Volume 8 (2007), Issue 1, Article 23, 8 pp.
BOUNDEDNESS OF THE WAVELET TRANSFORM IN CERTAIN FUNCTION SPACES R.S. PATHAK AND S.K. SINGH D EPARTMENT OF M ATHEMATICS , BANARAS H INDU U NIVERSITY, VARANASI - 221 005, I NDIA
[email protected]
Received 06 October, 2005; accepted 25 January, 2007 Communicated by L. Debnath
A BSTRACT. Using convolution transform theory boundedness results for the wavelet transform n are obtained in the Triebel space-LΩ,k p , Hörmander space-Bp,q (R ) and general function spaceL∞,k , where k denotes a weight function possessing specific properties in each case. Key words and phrases: Continuous wavelet transform, Distributions, Sobolev space, Besov space, Lizorkin-Triebel space. 2000 Mathematics Subject Classification. 42C40, 46F12.
1. I NTRODUCTION The wavelet transform W of a function f with respect to the wavelet ψ is defined by Z ˜ (1.1) f (a, b) = (Wψ f )(a, b) = f (t)ψa,b (t)dt = (f ∗ ha,0 )(b), Rn n
where ψa,b = a− 2 ψ( x−b ), h(x) = ψ(−x), b ∈ Rn and a > 0, provided the integral exists. In a view of (1.1) the wavelet transform (Wψ f )(a, b) can be regarded as the convolution of f and ha,0 . The existence of convolution f ∗g has been investigated by many authors. For this purpose Triebel [6] defined the space LΩ,k and showed that for certain weight functions k, f ∗ g ∈ LΩ,k p p , Ω,k where f, g ∈ Lp , 0 < p ≤ 1. Convolution theory has also been developed by Hörmander in the generalized Sobolev space Bp,q (Rn ), 1 ≤ p ≤ ∞. In Section 2 of the paper, a definition and properties of the space LΩ,k are given and a boundp edness result for the wavelet transform Wψ f is obtained. In Section 3 we recall the definition and properties of the generalized Sobolev space Bp,q (Rn ) due to Hörmander [1] and obtain a certain boundedness result for Wψ f . Finally, using Young’s inequality a third boundedness result is also obtained. 303-05
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R.S. PATHAK AND S.K. S INGH
2. B OUNDEDNESS OF W
IN
LΩ,k p
Let us recall the definition of the space LΩ,k by Triebel [6]. p Definition 2.1. Let Ω be a bounded C ∞ -domain in Rn . If k(x) is a non-negative weight function in Rn and 0 < p ≤ ∞, then Ω,k (2.1) Lp = f |f ∈ S 0 , supp F f ⊂ Ω; Z k f kLΩ,k =k kf kLp = p
p
p
p1
k (x)|f (x)| dx
0.
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 23, 8 pp.
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WAVELET T RANSFORM
3
Proof. For k(x) = |x|α , α > 0,we have k(az) = aα k(z) and p1 Z n x kha,0 kLkp = k p (x)(a− 2 |h( )|)p dx a Rn Z p1 n p p = a2 k (az)|h(z)| dz Rn
Z
n 2
=a
pα p
p1
p
a k (z)|h(z)| dz Rn
Z
n +α 2
=a
p
p
p1
k (z)|h(z)| dz Rn
n
= a 2 +α k h kLkp n
= a 2 +α k ψ kLkp . For k(x) =
Qn
j=1
|xj |αj , αj ≥ 0, we have k(az) = a|α| k(z) and p1 Z x p p −n kha,0 kLkp = k (x)(a 2 |h( )|) dx a Rn Z p1 n p p = a2 k (az)|h(z)| dz Rn
=a
Z
n 2
ap|α| k p (z)|h(z)|p dz
p1
Rn
=a
n +|α| 2
Z
p
p
p1
k (z)|h(z)| dz Rn
n
= a 2 +|α| k h kLkp n
= a 2 +|α| k ψ kLkp . γ
Next, for k(x) = kβ,γ (x) = eβ|x| , β ≥ 0, 0 ≤ γ ≤ 1, we have γ
γ |z|γ
kβ,γ (az) = eβ|az| = eβa
≤ eβ
a2γ +|z|2γ 2
1
2γ
1
= e 2 βa e 2 β|z|
2γ
1
2γ
= e 2 βa kβ,2γ (z),
and Z kha,0 k
k
Lpβ,γ
= Rn
=a
n 2
Z Rn
≤a
n 2
x p p1 a h dz a p1 p kβ,γ (az)|h(z)|p dz
p kβ,γ (x)
Z e
−n 2
1 pβa2γ 2
Rn n 2
1 βa2γ 2
n 2
1 βa2γ 2
n
1
=a e
Z
p kβ,2γ (z)|h(z)|p dz
Rn
=a e
2γ
= a 2 e 2 βa
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 23, 8 pp.
p kβ,2γ (z)|h(z)|p dz
khk
p1
p1
k
Lpβ,2γ
kψk
k
Lpβ,2γ
.
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R.S. PATHAK AND S.K. S INGH
The proofs of (2.7), (2.8) and (2.9) follow from (2.6). 3. B OUNDEDNESS OF W
IN
Bp,k
The space Bp,k (Rn ) was introduced by Hörmander [1], as a generalization of the Sobolev space H s (Rn ), in his study of the theory of partial differential equations. We recall its definition. Definition 3.1. A positive function k defined in Rn will be called a temperate weight function if there exist positive constants C and N such that (3.1)
k(ξ + η) ≤ (1 + C|ξ|)N k(η);
ξ, η ∈ Rn ,
the set of all such functions k will be denoted by K . Certain properties of the weight function k are contained in the following theorem whose proof can be found in [1]. Theorem 3.1. If k1 and k2 belong to K ,then k1 + k2 , k1 k2 , sup(k1 , k2 ), inf(k1 , k2 ), are also in K . If k ∈ K we have k s ∈ K for every real s,and if µ is a positive measure we have either µ ∗ k ≡ ∞ or else µ ∗ k ∈ K . Definition 3.2. If k ∈ K and 1 ≤ p ≤ ∞, we denote by Bp,k the set of all distributions u ∈ S 0 such that uˆ is a function and p1 Z −n p (3.2) k u kp,k = (2π) |k(ξ)ˆ u| dξ < ∞, 1 ≤ p < ∞; k u k∞,k = ess sup |k(ξ)ˆ u(ξ)|.
(3.3)
We need the following theorem [1, p.10] in the proof of our boundedness result. T Theorem 3.2 (Lars Hörmander). If u1 ∈ Bp,k1 E 0 and u2 ∈ B∞,k2 then u1 ∗ u2 ∈ B∞,k1 k2 , and we have the estimate (3.4)
k u1 ∗ u2 kp,k1 k2 ≤k u1 kp,k1 k u2 k∞,k2 ,
1 ≤ p < ∞.
Using the above theorem we obtain the following boundedness result. T Theorem 3.3. Let k1 and k2 belong to K . Assume that f ∈ Bp,k1 E 0 and ψ ∈ B∞,k2 then the wavelet transform (Wψ f )(a, b) = (f ∗ ha,0 )(b), defined by (1.1) is in Bp,k1 k2 , and
N
n 1 C 2
ˆ (3.5) k Wψ f (a, b) kp,k1 k2 ≤ a 2 k2 k f k 1 + t ψ(t)
. p,k 1
2a2 2 ∞
Proof. Since kha,0 k∞,k2
ˆ = ess sup k2 (ξ)ha,0 (ξ) n ˆ = ess sup k2 (ξ)a 2 ψ(aξ) n ˆ ≤ a 2 ess sup k2 ( at )ψ(t) N n 1 C 2 ˆ ess sup 1 + t ψ(t) ≤ a 2 k2 2a2 2
on using (3.1). Hence by Theorem 3.2 we have kWψ f (a, b)kp,k1 k2 =k (f ∗ ha,0 (b) kp,k1 k2 n 1 2 ≤ a k2 k f kp,k1 2a2
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 23, 8 pp.
N
C 2
ˆ ψ(t) .
1+ t
2 ∞
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WAVELET T RANSFORM
5
This proves the theorem. 4. A G ENERAL B OUNDEDNESS R ESULT
Using Young’s inequality for convolution we obtained a general boundedness result for the wavelet transform. In the proof of our result the following theorem will be used [3, p. 90]. Theorem 4.1. Let p, q, r ≥ 1 and p1 + 1q + 1r = 2. Let k ∈ Lp (Rn ), f ∈ Lq (Rn ) and g ∈ Lr (Rn ), then Z kf ∗ gk∞,k = k(x)(f ∗ g)(x)dx n ZR Z k(x)f (x − y)g(y)dxdy = Rn
Rn
≤ Cp,q,r;n k k kp k f kq k g kr . 1
The sharp constant Cp,q,r;n = (Cp Cq Cr )n , where Cp2 =
pp 0 10 p
p
with ( p1 +
1 p0
= 1). Using The-
orem 4.1 and following the same method of proof as for Theorem 3.3 we obtain the following boundedness result. Theorem 4.2. Let p, q, r ≥ 1, p1 + 1q + 1r = 2 and k ∈ Lp (Rn ). Let f ∈ Lq (Rn ) and ψ ∈ Lr (Rn ), then n
n
k Wψ f k∞,k ≤ Cp,q,r;n a r − 2 k k kp k f kq k ψ kr 1
where Cp,q,r;n = (Cp Cq Cr )n , Cp2 =
pp 1
p0 p0
with
1 p
+
1 p0
= 1.
R EFERENCES [1] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators II, Springer-Verlag, Berlin Heiedelberg New York, Tokyo, 1983. [2] T.H. KOORNWINDER, Wavelets, World Scientific Publishing Co. Pty. Ltd., Singapore , 1993. [3] E.H. LIEB AND M. LOSS, Analysis, Narosa Publishing House, 1997. ISBN: 978-81-7319-201-2. [4] R.S. PATHAK, Integral Transforms of Generalized Functions and Their Applications, Gordon and Breach Science Publishers, Amsterdam, 1997. [5] R.S. PATHAK, The continuous wavelet transform of distributions, Tohoku Math. J., 56 (2004), 411–421. [6] H. TRIEBEL, A note on quasi-normed convolution algebras of entire analytic functions of exponential type, J. Approximation Theory, 22(4) (1978), 368–373. [7] H. TRIEBEL, Multipliers in Bessov-spaces and in LΩ p - spaces (The cases 0 < p ≤ 1 and p = ∞), Math. Nachr., 75 (1976), 229–245. [8] H.J. SCHMEISSER AND H. TRIEBEL, Topics in Fourier Analysis and Function Spaces, John Wiley and Sons, Chichester, 1987.
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 23, 8 pp.
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