BOUNDEDNESS OF THE WAVELET TRANSFORM IN CERTAIN FUNCTION SPACES

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Wavelet Transform R.S. Pathak and S.K. Singh

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BOUNDEDNESS OF THE WAVELET TRANSFORM IN CERTAIN FUNCTION SPACES

R.S. PATHAK AND S.K. SINGH

Department of Mathematics, Banaras Hindu University, Varanasi - 221 005, India

EMail:[email protected]

Received: 06 October, 2005 Accepted: 25 January, 2007 Communicated by: L. Debnath 2000 AMS Sub. Class.: 42C40, 46F12.

Key words: Continuous wavelet transform, Distributions, Sobolev space, Besov space, Lizorkin-Triebel space.

Abstract: Using convolution transform theory boundedness results for the wavelet transform are obtained in the Triebel space-L^Ω,kp , Hörmander space-Bp,q(Rⁿ) and general function space-L∞,k,wherekdenotes a weight function possessing spe- cific properties in each case.

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1. Introduction

The wavelet transformW of a functionf with respect to the waveletψ is defined by

(1.1) f(a, b) = (W˜ ψf)(a, b) = Z

Rⁿ

f(t)ψ_a,b(t)dt = (f ∗ha,0)(b),

whereψ_a,b =a⁻ⁿ²ψ(^x−b_a ), h(x) =ψ(−x),b ∈ Rⁿanda > 0, provided the integral exists. In view of (1.1) the wavelet transform (W_ψf)(a, b) can be regarded as the convolution off andh_a,0. The existence of convolutionf ∗g has been investigated by many authors. For this purpose Triebel [6] defined the spaceL^Ω,k_p and showed that for certain weight functions k, f ∗g ∈ L^Ω,k_p , where f, g ∈ L^Ω,k_p , 0 < p ≤ 1. Convolution theory has also been developed by Hörmander in the generalized Sobolev spaceB_p,q(Rⁿ), 1≤p≤ ∞.

In Section2of the paper, a definition and properties of the spaceL^Ω,k_p are given and a boundedness result for the wavelet transformW_ψf is obtained. In Section3we recall the definition and properties of the generalized Sobolev space B_p,q(Rⁿ) due to Hörmander [1] and obtain a certain boundedness result for W_ψf. Finally, using Young’s inequality a third boundedness result is also obtained.

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2. Boundedness of W in L

^Ω,k_p

Let us recall the definition of the spaceL^Ω,k_p by Triebel [6].

Definition 2.1. Let Ω be a bounded C^∞-domain in Rⁿ. If k(x) is a non-negative weight function inRⁿand0< p≤ ∞, then

(2.1) L^Ω,k_p =

f|f ∈S⁰,suppF f ⊂Ω;

kf k_LΩ,k

p =kkf k_L_p= Z

Rⁿ

k^p(x)|f(x)|^pdx ¹_p

<∞

. Ifk(x) = 1thenL^Ω,k_p =L^Ω_p.

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

|x_j|^α^j, α_j ≥0

(2.4) k(x) = k_β,γ(x) =e^β|x|^γ, β ≥0,0≤γ ≤1 and0< p≤1, then

(2.5) L^Ω,k_p ∗L^Ω,k_p ⊂L^Ω,k_p

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and there exists a positive numberCsuch that for allf, g ∈L^Ω,k_p ,

(2.6) kf ∗gk_Lk

p ≤Ckfk_Lk pkgk_Lk

p.

Using the above theorem we obtain the following boundedness result for the wavelet transformW_ψf.

Theorem 2.2. Letf ∈L^Ω,k_p andψ ∈L^Ω,k_p ,0< p ≤1,then for the wavelet transform W_ψf we have the estimates:

(2.7) k(W_ψf)(a, b)k_L^k

p≤Ca^α+ⁿ² kf k_L^k

pkψ k_L^k

p for (2.2);

(2.8) k(W_ψf)(a, b)k_L^k

p≤Ca^|α|+ⁿ² kf k_L^k

pkψ k_L^k

p for (2.3);

(2.9) k(W_ψf)(a, b)k

L^kβ,γp

≤Caⁿ²e¹²^βa^2γ kf k

L^kβ,γp

kψ k

L^kβ,2γp

for (2.4), whereb ∈Rⁿanda >0.

Proof. Fork(x) = |x|^α, α >0,we havek(az) = a^αk(z)and kh_a,0k_Lk

p = Z

Rⁿ

k^p(x)(a⁻ⁿ²|h(x a)|)^pdx

¹_p

=aⁿ² Z

Rⁿ

k^p(az)|h(z)|^pdz ¹_p

=aⁿ² Z

Rⁿ

a^pαk^p(z)|h(z)|^pdz ¹_p

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=aⁿ²^+α Z

Rⁿ

k^p(z)|h(z)|^pdz ¹_p

=aⁿ²^+α khk_L^k_p

=aⁿ²^+α kψ k_L^k

p . Fork(x) = Qn

j=1|x_j|^α^j, α_j ≥0, we havek(az) = a^|α|k(z)and kh_a,0k_Lk

p = Z

Rⁿ

k^p(x)(a⁻ⁿ²|h(x a)|)^pdx

¹_p

=aⁿ² Z

Rⁿ

k^p(az)|h(z)|^pdz ¹_p

=aⁿ² Z

Rⁿ

a^p|α|k^p(z)|h(z)|^pdz ¹_p

=aⁿ²^+|α|

Z

Rⁿ

k^p(z)|h(z)|^pdz ¹_p

=aⁿ²^+|α|khk_L^k

p

=aⁿ²^+|α|kψ k_L^k_p .

Next, fork(x) = k_β,γ(x) =e^β|x|^γ, β ≥0,0≤γ ≤1, we have k_β,γ(az) = e^β|az|^γ =e^βa^γ^|z|^γ ≤e^β^a

2γ+|z|2γ

2 =e¹²^βa^2γe¹²^β|z|^2γ =e¹²^βa^2γk_β,2γ(z),

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and

kh_a,0k

L^kβ,γp

= Z

Rⁿ

k_β,γ^p (x) a⁻ⁿ²

hx

a

p

dz ¹_p

=aⁿ² Z

Rⁿ

k^p_β,γ(az)|h(z)|^pdz ¹_p

≤aⁿ² Z

Rⁿ

e¹²^pβa^2γk_β,2γ^p (z)|h(z)|^pdz ¹_p

=aⁿ²e¹²^βa^2γ Z

Rⁿ

k^p_β,2γ(z)|h(z)|^pdz ¹_p

=aⁿ²e¹²^βa^2γ kh k

L^kβ,2γp

=aⁿ²e¹²^βa^2γ kψ k

L^kβ,2γp

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

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3. Boundedness of W in B

_p,k

The space B_p,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 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 functionskwill be denoted byK . Certain properties of the weight functionkare contained in the following theorem whose proof can be found in [1].

Theorem 3.1. Ifk₁andk₂belong toK ,thenk₁+k₂, k₁k₂,sup(k₁, k₂),inf(k₁, k₂), are also inK . If k ∈ K we have k^s ∈ K for every reals,and ifµis a positive measure we have eitherµ∗k ≡ ∞or elseµ∗k ∈K .

Definition 3.2. Ifk ∈ K and1≤p≤ ∞, we denote byB_p,kthe set of all distribu- tionsu∈S⁰ such thatuˆis a function and

(3.2) kuk_p,k= (2π)⁻ⁿ Z

|k(ξ)ˆu|^pdξ ¹_p

<∞, 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). Ifu₁ ∈B_p,k₁TE⁰andu₂ ∈B∞,k₂ thenu₁∗u₂ ∈ B∞,k₁k2, and we have the estimate

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

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Using the above theorem we obtain the following boundedness result.

Theorem 3.3. Let k₁ and k₂ belong to K . Assume that f ∈ B_p,k₁TE⁰ andψ ∈ B∞,k₂ then the wavelet transform(W_ψf)(a, b) = (f∗h_a,0)(b),defined by (1.1) is in B_p,k₁_k₂, and

(3.5) kW_ψf(a, b)k_p,k₁_k₂≤aⁿ²k₂ 1

2a²

kf k_p,k₁

1 + C

2t² N

ψˆ(t) ∞

.

Proof. Since

kh_a,0k_∞,k

2 =esssup

k₂(ξ)ˆh_a,0(ξ)

=esssup

k₂(ξ)aⁿ²ψ(aξ)ˆ

≤aⁿ²esssup

k₂(_a^t) ˆψ(t)

≤aⁿ²k₂ 1

2a²

esssup

1 + C

2t² N

ψ(t)ˆ

on using (3.1). Hence by Theorem3.2we have kW_ψf(a, b)k_p,k

1k2 =k(f∗h_a,0(b)k_p,k₁_k₂

≤aⁿ²k₂ 1

2a²

kf k_p,k₁

1 + C

2t² N

ψ(t)ˆ _∞

. This proves the theorem.

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4. A General Boundedness Result

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 ≥1and ¹_p +¹_q +¹_r = 2. Letk ∈L^p(Rⁿ), f ∈L^q(Rⁿ)and g ∈L^r(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 constant C_p,q,r;n = (C_pC_qC_r)ⁿ, where C_p² = ^p

1 p

p⁰

1

p0 with (¹_p + _p¹0 = 1).

Using Theorem4.1and following the same method of proof as for Theorem3.3we obtain the following boundedness result.

Theorem 4.2. Letp, q, r ≥1, ¹_p + ¹_q +¹_r = 2andk∈L^p(Rⁿ). Letf ∈L^q(Rⁿ)and ψ ∈L^r(Rⁿ), then

kW_ψf k_∞,k ≤ C_p,q,r;naⁿ^r⁻ⁿ² kk k_pkf k_qkψ k_r whereC_p,q,r;n = (C_pC_qC_r)ⁿ,C_p² = ^p

p1

p⁰

1

p0 with ¹_p + _p¹0 = 1.

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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.

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