International Journal of Analysis and Applications
ISSN 2291-8639
Volume 7, Number 2 (2015), 171-178
http://www.etamaths.com

THE S-TRANSFORM ON HARDY SPACES AND ITS DUALS

SUNIL KUMAR SINGH∗ AND BABY KALITA

Abstract. In this paper, continuity and boundedness results for the contin-

uous S-transform in BMO and Hardy spaces are obtained. Furthermore, the

continuous S-transform is also studied on the weighted BMOk and weighted
Hardy spaces associated with a tempered weight function which was proposed

by L. Hörmander in the study of the theory of partial differential equations.

1. Introduction

The S-transform is a time-frequency localization technique that has characteris-
tics superior to both of the Fourier transform and the wavelet transform[12]. The
n-dimensional continuous S-transform of a function f with respect to the window
function ω is defined as [13]

(1.1) (Sωf)(τ,ξ) =

∫
Rn
f(t) ω(τ − t,ξ) e−i2π〈ξ,t〉 dt, for τ,ξ ∈ Rn,

provided the integral exists.
In signal analysis, at least in dimension n = 1, R2n is called the time-frequency

plane, and in physics R2n is called the phase space[11]. Equation(1.1) can be
rewritten as a convolution

(1.2) (Sωf)(τ,ξ) =
(
f(·)e−i2π〈ξ,·〉 ∗ω(·,ξ)

)
(τ).

Applying the convolution property for the Fourier transform in (1.2), we obtain

(1.3) (Sωf)(τ,ξ) = F
−1
{
f̂(· + ξ) ω̂(·,ξ)

}
(τ),

where f̂(η) = (Ff)(η) =
∫
Rn f(t) e

−i2π〈η,t〉dt, is the Fourier transform of f.

2. THE S-TRANSFORM ON BMO SPACES

The bounded mean oscillation space BMO(Rn) was first introduced by F. John
and L. Nirenberg in 1961 [3]. It is the dual space of the real Hardy space H1 and
serves in many ways as a substitute space for L∞. The BMO(Rn) space has become
extremely important in various areas of analysis including harmonic analysis, PDEs
and function theory.

2010 Mathematics Subject Classification. 65R10, 32A37, 30H10.

Key words and phrases. S-transform; BMO space; Hardy space.

c©2015 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

171



172 SINGH AND KALITA

Definition 2.1. The bounded mean oscillation space BMO(Rn) is defined as the
space of all locally Lebesgue integrable functions defined on Rn such that

‖ f ‖BMO= sup
B⊂Rn

1

|B|

∫
B

|f(x) −fB|dx < ∞,

here the supremum is taken over the ball B in Rn of measure |B| and fB stands
for the mean of f on B, namely

(2.1) fB :=
1

|B|

∫
B

f(x)dx ≤
1

|B|

∫
B

|f(x)|dx ≤ m < ∞.

Lemma 2.1. Let f ∈ L1(Rn), then e−i2π<ξ,·>f(·) ∈ L1(Rn) and

‖ e−i2π<ξ,·>f(·) ‖BMO ≤ ‖ f ‖BMO +2m

where m is a constant given in equation (2.1).

Proof.

‖ e−i2π<ξ,·>f(·) ‖BMO

= sup
B⊂Rn

1

|B|

∫
B

∣∣∣∣e−i2π<ξ,x>f(x) − 1|B|
∫
B

e−i2π<ξ,t>f(t)dt

∣∣∣∣dx
= sup

B⊂Rn

1

|B|

∫
B

∣∣∣∣e−i2π<ξ,x>f(x) − e−i2π<ξ,x>|B|
∫
B

f(t)dt

+
e−i2π<ξ,x>

|B|

∫
B

f(t)dt−
1

|B|

∫
B

e−i2π<ξ,t>f(t)dt

∣∣∣∣dx
≤ sup

B⊂Rn

1

|B|

∫
B

(∣∣∣∣e−i2π<ξ,x>
(
f(x) −

1

|B|

∫
B

f(t)dt

)∣∣∣∣
+

∣∣∣∣ 1|B|
∫
B

f(t)dt

∣∣∣∣ +
∣∣∣∣ 1|B|

∫
B

e−i2π<ξ,t>f(t)dt

∣∣∣∣
)
dx

≤ sup
B⊂Rn

1

|B|

∫
B

|f(x) −fB|dx + sup
B⊂Rn

1

|B|

∫
B

|fB|dx

+ sup
B⊂Rn

1

|B|

∫
B

(
1

|B|

∫
B

|f(t)|dt
)
dx

≤ ‖ f ‖BMO +
1

|B|
m|B| +

1

|B|
m|B|

= ‖ f ‖BMO + 2m.

�

Theorem 2.2. Suppose ω(·,ξ) ∈ L1(Rn)
⋂
L2(Rn), then, for any fixed ξ ∈ Rn0 =

Rn \{0}, the operator Sω : BMO(Rn) → BMO(Rn) is continuous. Furthermore,
we have

‖ (Sωf)(·,ξ) ‖BMO ≤ ‖ ω(·,ξ) ‖L1 (‖ f ‖BMO +2m) .

Proof. For any arbitrary ball B in Rn, we have

(Sωf)B(τ,ξ) =
1

|B|

∫
B

(Sωf)(τ,ξ)dτ

=
1

|B|

∫
B

∫
Rn
e−i2π<ξ,τ−x>f(τ −x)ω(x,ξ)dxdτ,



THE S-TRANSFORM ON HARDY SPACES AND BMO 173

and hence

|(Sωf)(τ,ξ) − (Sωf)B(τ,ξ)|

=

∣∣∣∣
∫
Rn
e−i2π<ξ,τ−x>f(τ −x)ω(x,ξ)dx

−
1

|B|

∫
B

∫
Rn
e−i2π<ξ,α−x>f(α−x)ω(x,ξ)dxdα

∣∣∣∣
=

∣∣∣∣
∫
Rn
e−i2π<ξ,τ−x>f(τ −x)ω(x,ξ)dx

−
∫
Rn
ω(x,ξ)

(
1

|B|

∫
B

e−i2π<ξ,α−x>f(α−x)dα
)
dx

∣∣∣∣
=

∣∣∣∣
∫
Rn
ω(x,ξ)

(
e−i2π<ξ,τ−x>f(τ −x)

−
1

|B|

∫
B

e−i2π<ξ,α−x>f(α−x)dα
)
dx

∣∣∣∣
≤

∫
Rn
|ω(x,ξ)|

∣∣∣∣e−i2π<ξ,τ−x>f(τ −x)
−

1

|B|

∫
B

e−i2π<ξ,α−x>f(α−x)dα
∣∣∣∣dx.

Therefore,

‖ (Sωf)(·,ξ) ‖BMO

= sup
B⊂Rn

1

|B|

∫
B

|(Sωf)(τ,ξ) − (Sωf)B(τ,ξ)|dτ

≤ sup
B⊂Rn

1

|B|

∫
B

(∫
Rn
|ω(x,ξ)|

∣∣∣∣e−i2π<ξ,τ−x>f(τ −x)
−

1

|B|

∫
B

e−i2π<ξ,α−x>f(α−x)dα
∣∣∣∣dx
)
dτ

=

∫
Rn
|ω(x,ξ)|

(
sup
K⊂Rn

1

|K|

∫
K

∣∣∣∣e−i2π<ξ,y>f(y)
−

1

|K|

∫
K

e−i2π<ξ,t>f(t)dt

∣∣∣∣dy
)
dx

≤ ‖ ω(·,ξ) ‖L1‖ e−i2π<ξ,·>f(·) ‖BMO,

here K = B −x for x ∈ Rn.
By using above lemma we get,

‖ (Sωf)(·,ξ) ‖BMO≤‖ ω(·,ξ) ‖L1 (‖ f ‖BMO +2m) .

�

3. THE S-TRANSFORM ON WEIGHTED BMO SPACES.

Definition 3.1. A positive function k defined on Rn is called a tempered weight
function[2] if there exists positive constants C and N such that

(3.1) k(ξ + η) ≤ (1 + C|ξ|)Nk(η) for all ξ,η ∈ Rn.



174 SINGH AND KALITA

Definition 3.2. For 1≤ p ≤∞, the weighted Lebesgue space Lpk(R
n) is defined as

the space of all measurable functions f on Rn such that

‖ f ‖Lp
k
=

(∫
Rn
|f(x)|pk(x)dx

)1
p

< ∞.

Definition 3.3. The weighted bounded mean oscillation space BMOk(Rn) is de-
fined as the space of all weighted Lebesgue integrable (locally) functions defined on
Rn such that

‖ f ‖BMOk = sup
B⊂Rn

1

|B|k

∫
B

|f(x) −fB|k(x)dx < ∞,

where the supremum is taken over the ball B in Rn and |B|k =
∫
B
k(x)dx.

Lemma 3.1. Let f ∈ L1k(R
n), then e−i2π<ξ,·>f(·) ∈ L1k(R

n) and

‖ e−i2π<ξ,·>f(·) ‖BMOk≤‖ f ‖BMOk + 2m,

where m is a constant defined in equation (2.1).

Proof.

‖ e−i2π<ξ,·>f(·) ‖BMOk

= sup
B⊂Rn

1

|B|k

∫
B

∣∣∣∣e−i2π<ξ,x>f(x) − 1|B|
∫
B

e−i2π<ξ,t>f(t)dt

∣∣∣∣k(x)dx
= sup

B⊂Rn

1

|B|k

∫
B

∣∣∣∣e−i2π<ξ,x>f(x) − e−i2π<ξ,x>|B|
∫
B

f(t)dt

+
e−i2π<ξ,x>

|B|

∫
B

f(t)dt−
1

|B|

∫
B

e−i2π<ξ,t>f(t)dt

∣∣∣∣k(x)dx
≤ sup

B⊂Rn

1

|B|k

∫
B

(∣∣∣∣e−i2π<ξ,x>
(
f(x) −

1

|B|

∫
B

f(t)dt

)∣∣∣∣
+

∣∣∣∣ 1|B|
∫
B

f(t)dt

∣∣∣∣ +
∣∣∣∣ 1|B|

∫
B

e−i2π<ξ,t>f(t)dt

∣∣∣∣
)
k(x)dx

≤ sup
B⊂Rn

1

|B|k

∫
B

|f(x) −fB|k(x)dx + sup
B⊂Rn

1

|B|k

∫
B

|fB|k(x)dx

+ sup
B⊂Rn

1

|B|k

∫
B

(
1

|B|

∫
B

|f(t)|dt
)
k(x)dx

≤ ‖ f ‖BMOk +
1

|B|k
m

∫
B

k(x)dx +
1

|B|k
m

∫
B

k(x)dx

= ‖ f ‖BMOk +
1

|B|k
m|B|k +

1

|B|k
m|B|k

= ‖ f ‖BMOk +2m.

�

Theorem 3.2. Suppose ω is a window function such that for any fixed ξ ∈ Rn0

(3.2)

∫
Rn
|ω(x,ξ)|(1 + C|x|)Ndx ≤ A < ∞,



THE S-TRANSFORM ON HARDY SPACES AND BMO 175

where A,C and N are positive constants. Then the operator Sω : BMOk(Rn) →
BMOk(Rn) is continuous. Furthermore, we have

‖ (Sωf)(·,ξ) ‖BMOk≤ A (‖ f ‖BMOk +2m)

where m is a constant given in equation (2.1).

Proof. By using the techniques of Theorem 2.2, for any arbitrary ball B in Rn, we
have

‖ (Sωf)(·,ξ) ‖BMOk = sup
B⊂Rn

1

|B|k

∫
B

|(Sωf)(τ,ξ) − (Sωf)B(τ,ξ)|k(τ)dτ

≤ sup
B⊂Rn

1

|B|k

∫
B

(∫
Rn
|ω(x,ξ)|

∣∣∣∣e−i2π<ξ,τ−x>f(τ −x)
−

1

|B|

∫
B

e−i2π<ξ,α−x>f(α−x)dα
∣∣∣∣dx
)
k(τ)dτ

≤ sup
K⊂Rn

1

|K|k

∫
K

(∫
Rn
|ω(x,ξ)|

∣∣∣∣e−i2π<ξ,y>f(y)
−

1

|K|

∫
K

e−i2π<ξ,t>f(t)dt

∣∣∣∣dx
)

(1 + C|x|)Nk(y)dy

=

∫
Rn
|ω(x,ξ)|(1 + C|x|)N

(
sup
K⊂Rn

1

|K|k

∫
K

∣∣∣∣e−i2π<ξ,y>f(y)
−

1

|K|

∫
K

e−i2π<ξ,t>f(t)dt

∣∣∣∣k(y)dy
)
dx

≤ A ‖ e−i2π<ξ,·>f(·) ‖BMOk,

here K = B −x for x ∈ Rn.
By using above lemma we get

‖ (Sωf)(·,ξ) ‖BMOk≤ A (‖ f ‖BMOk +2m) .

�

4. THE S-TRANSFORM ON HARDY SPACES.

Definition 4.1. The Hardy space is defined as the space of all functions f ∈ L1(Rn)
such that

‖ f ‖H1 =
∫
Rn

sup
t>0
|(f ∗φt) (x)|dx < ∞,

where φ is any test function with
∫
φ 6= 0 and φt(x) = t−nφ(x/t); t > 0,x ∈ Rn.

Theorem 4.1. Let f ∈ L1(Rn) such that

(4.1) sup
t>0

∣∣∣∣
∫
Rn
f(x−y)φt(y)dy

∣∣∣∣ = sup
t>0

∫
Rn
|f(x−y)φt(y)|dy < ∞.

Then for any fixed ξ ∈ Rn0 , the operator Sω : H1(Rn) → H1(Rn) is continuous.
Furthermore, we have

‖ (Sωf)(·,ξ) ‖H1≤ 3 ‖ ω(·,ξ) ‖L1‖ f ‖H1 .



176 SINGH AND KALITA

Proof. Since

((Sωf)(·,ξ) ∗φt) (τ)

=

((∫
Rn
e−i2π<ξ, ·−x>f(·−x)ω(x,ξ)dx

)
∗φt

)
(τ)

=

∫
Rn

(∫
Rn
e−i2π<ξ,τ−x−y>f(τ −x−y)ω(x,ξ) dx

)
φt(y) dy

=

∫
Rn
ω(x,ξ)

(∫
Rn
e−i2π<ξ,τ−x−y>f(τ −x−y)φt(y)dy

)
dx.

Thus

‖ (Sωf)(·,ξ) ‖H1

=

∫
Rn

sup
t>0
|((Sωf)(·,ξ) ∗φt) (τ)|dτ

=

∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
ω(x,ξ)

(∫
Rn
e−i2π<ξ,τ−x−y>f(τ −x−y)φt(y)dy

)
dx

∣∣∣∣dτ
≤

∫
Rn
|ω(x,ξ)|

(∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
e−i2π<ξ,τ−x−y>f(τ −x−y)φt(y)dy

∣∣∣∣dτ
)
dx

=

∫
Rn
|ω(x,ξ)|

(∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
e−i2π<ξ,η−y>f(η −y)φt(y)dy

∣∣∣∣dη
)
dx.

Also, ∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
e−i2π<ξ,η−y>f(η −y)φt(y)dy

∣∣∣∣dη
=

∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
e−i2π<ξ,η−y>f(η −y)φt(y)dy

−
∫
Rn
f(η −y)φt(y)dy +

∫
Rn
f(η −y)φt(y)dy

∣∣∣∣dη
=

∫
Rn

sup
t>0

∣∣∣∣
∫
Rn

(e−i2π<ξ,η−y> − 1)f(η −y)φt(y)dy

+

∫
Rn
f(η −y)φt(y)dy

∣∣∣∣dη
≤
∫
Rn

sup
t>0

∫
Rn

∣∣(e−i2π<ξ,η−y> − 1)∣∣ |f(η −y)φt(y)|dydη
+

∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
f(η −y)φt(y)dy

∣∣∣∣dη
≤ 2

∫
Rn

sup
t>0

∫
Rn
|f(η −y)φt(y)|dydη+ ‖ f ‖H1

= 2 ‖ f ‖H1 + ‖ f ‖H1
= 3 ‖ f ‖H1 .

Therefore,

‖ (Sωf)(·,ξ) ‖H1≤
∫
Rn
|ω(x,ξ)|3 ‖ f ‖H1 dx = 3 ‖ ω(·,ξ) ‖L1‖ f ‖H1 .

�



THE S-TRANSFORM ON HARDY SPACES AND BMO 177

5. THE S-TRANSFORM ON WEIGHTED HARDY SPACES.

Definition 5.1. The weighted Hardy space is defined as the space of all functions
f ∈ L1k(R

n) such that

‖ f ‖H1
k
=

∫
Rn

sup
t>0
|(f ∗φt) (x)|k(x)dx < ∞.

Theorem 5.1. Suppose ω is a window function and satisfies the condition (3.2).
Let f ∈ L1(Rn) and satisfies the condition (4.1). Then, for any fixed ξ ∈ Rn0 , the
operator Sω : H

1
k(R

n) → H1k(R
n) is continuous. Furthermore, we have

‖ (Sωf)(·,ξ) ‖H1
k
≤ 3A ‖ f ‖H1

k
.

Proof. Since

‖ (Sωf)(·,ξ) ‖H1
k

=

∫
Rn

sup
t>0
|((Sωf)(·,ξ) ∗φt) (τ)|k(τ)dτ

=

∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
ω(x,ξ)

(∫
Rn
e−i2π<ξ,τ−x−y>f(τ −x−y)φt(y)dy

)
dx

∣∣∣∣k(τ)dτ
≤
∫
Rn
|ω(x,ξ)|

(∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
e−i2π<ξ,τ−x−y>f(τ −x−y)φt(y)dy

∣∣∣∣k(τ)dτ
)
dx

≤
∫
Rn
|ω(x,ξ)|

(∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
e−i2π<ξ,η−y>f(η −y)φt(y)dy

∣∣∣∣ (1 + C|x|)Nk(η)dη
)
dx

=

∫
Rn
|ω(x,ξ)|(1 + C|x|)N

(∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
e−i2π<ξ,η−y>f(η −y)φt(y)dy

∣∣∣∣k(η)dη
)
dx.

And ∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
e−i2π<ξ,η−y>f(η −y)φt(y)dy

∣∣∣∣k(η)dη
=

∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
e−i2π<ξ,η−y>f(η −y)φt(y)dy

−
∫
Rn
f(η −y)φt(y)dy +

∫
Rn
f(η −y)φt(y)dy

∣∣∣∣k(η)dη
≤
∫
Rn

sup
t>0

∣∣∣∣
∫
Rn

(e−i2π<ξ,η−y> − 1)f(η −y)φt(y)dy
∣∣∣∣k(η)dη

+

∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
f(η −y)φt(y)dy

∣∣∣∣k(η)dη
≤ 2

∫
Rn

sup
t>0

∫
Rn
|f(η −y)φt(y)|dy k(η) dη

+

∫
Rn

sup
t>0

∣∣∣∣
∫
Rn
f(η −y)φt(y)dy

∣∣∣∣k(η)dη
= 2 ‖ f ‖H1

k
+ ‖ f ‖H1

k

= 3 ‖ f ‖H1
k
.

Therefore, using equation (3.2), we have

‖ (Sωf)(·,ξ) ‖H1
k
≤
∫
Rn
|ω(x,ξ)|(1 + C|x|)N 3 ‖ f ‖H1

k
dx ≤ 3A ‖ f ‖H1

k
.



178 SINGH AND KALITA

This completes the proof. �

References

[1] Chuong. N. M. and Duong. D. V., Boundedness of the wavelet integral operator on weighted

function spaces, Russian Journal of Mathematical Physics, 20(3) (2013), 268-275.

[2] Hörmander. L., The Analysis of Linear Partial Differential Operators II, Springer-Verlag,
Berlin Heidelberg New York, 1983.

[3] John. F. and Nirenberg. L., On functions of bounded mean oscillation , Communications on

Pure and Applied Mathematics, 14 (1961), 415-426.
[4] Pathak. R. S. and Singh. S. K., Boundedness of the wavelet transform in certain function

spaces, Journal of inequalities in pure and applied mathematics, 8(1) (2007), Article 23, 8

pages.
[5] Singh. S. K., The S-transform on spaces of type S, Integral Transforms and Special Functions,

23(7) (2012), 481-494.
[6] Singh. S. K., The S-transform on spaces of type W, Integral Transforms and Special Functions,

23(12) (2012), 891-899.

[7] Singh. S. K., The fractional S-transform of tempered ultradistibutions, Investigations in
Mathematical Sciences, 2(2) (2012), 315-325.

[8] Singh. S. K., The fractional S-transform on spaces of type S, Journal of Mathematics, 2013

(2013), Article ID 105848.
[9] Singh. S. K., The fractional S-transform on spaces of type W, Journal of Pseudo-Differential

Operators and Applications, 4(2) (2013), 251-265.

[10] Singh. S. K., A new integral transform: Theory part, Investigations in Mathematical Sciences,
3(1) (2013), 135-139.

[11] Singh. S. K., The S-Transform of distributions, The scientific world journal, 2014 (2014),

Article ID 623294.
[12] Stockwell. R. G., Mansinha. L. and Lowe. R. P., Localization of the complex spectrum: The

S transform, IEEE Trans. Signal Process., 44(4) (1996), 998-1001.

[13] Ventosa. S., Simon. C., Schimmel. M. , Dañobeitia. J. and Mànuel. A., The S-transform from
a wavelet point of view, IEEE Trans. Signal Process., 56(07) (2008), 2771-2780.

Department of Mathematics, Rajiv Gandhi University, Doimukh-791112, Arunachal

Pradesh, India

∗Corresponding author