T ( f

)( x )

>

)

c w

j

y

h

<

=

f ( y )

w ( y ) dy:

Next, by Chebyshev's inequality, Holder's inequality and the fact w ^u

A

we easily see that

w (

x

B ( h = 4) :

T ( f

)( x )

>

)

cJ r C

( w ^u )

^=u

j

y

h

<

=

f ( y )

w ( y ) dy:

Finally, if

x

h

= 4 and

y

h

5 = 4, then

x

y

, and so T ( f

)( x ) = 0.

Combining these results, we get (3.2). This completes the proof of Lemma 1.

Now we return to the proof of (3.1). If

x

1 and

y

2, then

exp( iR ( x y ))

exp

0

B

B

@

i

0

B

B

@

R

( x y ) +

j

j

j

k

a y

1

C

C

A 1

C

C

A

c

x

y

where c depends only on k j and n .

Hence, if

x

1,

j

S

( f )( x )

U

0

B

B

@

exp

0

B

B

@

i

j

j

j

k

a y

1

C

C

A

f ( y )

1

C

C

A

( x )

+ cI ( f )( x ) where

U ( f )( x ) = p : v :

e ^iR

^{x y}

K

( x

y ) f ( y ) dy I ( f )( x ) =

Z

j

x

y

<

x

y

L ( x

y )

f ( y )

dy:

Note that U ( f )( x ) = U ( f _B

)( x ), I ( f )( x ) = I ( f _B

)( x ) if

x

< 1. By the induction hypothesis A ( j k ) and Lemma 1, we see that U is bounded from L

_w to L

_w

. On the other hand, it is easy to see that

Z

j

x

y

<

x

y

L ( x

y ) w ( x ) dx

j

2 ^j

Z

2 j;1

j

x

y

L ( x

y ) w ( x ) dx

cJ r M u ( w )( y ) where M u ( w ) = M ( w ^u )

^=u . Thus, by Chebyshev's inequality and the fact w ^u

A

we have

w (

x

B (0 1) : I ( f )( x ) >

)

cJ r C

( w ^u )

^=u

j

y

<

f ( y )

w ( y ) dy:

Combining these results, we get (3.1).

Similarly we can prove

(3.3) w (

x

B ( h 1) :

S

( f )( x )

>

)

c

j

y

h

<

f ( y )

w ( y ) dy where c is independent of h

R ⁿ . To see this, we rst note that

S

( f )( x + h ) = p : v :

e ^iR

^x

^{h y}

^h

K

( x

y ) f ( y + h ) dy and R ( x + h y + h ) = R

( x y h ) +

j

j

j

k

a x y :

We can apply the induction hypothesis A ( j k ) to the operator p : v :

e ^iR

^{x y h}

K ( x

y ) f ( y ) dy

to get its boundedness from L

_w to L

_w

. Thus, by the same argument that leads to (3.1) we get

w (

x

B ( h 1) :

S

( f )( x )

>

) = _h w (

x

B (0 1) :

S

( f )( x + h )

>

)

c

j

y

<

f ( y + h )

w ( y + h ) dy

c

j

y

h

<

f ( y )

w ( y ) dy

where h w ( x ) = w ( x + h ), and we have used the translation invariance of the class A

. Integrating both sides of the inequality (3.3) with respect to h , we get (2.2).

4. Outline of proof of Proposition 1

Let f

(

ⁿ ). By Calderon-Zygmund decomposition at height > 0 we have a collection

Q

of non-overlapping closed dyadic cubes and functions g b such that

f = g + b (4.1)

Q

Q

f

c (4.2)

v (

Q )

c v

f

_L

= for all v

A

(4.3)

k

g

c

g

_L

c v

f

_L

for all v

A

(4.4)

b =

Q b Q supp( b Q )

Q

b Q

L

c

Q

: (4.5)

Let a polynomial P be as in Proposition 1. We assume as we may that M

1 as in the outline of the proof of Theorem 1 in

2. We write P as in (1.5). Then, let q ( y ) =

P

j

L c y be the coecient of x ^M

. By a rotation of coordinates and a normalization, to prove Proposition 1 we may assume max

L

c

= 1 (see 1, p. 151] and Sublemma 2 in

6).

We take a non-negative '

C

( R ⁿ ) such that supp( ' )

1 = 2

x

2

j

' (2

^j x ) = 1 if

x

1 :

Put K j ( x y ) = ' (2

^j ( x

y )) K

( x y ), where K

( x y ) = e ^iP

^{x y}

K

( x

y ) ( K

( x ) is as in

2) and decompose K

( x y ) as K

( x y ) =

_j

K j ( x y ).

Dene V j ( f )( x ) =

Z

K j ( x y ) f ( y ) dy for j

0

and put

V ( f )( x ) =

j

V j ( f )( x ) :

Then T

= V

+ V . We have only to deal with V since we easily see that V

is bounded on L

_w ( w ^u

A

).

We set (see 3, 4])

B i =

j

Q

b Q ( i

1) B

=

j

Q

b Q :

Put

=

Q ~ , where ~ Q denotes the cube with the same center as Q and with sidelength 100 times that of Q . (Throughout this note we consider the cubes with sides parallel to the coordinate axes.)

When x

R ⁿ

, we observe that (4.6) V ( b )( x ) = V

0

@ X

i

B _i

1

A

( x )

=

i

X

j

Z

K j ( x y ) B i ( y ) dy =

i

X

j

i

Z

K j ( x y ) B i ( y ) dy

=

s

X

j

s

Z

K j ( x y ) B j

s ( y ) dy =

s

X

j

s V j ( B j

s )( x ) : To prove Proposition 1 we need the following results (Lemmas 2, 3 and 4).

Lemma 2. ^{Suppose w}

^A

^{. Let}

^L j

j

be a family of kernels satisfying

supp( L j )

2 ^j

x

2 ^j

L j ( x )

c

x

ⁿ

L j ( x )

c

x

ⁿ

:

Let G _j ( f )( x ) =

Z

R n

e ^iP

^{x y}

L _j ( x

y ) f ( y ) dy:

Put

E _s =

8

<

:

x

R ^{n :}

X

j

s G j ( B j

s )( x )

>

9

=

: Then there exists > 0 such that, for any positive integer s ,

w

E _sc

c 2

^s

f

_L

where c is a positive constant satisfying

_s

c 2

^s=

= 1.

We shall prove this in

5.

Lemma 3. ^{Let L} j and G j be as in Lemma 2. Then, for j

1,

k

G j

c 2

^j for some > 0, where

G j

denotes the operator norm on L

.

This follows from Ricci-Stein 5]. See also 8] for an alternative proof.

Lemma 4. If w ^u

A

, then the operator V is bounded on L

_w . Proof. Let

N j ( x ) = ' (2

^j x ) K ( x ) L j ( x ) = N j

( x ) ( > 0)

where

C

( R ⁿ ) which is supported in

x

< 2

and satisfying

= 1. Then L j

satises all the conditions of Lemma 2 with c

= c 2 ^{n j} , c

= c 2

ⁿ

^j , and we nd

k

L _j

_L

cC

(4.7)

k

L j

L

cC r 2

^jn=u : (4.8)

Put R _j ( x ) = N _j ( x )

L _j ( x ) =

Z

( N _j ( x )

N _j ( x

y ))

( y ) dy:

Then, it is easy to see that

k

R j

L

c!

(2

^j ) + c 2

^j

c! r (2

^j ) + c 2

^j (4.9)

k

R j

L

c ( ! r (2

^j ) + c 2

^j )2

^jn=u : (4.10)

Put

U j ( f )( x ) =

Z

R n

e ^iP

^{x y}

L j ( x

y ) f ( y ) dy W j ( f )( x ) =

Z

R n

e ^iP

^{x y}

R j ( x

y ) f ( y ) dy:

First we estimate U j . By Holder's inequality and (4.7), (4.8) we have (4.11)

U j ( f )

_L

c

Z

j

L j ( x

y )

w ( x ) dx

j

f ( y )

dy

c

f ( y )

M u ( w )( y ) dy:

On the other hand, if is small enough, by Lemma 3

(4.12)

U _j ( f )

_L

c 2

^j

f

for some > 0 : Interpolating between the estimates (4.11) and (4.12), we get

k

U j ( f )

_L

w

c 2

^j

Z

j

f ( y )

M u ( w )( y ) dy for

(0 1). Substituting w

⁼ for w , we have

(4.13)

U j ( f )

_L

c 2

^s

^u

^j=s

Z

j

f ( y )

M s ( w )( y ) dy for all s > u:

Next we estimate W j . By Holder's inequality and (4.9), (4.10)

k

W j ( f )

_L

c ( ! r (2

^j ) + c 2

^j )

Z Z

j

R j ( x

y )

w ( x ) dx

f ( y )

dy (4.14)

c ( ! r (2

^j ) + c 2

^j )

Z

j

f ( y )

M u ( w )( y ) dy:

By (4.13) and (4.14), for all s > u ,

k

V ( f )

L

c

j

( ! r (2

^j ) + 2

^j + 2

^s

^u

^j=

^s

)

f

_L

c s

f

_L

: From this we get the conclusion of Lemma 4, since w ^s

A

for some s > u .

Using these results, we can prove Proposition 1. Let N j and be as in the proof of Lemma 4. For a positive integer s let

L

_j ^s

( x ) = N j

( x ) ( > 0) : Put R _j

^s

( x ) = N j ( x )

L

_j ^s

( x ) =

Z

( N j ( x )

N j ( x

y ))

( y ) dy:

Then L

_j ^s

is supported in

2 ^j

x

2 ^j

and satises

j

L

_j ^s

( x )

c 2 ^{n s}

x

ⁿ

L

_j ^s

( x )

c 2

ⁿ

^s

x

ⁿ

: Set

U _j

^s

( f )( x ) =

Z

R n

e ^iP

^{x y}

L

_j ^s

( x

y ) f ( y ) dy W _j

^s

( f )( x ) =

Z

R n

e ^iP

^{x y}

R

_j ^s

( x

y ) f ( y ) dy:

Put

F _s =

8

<

:

x

R ⁿ :

X

j

s U _j

^s

( B j

s )( x )

>

9

=

: Then, if ( n + 1) < = 2 by Lemma 2

(4.15) w

F _sc

c 2

^s

f

_L

where , and c are as in Lemma 2. Since

_s

c 2

^s=

= 1, we have

8

<

:

x

R ^{n :}

X

s

X

j

s U _j

^s

( B j

s )( x )

>

9

=

s

F _sc

:

Thus by (4.15)

w

0

@ 8

<

:

x

R ^{n :}

X

s

X

j

s U _j

^s

( B j

s )( x )

>

9

= 1

A

X

s

w

F _sc

(4.16)

c

f

_L

: Since

k

R

_j ^s

L

c ( ! r (2

^s ) + 2

^s )2

^jn=u by Holder's inequality and the condition that w ^u

A

we nd

X

j

s W _j

^s

( B j

s )

L

c

! r (2

^s ) + 2

^s

f

_L

: Thus, by Chebyshev's inequality we have

(4.17) w

0

@ 8

<

:

x

R ^{n :}

X

s

X

j

s W _j

^s

( B j

s )( x )

>

9

= 1

A

c

0

@ X

s

;

! r (2

^s ) + 2

^s

1

A

f

_L

: By (4.6), (4.16) and (4.17) we have

(4.18) w (

x

R ⁿ

^:

^{V ( b )( x )}

^{> 2}

⁾

^c

^f

L

: By (4.3) we see that

(4.19) w (

)

c w

f

_L

: By Lemma 4 and (4.4)

(4.20) w (

x

R ^{n :}

^{V ( g )( x )}

^>

⁾

^c

^f

L

:

Combining (4.18), (4.19) and (4.20), we conclude the proof of Proposition 1.

5. Proof of Lemma 2

In this section we shall prove Lemma 2 in

4. For k m

1, put (5.1) H km ( x y ) =

Z

e

^iP

^{z x}

^iP

^{z y}

L k ( z

x ) L m ( z

y ) dz:

Then G

_k G m ( f )( x ) =

H km ( x y ) f ( y ) dy , where G

_k denotes the adjoint of G k .