Bulletin T.CXXIII de l’Académie Serbe des Sciences et des Arts - 2002
Classe des Sciences mathématiques et naturelles
Sciences mathématiques, No 27
AN EXTENSION ON TRILINEAR HILBERT TRANSFORM
ANETA BUČKOVSKA
(Presented at the 5th Meeting, held on June 21, 2002)
A b s t r a c t. The Trilinear Hilbert transform H : Lp × Lq × A → Lr
is extended to DLp × DLq × DA → DLr as a continuous mapping.
AMS Mathematics Subject Classification (2000): 44A15, 46F12
Key Words: trilinear Hilbert transform, multi-linear operators
1. Introduction
In several papers Lacey and Thiele ([3]-[5]) had studied the continuity
of the bilinear Hilbert transform
Z
Hα (f, g)(x) = p.v.
f (x − y)g(x + αy)
dy
,
y
α ∈ R \ {0, −1} ,
where f ∈ L2 (R) and g ∈ L∞ (R), respectively f ∈ Lp1 (R) and g ∈ Lp2 (R),
2/3 < p =
p1 p2
, 1 < p1 , p2 < ∞, q = p/(p − 1), q1 = p1 /(p1 − 1). (1)
p1 + p2
Their main result is the affirmative answer on the Calderon conjecture, first
for p1 = 2, p2 = ∞ ([3]), then for p1 and p2 with the assumptions given
above ([5]). Their main result is
||Hα (f, a)||Lp ≤ C||f ||Lp1 ||a||Lp2 , f ∈ Lp1 , a ∈ Lp2 ,
(2)
70
Aneta Bučkovska
where C > 0 depends on α, p1 , p2 (for p1 = 2, p2 = ∞, p equals 2).
In [1] the bilinear Hilbert transform Hα : L2 × L∞ → L2 respectively,
0 × D ∞ → D 0 , respectively,
Hα : Lp1 × Lp2 → Lp , was extended to DL
2
L
L2
0
0
DLq × DLp2 → Dq1 , (with suitable parameters) as a hypocontinuous, respectively, continuous mapping and in [2] the Bilinear Hilbert transform of
ultradistributions was defined.
In this paper we extend the trilinear Hilbert transform on Lp × Lq × A to
r
L whenever 1 < p, q ≤ ∞, p1 + 1q = p1 , and 23 < r < ∞ to DLp × DLq × DA →
DLr as a continuous mapping.
2. Preliminaries
Denote by A the Wiener algebra: the space of functions whose Fourier
transform is in L1 . Denote by DA a space of smooth functions ϕ on R such
that for every α ∈ N0 hold
F(ϕ(α) ) = ξ α ϕ̂ ∈ L1 .
Define ||ϕ||k = sup||ξ α ϕ̂||L1 , k ∈ N0 . In [7] it is proved that the mapping
0
T : Lp1 × · · · × Lpn−s−1 × A × · · · × A → Lpn−s
1
is continuous, where p11 + · · · + pn−1
, 1 < pi ≤ ∞, for i = 1, 2, ..., n − s − 1
and
(n − s) − 2(k − s) + r
1
1
<
+ ··· +
pi1
pir
2
for all 1 ≤ i1 < · · · < ir ≤ n − s, 1 ≤ r ≤ n − s.
Thus, for instance Trilinear Hilbert Transform
Z
T (f, g, h) =
f (x − t)g(x + t)h(x + 2t)
maps Lp × Lq × A to Lr whenever 1 < p, q ≤ ∞,
That is the case when n = 4, k = 2 and s = 1.
1
p
dt
t
+ 1q = p1 , and
2
3
< r < ∞.
3. Mappings Tf,g , Tf,h and Tg,h
Theorem 3.1Let f ∈ DLp , g ∈ DLq and h ∈ DA . Then for 1 < p, q < ∞
mappings Tf,g , Tf,h and Tg,h from DA to DLr , from DLq to DLr and from
DLp to DLr respectively are linear and continuous.
71
An extension on Trilinear Hilbert transform
P r o o f. Let f ∈ DLp , g ∈ DLq and h ∈ DA .
Z
T (f, g, h)(x) = lim
ε→0+ |t|>ε
Z
=
|t|>N
f (x − t)g(x + t)h(x + 2t)
Z
+ lim
Z
f (x − t)g(x + t)h(x + 2t)
ε→0+
N ≥|t|>ε
dt
t
dt
t
f (x − t)g(x + t)h(x + 2t)
dt
t
dt
t
|t|>N
Z
g(x + t) − g(x)
f (x − t)
+ lim
h(x + 2t)dt
+
t
ε→0
N ≥|t|>ε
Z
dt
+ lim
f (x − t)g(x)h(x + 2t)
+
t
ε→0
N ≥|t|>ε
Z
dt
=
f (x − t)g(x + t)h(x + 2t)
t
|t|>N
Z
g(x + t) − g(x)
+ lim
f (x − t)
+
t
ε→0
N ≥|t|>ε
Z
dy
y
·h(x + 2t)dt + lim
f (x − )g(x)h(x + y)
2
y
ε→0+
2N ≥|y|>2ε
Z
dt
=
f (x − t)g(x + t)h(x + 2t)
t
|t|>N
Z
g(x + t) − g(x)
+ lim
h(x + 2t)dt
f (x − t)
+
t
ε→0
N ≥|t|>ε
Z
y
h(x + y) − h(x)
+ lim
f (x − )g(x)
dy
+
2
y
ε→0
2N ≥|y|>2ε
Z
y
dy
+ lim
f (x − )g(x)h(x)
2
y
ε→0+
2N ≥|y|>2ε
Z
dt
=
f (x − t)g(x + t)h(x + 2t)
t
|t|>N
Z
g(x + t) − g(x)
+ lim
f (x − t)
h(x + 2t)dt
t
ε→0+
N ≥|t|>ε
Z
y
h(x + y) − h(x)
+ lim
f (x − )g(x)
dy
2
y
ε→0+ 2N ≥|y|>2ε
=
f (x − t)g(x + t)h(x + 2t)
72
Aneta Bučkovska
Z
− lim
Z
=
ε→0+
N ≥|z|>ε
f (x + z)g(x)h(x)
dz
z
dt
t
|t|>N
Z
g(x + t) − g(x)
+ lim
f (x − t)
h(x + 2t)dt
t
ε→0+
N ≥|t|>ε
Z
y
h(x + y) − h(x)
f (x − )g(x)
+ lim
dy
2
y
ε→0+ 2N ≥|y|>2ε
Z
f (x + z) − f (x)
− lim
· g(x)h(x)dz
+
z
ε→0
N ≥|z|>ε
f (x − t)g(x + t)h(x + 2t)
Z
dz
z
N ≥|z|>ε
Z
dt
=
f (x − t)g(x + t)h(x + 2t)
t
|t|>N
Z
g(x + t) − g(x)
+ lim
f (x − t)
h(x + 2t)dt
t
ε→0+
N ≥|t|>ε
Z
h(x + y) − h(x)
y
+ lim
dy
f (x − )g(x)
+
2
y
ε→0
2N ≥|y|>2ε
Z
f (x + z) − f (x)
− lim
g(x)h(x)dz
+
z
ε→0
N ≥|z|>ε
Z
dt
=
f (x − t)g(x + t)h(x + 2t)
t
|t|>N
f (x)g(x)h(x)
− lim
ε→0+
Z
+ lim
ε→0+
f (x − t)G(x, t)h(x + 2t)dt
Z
+ lim
ε→0+
− lim
ε→0+
N ≥|t|>ε
y
f (x − )g(x)H(x, y)dy
2
Z 2N ≥|y|>2ε
N ≥|z|>ε
where
F (x, z)g(x)h(x)dz
(
G(x, t) =
(
H(x, y) =
g(x+t)−g(x)
,
t
d
dx g(x),
t 6= 0
t = 0, x ∈ R,
h(x+y)−h(x)
,
y
d
dx h(x),
y 6= 0
.
y = 0, x ∈ R
73
An extension on Trilinear Hilbert transform
(
F (x, z) =
f (x+z)−f (x)
,
z
d
dx f (x),
z=
6 0
z = 0, x ∈ R.
It is obvious that F (x, z), G(x, t) and H(x, y), along with all of their
partial derivatives are continuous functions of x, y, z, t, respectively.
Let κN be characteristic function of (−∞, −N ]∪[N, ∞). Then, according
to Muscalu’s proof in [7] we have
∂
1
∂
f (x − t)g(x + t)h(x + 2t) ||r ≤ C||κN f (x − ·)||p ||g||q ||h||A ,
∂x
t
∂x
1
∂
∂
||κN (t)f (x − t) g(x + t)h(x + 2t) ||r ≤ C||κN f (x − ·)||p || g||q ||h||A ,
∂x
t
∂x
∂
1
∂
||κN (t)f (x − t)g(x + t) h(x + 2t) ||r ≤ C||κN f (x − ·)||p ||g||q || h||A .
∂x
t
∂x
So we get
||κN (t)
Z
∂
dt
f (x − t)g(x + t)h(x + 2t)
t
|t|>N ∂x
Z
∂
dt
+
f (x − t) g(x + t)h(x + 2t)
∂x
t
|t|>N
Z
dt
∂
+
f (x − t)g(x + t) h(x + 2t)
∂x
t
|t|>N
Z
∂
+ lim
[f (x − t)G(x, t)h(x + 2t)]dt
+
ε→0
N ≥|t|>ε ∂x
Z
∂
[f (x − y/2)g(x)H(x, y)]dy
+ lim
∂x
ε→0+
2N ≥|y|>2ε
Z
∂
− lim
[F (x, z)g(x)h(x)]dz
+
ε→0
N ≥|z|>ε ∂x
Z
∂
dt
=
f (x − t)g(x + t)h(x + 2t)
t
|t|>N ∂x
Z
∂
dt
+
f (x − t) g(x + t)h(x + 2t)
∂x
t
|t|>N
Z
∂
dt
+
f (x − t)g(x + t) h(x + 2t)
∂x
t
|t|>N
Z
∂
+ lim
[f (x − t)G(x, t)h(x + 2t)]dt
∂x
ε→0+
N ≥|t|>ε
d
T (f, g, h)(x) =
dx
74
Aneta Bučkovska
Z
+ lim
ε→0+
∂
f (x − t)G(x, t)h(x + 2t)dt
∂x
N ≥|t|>ε
Z
∂
∂
g(x + t) − ∂x
g(x)
h(x + 2t)dt
+
t
ε→0
N ≥|t|>ε
Z
∂
+ lim
f (x − t)G(x, t) h(x + 2t)dt
∂x
ε→0+
N ≥|t|>ε
Z
∂
y
+ lim
f (x − )g(x)H(x, y)dy
+
2
ε→0
2N ≥|y|>2ε ∂x
Z
y ∂
f (x − ) g(x)H(x, y)dy
+ lim
2 ∂x
ε→0+
2N ≥|y|>2ε
f (x − t) ∂x
+ lim
Z
+ lim
ε→0+
Z
∂
h(x + y) −
y
f (x − )g(x) ∂x
2
y
2N ≥|y|>2ε
∂
∂x h(x)
dy
∂
+ z) − ∂x
f (x)
g(x)h(x)dz
+
z
ε→0
N ≥|z|>ε
Z
∂
− lim
F (x, z) g(x)h(x)dz
+
∂x
ε→0
N ≥|z|>ε
Z
∂
− lim
F (x, z)g(x) h(x)dz
∂x
ε→0+ N ≥|z|>ε
Z
dt
∂
f (x − t)g(x + t)h(x + 2t)
=
t
|t|>N ∂x
Z
∂
dt
+
f (x − t) g(x + t)h(x + 2t)
∂x
t
|t|>N
Z
dt
∂
+
f (x − t)g(x + t) h(x + 2t)
∂x
t
|t|>N
Z
∂
g(x + t) − g(x)
+ lim
f (x − t)
h(x + 2t)dt
+
t
ε→0
N ≥|t|>ε ∂x
Z
dt
∂
+ lim
f (x − t) g(x + t)h(x + 2t)
∂x
t
ε→0+ N ≥|t|>ε
Z
∂
dt
− g(x) lim
f (x − t)h(x + 2t)
+
∂x
t
ε→0
N ≥|t|>ε
Z
g(x + t) − g(x) ∂
·
h(x + 2t)dt
+ lim
f (x − t)
+
t
∂x
ε→0
N ≥|t|>ε
Z
∂
y
h(x + y) − h(x)
+ lim
f (x − )g(x)
dy
2
y
ε→0+ 2N ≥|y|>2ε ∂x
− lim
∂
∂x f (x
An extension on Trilinear Hilbert transform
∂
g(x) lim
∂x
ε→0+
Z
y h(x + y) − h(x)
f (x − )
dy
2
y
2N ≥|y|>2ε
Z
y ∂
dy
+g(x) lim
f (x − ) h(x + y)
+
2 ∂x
y
ε→0
2N ≥|y|>2ε
Z
y dy
∂
f (x − )
−g(x) h(x) lim
∂x
2 y
ε→0+ 2N ≥|y|>2ε
Z
∂
dz
−g(x)h(x) lim
f (x + z)
+
z
ε→0
N ≥|z|>ε ∂x
Z
dz
∂
+ f (x)g(x)h(x) lim
∂x
ε→0+ N ≥|z|>ε z
Z
∂
f (x + z) − f (x)
− g(x)h(x) lim
dz
∂x
z
ε→0+ N ≥|z|>ε
Z
∂
f (x + z) − f (x)
−g(x) h(x) lim
dz
+
∂x
z
ε→0
N ≥|z|>ε
Z
dt
∂
f (x − t)g(x + t)h(x + 2t)
=
∂x
t
|t|>N
Z
∂
dt
+
f (x − t) g(x + t)h(x + 2t)
∂x
t
|t|>N
Z
∂
dt
+
f (x − t)g(x + t) h(x + 2t)
∂x
t
|t|>N
Z
dt
∂
f (x − t)g(x + t)h(x + 2t)
+ lim
t
ε→0+
N ≥|t|>ε ∂x
Z
dt
∂
+ lim
f (x − t) g(x + t)h(x + 2t)
+
∂x
t
ε→0
N ≥|t|>ε
∂
dt
+ lim ∈tN ≥|t|>ε f (x − t)g(x + t) h(x + 2t)
∂x
t
ε→0+
Z
∂
dt
= lim
f (x − t)g(x + t)h(x + 2t)
+
t
ε→0
|t|>ε ∂x
Z
dt
∂
+ lim
f (x − t) g(x + t)h(x + 2t)
+
∂x
t
ε→0
|t|>ε
Z
∂
dt
+ lim
f (x − t)g(x + t) h(x + 2t)
∂x
t
ε→0+ |t|>ε
+
= T (f 0 , g, h)(x) + T (f, g 0 , h)(x) + T (f, g, h0 )(x)
75
76
Aneta Bučkovska
Using a similar technique, we can show by induction that
[T (f, g, h)]
(n)
(x) =
à !à !
n X
k
X
n
k
k=0 m=0
k
m
T (f (n−k) , g (k−m) , h(m) )(x)
for all n ∈ N .
Now we have
(n)
||[T (f, g, h)]
(x)||r ≤
à !à !
n X
k
X
n
k
k=0 m=0
≤
n X
k
X
k=0 m=0
k
m
||T (f (n−k) , g (k−m) , h(m) )(x)||r
à !à !
n
k
k
||f (n−k) ||p · ||g (k−m) ||q · ||h(m) ||A
m
Theorem 3.2Bilinear mappings
(f, g) 7→ Th (f, g)f romDLp × DLq → DLr ,
(f, h) 7→ Tg (f, h)f romDLp × DA → DLr ,
(g, h) 7→ Tf (g, h)f romDLq × DA → DLr ,
are continuous.
P r o o f. According to Theorem 1 we have that these mappings are
separately continuous. Then Corollary of Theorem 34.1 from [11] implies
that these mappings are continuous.
2
Corollary 3.3Mapping
T : DLp × DLq × DA → DLr ,
is continuous.
REFERENCES
[1] A. B u č k o v s k a, S. P i l i p o v ić, An Extension of Bilinear Hilbert Transform
to Distributions, Integral Transforms and Special Functions, Taylor and Francis, accepted for publication.
[2] A. B u č k o v s k a, S. P i l i p o v i ć, Bilinear Hilbert Transform of Ultradistributions, Integral Transforms and Special Functions, Taylor and Francis, accepted for
publication.
An extension on Trilinear Hilbert transform
77
[3] M. T. L a c e y and C. M. T h i e l e, Lp estimates on the bilinear Hilbert transform
for 2 < p < ∞, Annals of Math.,146(1997), 693-724.
[4] M. T. L a c e y and C. M. T h i e l e, On Calderon’s Conjecture for the Bilinear
Hilbert Transform, Proc.Natl.Acad.Sci USA, 95(1998), 4828-4830.
[5] M. T. L a c e y and C. M. T h i e l e, On Calderon’s Conjecture, Annals of Math.,
149(1999), 475-496.
[6] J. H o r v a t, Topological Vector Spaces and Distributions, Vol. 1, Addison Wesley,
Massachusetts, 1966.
[7] C. M u s c a l u, T. T a o, C. T h i e l e, Multi-linear Operators Given by Singular
Multipliers, Journal of AMS, 2001.
[8] W. R u d i n, Real and Complex Analysis, McGraw-Hill, 1987
[9] L. S c h w a r t z, Théorie des distributions, I.2nd ed., Hermann, Paris, 1957.
[10] E. M. S t e i n and G. W e i s s, Introduction to Fourier Analysis on Euclidean Spaces,
Princeton University Press, 1971
[11] F. T r e v e s, Topological Vector Spaces, Distributions and Kernels, Acad. Press,
New York, 1967
Faculty of Electrical Engineering
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