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Weighted norm inequalities for the Dirichlet transform
J. Math. Anal. Appl. 359 (2009) 637–641
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Weighted norm inequalities for the Dirichlet transform Ronald Kerman a,1 , Colin Phipps b,∗,2 a b
Department of Mathematics, Brock University, 500 Glenridge Ave., St. Catharines, Ontario L2S 3A1, Canada Department of Applied Mathematics, University of Waterloo, 200 University Ave. W., Waterloo, Ontario N2L 3G1, Canada
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 11 September 2008 Available online 12 June 2009 Submitted by L. Grafakos
The Dirichlet transform is defined for suitable functions by
( D f )(x) :=
Keywords: Dirichlet transform Weights
∞
1
π
sin(x − y )
−∞
x− y
f ( y ) dy ,
x ∈ R.
We show that for 1 < p < ∞ and nonnegative w (x) on R
( D f )(x) p w (x) dx C pp
R
f ( y ) p w ( y ) dy ,
R
with C p > 0 independent of f , if and only if there exists K > 0 such that
w (x) dx I
w (x)
1 − p− 1
p −1 dx
K |I |p
I
for all intervals I ⊂ R with length | I | π .
© 2009 Elsevier Inc. All rights reserved.
1. Introduction The Dirichlet transform defined for suitable functions by
( D f )(x) :=
1
π
∞
sin(x − y )
−∞
x− y
f ( y ) dy ,
x ∈ R,
is central to the inversion theory of the Fourier transform on R. In this paper, we characterize, for fixed p, 1 < p < ∞, the nonnegative measurable (weight) functions w on R for which
( D f )(x) p w (x) dx C pp
R
f ( y ) p w ( y ) dy ,
R
with C p > 0 being independent of f . Our result is given in
* 1 2
Corresponding author. Fax: +1 519 746 4319. E-mail addresses: [email protected] (R. Kerman), [email protected] (C. Phipps). Supported by NSERC grant A4021. Supported in part by an NSERC USRA and NSERC grant A4021.
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2009.06.028
©
2009 Elsevier Inc. All rights reserved.
(1.1)
638
R. Kerman, C. Phipps / J. Math. Anal. Appl. 359 (2009) 637–641
Theorem 1.1. Fix an index p, 1 < p < ∞ and let w be a weight function on R. Then, (1.1) holds if and only if there exists a constant K > 0 such that
w (x) dx
1 − p− 1
w (x)
I
p −1 K |I |p ,
dx
(1.2)
I
for all intervals I ⊂ R with length | I | π . If (1.2) holds for all intervals I ⊂ R, then w is said to satisfy the A p condition introduced in [3]. Thus, we will say the 1 satisfies the global, but not the weights in Theorem 1.1 satisfy the global A p condition. Observe that w (x) = max 1, 1 1+log |x|
unrestricted A p condition. Global A p may be thought of as a discrete version of the A p condition. Indeed, fix p, 1 < p < ∞, and consider a doubly infinite (weight) sequence { w k }k∈Z , w k 0. The discrete A p condition for such a sequence requires the existence of a constant K > 0 such that
n
wk
k=m
n
1 − p− 1
p −1 K (n − m + 1) p ,
wk
(1.3)
k=m
for all m, n ∈ Z with m n. It is a straightforward exercise to verify that (1.2) for a weight w on R is equivalent to (1.3) for
(k+1)π
the sequence { w k }k∈Z , when w k = kπ w (x) dx, k ∈ Z. Now, it is shown in [2, Theorem 10], that for suitable sequences a = {ak }k∈Z , the discrete Hilbert transform
(ha)n :=
1
π
∞
ak
k=−∞ k=n
k−n
n ∈ Z,
,
satisfies ∞ ∞ (ha)n p w n C pp |ak | p w k , n=−∞
k=−∞
where C p > 0 is independent of {ak }k∈Z , if and only if (1.3) holds (see [1]). This will be crucial for proving the sufficiency of (1.2) in Theorem 1.1. Also important is the fact that (1.3) implies the existence of C p > 0, independent of {ak }, with ∞ ∞ (pa)n p w n C pp |ak | p w k , n=−∞
k=−∞
in which
(pa)n :=
∞ k=−∞
ak 1 + (n − k)2
n ∈ Z,
,
is the discrete Poisson transform of a = {ak }k∈Z . The latter is readily proved using the methods of [4]. The proof of Theorem 1.1 is given in the next section. In conclusion, we mention that our methods yield the expected: Theorem 1.2. Fix an index p, 1 < p < ∞ and let w be a weight function on R. Then, the weak-type inequality
w (x) dx {x∈R: |( D f )(x)|>t }
Cp t
p
f (x) p w (x) dx,
R
holds, with C p > 0 independent of f , if and only if there exists a constant K > 0 such that for all intervals I ⊂ R, | I | π , w satisfies (1.2) when p > 1 and
w (x) dx ess sup
I
when p = 1.
x∈ I
1 w (x)
K | I |,
R. Kerman, C. Phipps / J. Math. Anal. Appl. 359 (2009) 637–641
639
2. Proof of Theorem 1.1 We begin by showing that a weight w for which (1.1) holds must satisfy the doubling property
w ( y ) dy C
w ( y ) dy .
(2.1)
I
2I
Here, I ⊂ R is any interval with | I | π4 and 2I is the interval having the same centre as I and twice the length. j
j
To this end, set I k,l = [k π4 , (k + l) π4 ], I k = [(2k + j − 1) π8 , (2k + j ) π8 ] and I k,l = I k,l ∩ and j = 1, 2, . . . , min{8, 2l}. We claim
∞
j m=0 [ I k
+ mπ ], for k ∈ Z, l ∈ Z+
w ( y ) dy ≈ I k,l
w ( y ) dy ,
(2.2)
I k+l,l
where ‘≈’ indicates that both sides of the equation are within constant multiples of one another, the constants being independent of k and l. Given (2.2), (2.1) is readily seen to hold. j We will first consider the claim in the case l = 1. Indeed, when x ∈ I k+1,1 (for a fixed k ∈ Z and j = 1, 2),
π sin(x − y ) sin( 2 ) j 1 ( D χ j )(x) = I = . dy π k,1 I k,1 x− y 4 2 j
I k,1
So, we have
w ( y ) dy .
k,1
R
j
I k+1,1
( D χ j )(x) p w (x) dx C p I
w ( y ) dy 4 p
j
I k,1 j
j
Interchanging the roles of I k,1 and I k+1,1 , we get (2.2) for l = 1. In particular, since I k,1 = I k1,1 ∪ I k2,1 and I k+1,1 = I k1+1,1 ∪ I k2+1,1 ,
w ( y ) dy = I k+1,1
w ( y ) dy + I k1+1,1
w ( y ) dy ≈ I k2+1,1
w ( y ) dy + I k1,1
w ( y ) dy =
w ( y ) dy . I k,1
I k2,1
j Now for l > 1, there corresponds to each j a j so that for x ∈ I k+l,l
( D χ
j
I k,l
1 sin( π8 ) j I 4 · l π = 1 . )(x) k,l 2| I k,l | 2l π4 8 8 128
We must choose j so that the difference x − y avoids multiples of π . Indeed, this is achieved by choosing j = j when l is not a multiple of 4 and by taking j = ( j + 2) mod 8 when l is a multiple of 4 (the closest it comes is π8 ). So,
w ( y ) dy 128 p R
j I k+l,l
( D χ
j
I k,l
p )(x) w (x) dx C p
w ( y ) dy =
j
I k+l,l
w ( y ) dy . j
I k,l
We thus obtain (2.2), since
w ( y ) dy ≈
j
j
I k+l,l
w ( y ) dy =
w ( y ) dy . I k,l
j
I k,l
As (1.1) is equivalent to the dual inequality
( D f )(x) p w (x)1− p dx C
R
2I
f ( y ) p w ( y )1− p dy ,
R
it also implies
w ( y )1− p dy C
I
w ( y )1− p dy
for | I |
π 4
.
p =
p p−1
,
(2.3)
640
R. Kerman, C. Phipps / J. Math. Anal. Appl. 359 (2009) 637–641
We are now in a position to prove (1.1) implies (1.2). In view of the doubling properties (2.1) and (2.3), it suffices to show that (1.1) yields
1− p
w ( y ) dy
w ( y)
I +lπ
p −1
I
for I = [kπ , (k + l)π ], or, equivalently,
K |I |p
dy
w ( y )1− p dy
w ( y ) dy
p −1
K | J |p ,
(2.4)
J
J +(l+ 12 )π 1 2 for J = I 4k ,4l ∪ I 4k,4l .
Let f = w 1− p
χ J and suppose x ∈ J + (l + 12 )π . Then,
( D f )(x) = sin(x − y ) w ( y )1− p dy c w ( y )1− p dy . x− y 2| J | J
Hence,
1
| J|
J
w ( y )1− p dy
J
p
( D f )(x) p w (x) dx
w ( y ) dy K R
J +(l+ 12 )π
w ( y )(1− p ) p w ( y ) dy
K
by (1.1)
J
w ( y )1− p dy
K J
and we have (2.4). Observe that the translation by (l +
∈ [ π4 , 34π ].
x − y mod π Next, we prove the sufficiency of (1.2) for (1.1). Now, +1 ) π ∞ (k
( D f )(x) p w (x) dx =
π ensures that x − y avoids integer multiples of π . Indeed,
( D f )(x) p w (x) dx,
k=−∞ kπ
R
with (k +1 ) π
( D f )(x) p w (x) dx =
kπ
(k +1)π
( j +1 ) π
j ∈Z |k− j |2
kπ
kπ
where ( A f )(x) = 41π ( j +1 ) π
x+2π x−2π
sin(x − y )
jπ
x− y
sin(x − y ) x− y
jπ
(k+1)π C
( j +1 ) π
j ∈Z |k− j |2
(k +2 ) π
f ( y ) dy + (k−1)π
sin(x − y ) x− y
jπ
sin(x − y ) x− y
( j +1 ) π
f ( y ) cos y
f ( y ) dy = sin x
x− y
jπ
( D f )(x) p w (x) dx C
R
f c ( y) x− y
R |x− y |2π
+
R |x− y |2π
f s ( y)
kπ
x− y
with f c ( y ) = f ( y ) cos y and f s ( y ) = f ( y ) sin y.
( j +1 ) π
dy − cos x jπ
f ( y ) sin y x− y
dy .
p
dy w (x) dx
p
dy w (x) dx +
R
p f ( y ) dy w (x) dx
p (k +1 ) π p ( A f )(x) w (x) dx , f ( y ) dy w (x) dx +
| f ( y )| dy. Again,
Altogether, then,
1 ) 2
( A f )(x) p w (x) dx
R. Kerman, C. Phipps / J. Math. Anal. Appl. 359 (2009) 637–641
641
But, for x ∈ [kπ , (k + 1)π ], k ∈ Z,
|x− y |2π
g ( y) x− y
dy =
( j +1 ) π
|k− j |3
=
π
( j +1 ) π
|k− j |3
g ( y)
(k +3 ) π
dy +
g ( y ) dy
+O
+
Also,
∞
( Ag )(x) = O
j =−∞
π
jπ
( j +1 ) π jπ
| g ( y )| dy
1 + (k − j )2
( D f )(x) p w (x) dx C
jπ
g ( y ) dy
+O
∞
( j +1 ) π jπ
j =−∞
x− y
−
1
π (k − j )
dy
| g ( y )| dy
1 + (k − j )2
.
.
∞ ∞ ∞ p (h F c )(k) p w k + (h F s )(k) p w k + (p F )(k) w k ,
(k+1)π
k=−∞
(k+1)π
∞ ( D f )(x) p w (x) dx C
R
1
where F (k) = kπ f ( y ) dy, F c (k) = kπ f ( y ) cos y dy, F s (k) = global A p for w is equivalent to discrete A p for { w k }, we have
dy
k=−∞
R
g ( y)
|k− j |3
k− j
|k− j |1
We conclude
( j +1 ) π
1
( j +1 ) π
(k−3)π
=
g ( y) x− y
x+2π
(k−3)π
k− j
jπ
(k +3)π
x −2 π
dy +
x− y
jπ
1
g ( y)
k=−∞
(k+1)π kπ
f ( y ) sin y dy and w k =
(k+1)π kπ
w ( y ) dy. As
(k+1)π p (k+1)π f ( y ) dy w ( y ) dy
k=−∞
kπ
kπ
(k+1)π (k+1)π p −1 (k+1)π p 1− p
C f ( y ) w ( y ) dy w ( y) dy w ( y ) dy C
kπ
f ( y ) p w ( y ) dy .
kπ
kπ
R
Acknowledgment The first author would like to express his gratitude to Jan Lang for valuable discussions about the Dirichlet transform.
References [1] Kenneth F. Andersen, Inequalities with weights for discrete Hilbert transforms, Canad. Math. Bull. 20 (1) (1977) 9–16. [2] Richard Hunt, Benjamin Muckenhoupt, Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973) 227–251. [3] Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972) 207–226. [4] Benjamin Muckenhoupt, Two weight function norm inequalities for the Poisson integral, Trans. Amer. Math. Soc. 210 (1975) 225–231.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Weighted norm inequalities for the Dirichlet transform Ronald Kerman a,1 , Colin Phipps b,∗,2 a b
Department of Mathematics, Brock University, 500 Glenridge Ave., St. Catharines, Ontario L2S 3A1, Canada Department of Applied Mathematics, University of Waterloo, 200 University Ave. W., Waterloo, Ontario N2L 3G1, Canada
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 11 September 2008 Available online 12 June 2009 Submitted by L. Grafakos
The Dirichlet transform is defined for suitable functions by
( D f )(x) :=
Keywords: Dirichlet transform Weights
∞
1
π
sin(x − y )
−∞
x− y
f ( y ) dy ,
x ∈ R.
We show that for 1 < p < ∞ and nonnegative w (x) on R
( D f )(x) p w (x) dx C pp
R
f ( y ) p w ( y ) dy ,
R
with C p > 0 independent of f , if and only if there exists K > 0 such that
w (x) dx I
w (x)
1 − p− 1
p −1 dx
K |I |p
I
for all intervals I ⊂ R with length | I | π .
© 2009 Elsevier Inc. All rights reserved.
1. Introduction The Dirichlet transform defined for suitable functions by
( D f )(x) :=
1
π
∞
sin(x − y )
−∞
x− y
f ( y ) dy ,
x ∈ R,
is central to the inversion theory of the Fourier transform on R. In this paper, we characterize, for fixed p, 1 < p < ∞, the nonnegative measurable (weight) functions w on R for which
( D f )(x) p w (x) dx C pp
R
f ( y ) p w ( y ) dy ,
R
with C p > 0 being independent of f . Our result is given in
* 1 2
Corresponding author. Fax: +1 519 746 4319. E-mail addresses: [email protected] (R. Kerman), [email protected] (C. Phipps). Supported by NSERC grant A4021. Supported in part by an NSERC USRA and NSERC grant A4021.
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2009.06.028
©
2009 Elsevier Inc. All rights reserved.
(1.1)
638
R. Kerman, C. Phipps / J. Math. Anal. Appl. 359 (2009) 637–641
Theorem 1.1. Fix an index p, 1 < p < ∞ and let w be a weight function on R. Then, (1.1) holds if and only if there exists a constant K > 0 such that
w (x) dx
1 − p− 1
w (x)
I
p −1 K |I |p ,
dx
(1.2)
I
for all intervals I ⊂ R with length | I | π . If (1.2) holds for all intervals I ⊂ R, then w is said to satisfy the A p condition introduced in [3]. Thus, we will say the 1 satisfies the global, but not the weights in Theorem 1.1 satisfy the global A p condition. Observe that w (x) = max 1, 1 1+log |x|
unrestricted A p condition. Global A p may be thought of as a discrete version of the A p condition. Indeed, fix p, 1 < p < ∞, and consider a doubly infinite (weight) sequence { w k }k∈Z , w k 0. The discrete A p condition for such a sequence requires the existence of a constant K > 0 such that
n
wk
k=m
n
1 − p− 1
p −1 K (n − m + 1) p ,
wk
(1.3)
k=m
for all m, n ∈ Z with m n. It is a straightforward exercise to verify that (1.2) for a weight w on R is equivalent to (1.3) for
(k+1)π
the sequence { w k }k∈Z , when w k = kπ w (x) dx, k ∈ Z. Now, it is shown in [2, Theorem 10], that for suitable sequences a = {ak }k∈Z , the discrete Hilbert transform
(ha)n :=
1
π
∞
ak
k=−∞ k=n
k−n
n ∈ Z,
,
satisfies ∞ ∞ (ha)n p w n C pp |ak | p w k , n=−∞
k=−∞
where C p > 0 is independent of {ak }k∈Z , if and only if (1.3) holds (see [1]). This will be crucial for proving the sufficiency of (1.2) in Theorem 1.1. Also important is the fact that (1.3) implies the existence of C p > 0, independent of {ak }, with ∞ ∞ (pa)n p w n C pp |ak | p w k , n=−∞
k=−∞
in which
(pa)n :=
∞ k=−∞
ak 1 + (n − k)2
n ∈ Z,
,
is the discrete Poisson transform of a = {ak }k∈Z . The latter is readily proved using the methods of [4]. The proof of Theorem 1.1 is given in the next section. In conclusion, we mention that our methods yield the expected: Theorem 1.2. Fix an index p, 1 < p < ∞ and let w be a weight function on R. Then, the weak-type inequality
w (x) dx {x∈R: |( D f )(x)|>t }
Cp t
p
f (x) p w (x) dx,
R
holds, with C p > 0 independent of f , if and only if there exists a constant K > 0 such that for all intervals I ⊂ R, | I | π , w satisfies (1.2) when p > 1 and
w (x) dx ess sup
I
when p = 1.
x∈ I
1 w (x)
K | I |,
R. Kerman, C. Phipps / J. Math. Anal. Appl. 359 (2009) 637–641
639
2. Proof of Theorem 1.1 We begin by showing that a weight w for which (1.1) holds must satisfy the doubling property
w ( y ) dy C
w ( y ) dy .
(2.1)
I
2I
Here, I ⊂ R is any interval with | I | π4 and 2I is the interval having the same centre as I and twice the length. j
j
To this end, set I k,l = [k π4 , (k + l) π4 ], I k = [(2k + j − 1) π8 , (2k + j ) π8 ] and I k,l = I k,l ∩ and j = 1, 2, . . . , min{8, 2l}. We claim
∞
j m=0 [ I k
+ mπ ], for k ∈ Z, l ∈ Z+
w ( y ) dy ≈ I k,l
w ( y ) dy ,
(2.2)
I k+l,l
where ‘≈’ indicates that both sides of the equation are within constant multiples of one another, the constants being independent of k and l. Given (2.2), (2.1) is readily seen to hold. j We will first consider the claim in the case l = 1. Indeed, when x ∈ I k+1,1 (for a fixed k ∈ Z and j = 1, 2),
π sin(x − y ) sin( 2 ) j 1 ( D χ j )(x) = I = . dy π k,1 I k,1 x− y 4 2 j
I k,1
So, we have
w ( y ) dy .
k,1
R
j
I k+1,1
( D χ j )(x) p w (x) dx C p I
w ( y ) dy 4 p
j
I k,1 j
j
Interchanging the roles of I k,1 and I k+1,1 , we get (2.2) for l = 1. In particular, since I k,1 = I k1,1 ∪ I k2,1 and I k+1,1 = I k1+1,1 ∪ I k2+1,1 ,
w ( y ) dy = I k+1,1
w ( y ) dy + I k1+1,1
w ( y ) dy ≈ I k2+1,1
w ( y ) dy + I k1,1
w ( y ) dy =
w ( y ) dy . I k,1
I k2,1
j Now for l > 1, there corresponds to each j a j so that for x ∈ I k+l,l
( D χ
j
I k,l
1 sin( π8 ) j I 4 · l π = 1 . )(x) k,l 2| I k,l | 2l π4 8 8 128
We must choose j so that the difference x − y avoids multiples of π . Indeed, this is achieved by choosing j = j when l is not a multiple of 4 and by taking j = ( j + 2) mod 8 when l is a multiple of 4 (the closest it comes is π8 ). So,
w ( y ) dy 128 p R
j I k+l,l
( D χ
j
I k,l
p )(x) w (x) dx C p
w ( y ) dy =
j
I k+l,l
w ( y ) dy . j
I k,l
We thus obtain (2.2), since
w ( y ) dy ≈
j
j
I k+l,l
w ( y ) dy =
w ( y ) dy . I k,l
j
I k,l
As (1.1) is equivalent to the dual inequality
( D f )(x) p w (x)1− p dx C
R
2I
f ( y ) p w ( y )1− p dy ,
R
it also implies
w ( y )1− p dy C
I
w ( y )1− p dy
for | I |
π 4
.
p =
p p−1
,
(2.3)
640
R. Kerman, C. Phipps / J. Math. Anal. Appl. 359 (2009) 637–641
We are now in a position to prove (1.1) implies (1.2). In view of the doubling properties (2.1) and (2.3), it suffices to show that (1.1) yields
1− p
w ( y ) dy
w ( y)
I +lπ
p −1
I
for I = [kπ , (k + l)π ], or, equivalently,
K |I |p
dy
w ( y )1− p dy
w ( y ) dy
p −1
K | J |p ,
(2.4)
J
J +(l+ 12 )π 1 2 for J = I 4k ,4l ∪ I 4k,4l .
Let f = w 1− p
χ J and suppose x ∈ J + (l + 12 )π . Then,
( D f )(x) = sin(x − y ) w ( y )1− p dy c w ( y )1− p dy . x− y 2| J | J
Hence,
1
| J|
J
w ( y )1− p dy
J
p
( D f )(x) p w (x) dx
w ( y ) dy K R
J +(l+ 12 )π
w ( y )(1− p ) p w ( y ) dy
K
by (1.1)
J
w ( y )1− p dy
K J
and we have (2.4). Observe that the translation by (l +
∈ [ π4 , 34π ].
x − y mod π Next, we prove the sufficiency of (1.2) for (1.1). Now, +1 ) π ∞ (k
( D f )(x) p w (x) dx =
π ensures that x − y avoids integer multiples of π . Indeed,
( D f )(x) p w (x) dx,
k=−∞ kπ
R
with (k +1 ) π
( D f )(x) p w (x) dx =
kπ
(k +1)π
( j +1 ) π
j ∈Z |k− j |2
kπ
kπ
where ( A f )(x) = 41π ( j +1 ) π
x+2π x−2π
sin(x − y )
jπ
x− y
sin(x − y ) x− y
jπ
(k+1)π C
( j +1 ) π
j ∈Z |k− j |2
(k +2 ) π
f ( y ) dy + (k−1)π
sin(x − y ) x− y
jπ
sin(x − y ) x− y
( j +1 ) π
f ( y ) cos y
f ( y ) dy = sin x
x− y
jπ
( D f )(x) p w (x) dx C
R
f c ( y) x− y
R |x− y |2π
+
R |x− y |2π
f s ( y)
kπ
x− y
with f c ( y ) = f ( y ) cos y and f s ( y ) = f ( y ) sin y.
( j +1 ) π
dy − cos x jπ
f ( y ) sin y x− y
dy .
p
dy w (x) dx
p
dy w (x) dx +
R
p f ( y ) dy w (x) dx
p (k +1 ) π p ( A f )(x) w (x) dx , f ( y ) dy w (x) dx +
| f ( y )| dy. Again,
Altogether, then,
1 ) 2
( A f )(x) p w (x) dx
R. Kerman, C. Phipps / J. Math. Anal. Appl. 359 (2009) 637–641
641
But, for x ∈ [kπ , (k + 1)π ], k ∈ Z,
|x− y |2π
g ( y) x− y
dy =
( j +1 ) π
|k− j |3
=
π
( j +1 ) π
|k− j |3
g ( y)
(k +3 ) π
dy +
g ( y ) dy
+O
+
Also,
∞
( Ag )(x) = O
j =−∞
π
jπ
( j +1 ) π jπ
| g ( y )| dy
1 + (k − j )2
( D f )(x) p w (x) dx C
jπ
g ( y ) dy
+O
∞
( j +1 ) π jπ
j =−∞
x− y
−
1
π (k − j )
dy
| g ( y )| dy
1 + (k − j )2
.
.
∞ ∞ ∞ p (h F c )(k) p w k + (h F s )(k) p w k + (p F )(k) w k ,
(k+1)π
k=−∞
(k+1)π
∞ ( D f )(x) p w (x) dx C
R
1
where F (k) = kπ f ( y ) dy, F c (k) = kπ f ( y ) cos y dy, F s (k) = global A p for w is equivalent to discrete A p for { w k }, we have
dy
k=−∞
R
g ( y)
|k− j |3
k− j
|k− j |1
We conclude
( j +1 ) π
1
( j +1 ) π
(k−3)π
=
g ( y) x− y
x+2π
(k−3)π
k− j
jπ
(k +3)π
x −2 π
dy +
x− y
jπ
1
g ( y)
k=−∞
(k+1)π kπ
f ( y ) sin y dy and w k =
(k+1)π kπ
w ( y ) dy. As
(k+1)π p (k+1)π f ( y ) dy w ( y ) dy
k=−∞
kπ
kπ
(k+1)π (k+1)π p −1 (k+1)π p 1− p
C f ( y ) w ( y ) dy w ( y) dy w ( y ) dy C
kπ
f ( y ) p w ( y ) dy .
kπ
kπ
R
Acknowledgment The first author would like to express his gratitude to Jan Lang for valuable discussions about the Dirichlet transform.
References [1] Kenneth F. Andersen, Inequalities with weights for discrete Hilbert transforms, Canad. Math. Bull. 20 (1) (1977) 9–16. [2] Richard Hunt, Benjamin Muckenhoupt, Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973) 227–251. [3] Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972) 207–226. [4] Benjamin Muckenhoupt, Two weight function norm inequalities for the Poisson integral, Trans. Amer. Math. Soc. 210 (1975) 225–231.