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BOUNDEDNESS OF THE WAVELET TRANSFORM IN CERTAIN FUNCTION SPACES

R.S. PATHAK AND S.K. SINGH DEPARTMENT OFMATHEMATICS,

BANARASHINDUUNIVERSITY, VARANASI- 221 005, INDIA

ramshankarpathak@yahoo.co.in

Received 06 October, 2005; accepted 25 January, 2007 Communicated by L. Debnath

ABSTRACT. Using convolution transform theory boundedness results for the wavelet transform are obtained in the Triebel space-LΩ,kp , Hörmander space-Bp,q(Rn)and general function space- L∞,k,wherekdenotes 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. INTRODUCTION

The wavelet transformW of a functionf with respect to the waveletψis defined by (1.1) f(a, b) = (W˜ ψf)(a, b) =

Z

Rn

f(t)ψa,b(t)dt= (f∗ha,0)(b),

whereψa,b = an2ψ(x−ba ), h(x) = ψ(−x), b ∈ Rn anda > 0, provided the integral exists. In view of (1.1) the wavelet transform (Wψf)(a, b)can be regarded as the convolution of f and ha,0. The existence of convolutionf∗ghas been investigated by many authors. For this purpose Triebel [6] defined the spaceLΩ,kp and showed that for certain weight functionsk, f ∗g ∈LΩ,kp , wheref, g ∈ LΩ,kp , 0< p ≤1. Convolution theory has also been developed by Hörmander in the generalized Sobolev spaceBp,q(Rn), 1≤p≤ ∞.

In Section 2 of the paper, a definition and properties of the spaceLΩ,kp are given and a bound- edness result for the wavelet transform Wψf 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 Wψf. Finally, using Young’s inequality a third boundedness result is also obtained.

303-05

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2. BOUNDEDNESS OF W INLΩ,kp Let us recall the definition of the spaceLΩ,kp by Triebel [6].

Definition 2.1. LetΩbe a boundedC-domain inRn. Ifk(x)is a non-negative weight function inRnand0< p ≤ ∞, then

(2.1) LΩ,kp =

f|f ∈S0,suppF f ⊂Ω;

kf kLΩ,k

p =kkf kLp= Z

Rn

kp(x)|f(x)|pdx 1p

<∞

.

Ifk(x) = 1thenLΩ,kp =Lp.

We need the following theorem [6, p. 369] in the proof of our boundedness result.

Theorem 2.1 (Hans Triebel). Ifkis one of the following weight functions:

(2.2) k(x) = |x|α, α ≥0

(2.3) k(x) =

n

Y

j=1

|xj|αj, αj ≥0

(2.4) k(x) =kβ,γ(x) =eβ|x|γ, β ≥0,0≤γ ≤1 and0< p ≤1, then

(2.5) LΩ,kp ∗LΩ,kp ⊂LΩ,kp

and there exists a positive numberCsuch that for allf, g∈LΩ,kp ,

(2.6) kf ∗gkLk

p ≤CkfkLk pkgkLk

p.

Using the above theorem we obtain the following boundedness result for the wavelet trans- formWψf.

Theorem 2.2. Letf ∈LΩ,kp andψ ∈LΩ,kp ,0< p≤1,then for the wavelet transformWψf we have the estimates:

(2.7) k(Wψf)(a, b)kLkp≤Caα+n2 kf kLkpkψ kLkp for (2.2);

(2.8) k(Wψf)(a, b)kLk

p≤Ca|α|+n2 kf kLk

pkψ kLk

p for (2.3);

(2.9) k(Wψf)(a, b)k

Lkβ,γp

≤Can2e12βa kf k

Lkβ,γp

kψ k

Lkβ,2γp

for (2.4), whereb∈Rnanda >0.

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Proof. Fork(x) = |x|α, α >0,we havek(az) =aαk(z)and kha,0kLk

p = Z

Rn

kp(x)(an2|h(x a)|)pdx

1p

=an2 Z

Rn

kp(az)|h(z)|pdz 1p

=an2 Z

Rn

akp(z)|h(z)|pdz 1p

=an2 Z

Rn

kp(z)|h(z)|pdz 1p

=an2 khkLk p

=an2 kψ kLk p . Fork(x) =Qn

j=1|xj|αj, αj ≥0, we havek(az) = a|α|k(z)and kha,0kLk

p = Z

Rn

kp(x)(an2|h(x a)|)pdx

1p

=an2 Z

Rn

kp(az)|h(z)|pdz 1p

=an2 Z

Rn

ap|α|kp(z)|h(z)|pdz 1p

=an2+|α|

Z

Rn

kp(z)|h(z)|pdz p1

=an2+|α|khkLk

p

=an2+|α|kψ kLk

p .

Next, fork(x) = kβ,γ(x) = eβ|x|γ, β ≥0,0≤γ ≤1, we have kβ,γ(az) = eβ|az|γ =eβaγ|z|γ ≤eβa

+|z|

2 =e12βae12β|z| =e12βakβ,2γ(z), and

kha,0k

Lkβ,γp

= Z

Rn

kβ,γp (x) an2

hx

a

p

dz 1p

=an2 Z

Rn

kpβ,γ(az)|h(z)|pdz 1p

≤an2 Z

Rn

e12pβakβ,2γp (z)|h(z)|pdz 1p

=an2e12βa Z

Rn

kpβ,2γ(z)|h(z)|pdz 1p

=an2e12βa khk

Lkβ,2γp

=an2e12βa kψ k

Lkβ,2γp

.

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The proofs of (2.7), (2.8) and (2.9) follow from (2.6).

3. BOUNDEDNESS OFW INBp,k

The space Bp,k(Rn) was introduced by Hörmander [1], as a generalization of the Sobolev spaceHs(Rn), in his study of the theory of partial differential equations. We recall its definition.

Definition 3.1. A positive functionkdefined inRnwill be called a temperate weight function if there exist positive constantsCandN such that

(3.1) k(ξ+η)≤(1 +C|ξ|)Nk(η); ξ, η∈Rn,

the set of all such functionsk will be denoted byK . Certain properties of the weight function kare contained in the following theorem whose proof can be found in [1].

Theorem 3.1. If k1 andk2 belong to K ,thenk1 +k2, k1k2, sup(k1, k2), inf(k1, k2), are also inK . Ifk ∈K we haveks ∈K for every reals,and ifµis a positive measure we have either µ∗k≡ ∞or elseµ∗k ∈K.

Definition 3.2. Ifk ∈K and1≤p≤ ∞, we denote byBp,kthe set of all distributionsu∈S0 such thatuˆis a function and

(3.2) kukp,k= (2π)−n Z

|k(ξ)ˆu|p1p

<∞, 1≤p < ∞;

(3.3) kuk∞,k=esssup|k(ξ)ˆu(ξ)|.

We need the following theorem [1, p.10] in the proof of our boundedness result.

Theorem 3.2 (Lars Hörmander). If u1 ∈ Bp,k1TE0andu2 ∈ B∞,k2 then u1 ∗u2 ∈ B∞,k1k2, and we have the estimate

(3.4) ku1∗u2 kp,k1k2≤ku1 kp,k1ku2 k∞,k2, 1≤p < ∞.

Using the above theorem we obtain the following boundedness result.

Theorem 3.3. Letk1 and k2 belong to K . Assume thatf ∈ Bp,k1

TE0 and ψ ∈ B∞,k2 then the wavelet transform(Wψf)(a, b) = (f∗ha,0)(b),defined by (1.1) is inBp,k1k2, and

(3.5) kWψf(a, b)kp,k1k2≤an2k2 1

2a2

kf kp,k1

1 + C

2t2 N

ψ(t)ˆ

.

Proof. Since

kha,0k∞,k

2 =esssup

k2(ξ)ˆha,0(ξ)

=esssup

k2(ξ)an2ψ(aξ)ˆ

≤an2esssup

k2(at) ˆψ(t)

≤an2k2 1

2a2

esssup

1 + C

2t2 N

ψ(t)ˆ on using (3.1). Hence by Theorem 3.2 we have

kWψf(a, b)kp,k

1k2 =k(f∗ha,0(b)kp,k1k2

≤an2k2

1 2a2

kf kp,k1

1 + C

2t2 N

ψ(t)ˆ

.

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This proves the theorem.

4. A GENERAL BOUNDEDNESSRESULT

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. Letp, q, r≥1and1p+1q+1r = 2. Letk ∈Lp(Rn), f ∈Lq(Rn)andg ∈Lr(Rn), then

kf ∗gk∞,k = Z

Rn

k(x)(f ∗g)(x)dx

= Z

Rn

Z

Rn

k(x)f(x−y)g(y)dxdy

≤Cp,q,r;n kkkpkf kqkg kr. The sharp constantCp,q,r;n = (CpCqCr)n, whereCp2 = p

1p

p0

1

p0 with (1p + p10 = 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. Letp, q, r ≥1, 1p+1q+1r = 2andk ∈Lp(Rn). Letf ∈Lq(Rn)andψ ∈Lr(Rn), then

kWψf k∞,k ≤ Cp,q,r;nanrn2 kk kpkf kqkψ kr whereCp,q,r;n = (CpCqCr)n,Cp2 = p

1p

p0

1

p0 with 1p +p10 = 1.

REFERENCES

[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. LIEBANDM. 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 expo- nential type, J. Approximation Theory, 22(4) (1978), 368–373.

[7] H. TRIEBEL, Multipliers in Bessov-spaces and inLp- spaces (The cases0< p≤1andp=∞), 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.

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