IN WEIGHTED SOBOLEV SPACES

IN WEIGHTED SOBOLEV SPACES

NGUYEN MINH CHUONG AND TA NGOC TRI Received 27 September 2001

The integral wavelet transform is defined in weighted Sobolev spaces, in which some properties of the transform as well as its asymptotical behaviour for small dilation parameter are studied.

1. Preliminaries and notations

It is well known that the integral wavelet transform is a very powerful tool to study sciences and technology. In [5] wavelet theory has been investigated in very much functional spaces, even in BMO, VMO (for further details of those spaces, please refer to [5] and the references therein) even for pseudodifferential operators, however, the integral wavelet transform in weighted Sobolev spaces has not been studied yet neither in [5] nor in any other work.

The aim of this paper is to study this unsolved problem.

Letωµ(x)∈L^∞(Rⁿ),ωµ(x)>0, for almost allx∈Rⁿand for eachx,

ωµ(x+y)≤C1,µωµ(x), (1.1) for almost ally∈Rⁿ, whereµis a multi-index.

We use the Sobolev space with weighted norm defined as follows:

W_ω^m,pRⁿ

=

f ∈L^pRⁿ

|∂^kf ∈L^pRⁿ

,|k| ≤m (1.2) equipped with the norm

fm,p,ω=

|µ|≤m

Rⁿωµ(x)∂^µf(x)^pdx

1/ p

<∞, (1.3) whereµ₌(µ1, . . . , µ_n),|µ_|=µ1+···+µ_n,µ_i≥0.

Copyright©2002 Hindawi Publishing Corporation Abstract and Applied Analysis 7:3 (2002) 135–142

2000 Mathematics Subject Classification: 42Cxx, 42C40, 65Txx, 65T60 URL:http://dx.doi.org/10.1155/S1085337502000775

Let᏿(Rⁿ) be the Schwartz space of all differentiable functionsϕonRⁿsuch that for all multi-indicesαandβ

x∈Rsupⁿ

x^αD^βϕ(x)<∞. (1.4)

The Fourier transformᏲ:f →fˆis given by fˆ(y)=(2π)^−n/2

Rⁿ f(x)e^−i(x,y)dx, (1.5)

where (x, y)₌x1y1+···+x_ny_n,x₌(x1, . . . , x_n),y₌(y1, . . . , y_n) (see [1,2,4]).

As traditionally, it is not difficult to prove that᏿(Rⁿ) is dense inW_ω^m,p(Rⁿ).

Now we recall that a basic wavelet is a nontrivial functionψ_∈L¹(Rⁿ) such that its integral onRⁿis 0 and its Fourier transform ˆψ(ξ) satisfies the condition

(2π)ⁿ _∞

0

ψ(aξ)ˆ ²

a da, (1.6)

denoted byCψ which is a constant for everyξ=0 andCψ=0.

With a basic waveletψand a function f _∈᏿(Rⁿ), we define the following in- tegral:

Lψf(b, a)= 1

2n C_ψ

1

|a_|ⁿ

Rⁿψ¯ t−b

a f(t)dt, (1.7)

whereb_∈Rⁿanda_∈R\{0_}(see [3,5,6]).

2. Some properties

Proposition2.1. If f ∈᏿(Rⁿ)then

Lψf(·, a)_m,p,ω≤Cfm,p,ω, (2.1) wherea∈R,a=0and fixed,Cis a constant independent of f.

Proof. Obviously we have

L_ψf(·, a)₌ 1

2n Cψ

f_∗D^−aψ¯(·), (2.2)

whereD^a:L²(Rⁿ)→L²(Rⁿ) and (D^aψ)(a)_=|a_|^−n/2ψ(x/a);a₌0.

Since f ∈᏿(Rⁿ) the differentiation and integration can be interchanged

∂^µLψf(·, a)= 1

2n C_ψ

D^−aψ¯∗∂^µf(·). (2.3)

It is not difficult to see that L_ψf(_·, a)_m,p,ω

=

|µ|≤m

Rⁿωµ(·)∂^µLψf(·, a)^pd(·)

1/ p

=

|µ|≤m

1

2n Cψ

Rⁿω_µ(x)

Rⁿ

∂^µf(x−y)D^−aψ¯(y)d y

p

dx

1/ p

≤ 1

2n C_ψ

|µ|≤m

Rⁿ

Rⁿ

∂^µf(x−y)^pD^−aψ¯(y)^pωµ(x)dx

1/ p

d y

≤ 1

2n C_ψ

|µ|≤m

Rⁿ

D^−aψ(y)¯

Rⁿωµ(x)∂^µf(x−y)^pdx

1/ p

d y.

(2.4) However,

Rⁿω_µ(x)∂^µf(x₋y)^pdx₌

Rⁿω_µ(u+y)∂^µf(u)^pdu

≤C1,µ

Rⁿωµ(u)∂^µf(u)^pdu.

(2.5)

Consequently,

Rⁿω_µ(_·)∂^µL_ψf(_·, a)^pd(_·)

1/ p

≤

C1,µ1/ p

2n Cψ

Rⁿ

D^−aψ(y)¯ d y

Rⁿω_µ(x)∂^µf(x)^pdx

1/ p

≤Cµ

Rⁿωµ(x)∂^µf(x)^pdx

1/ p

,

(2.6)

where

C_µ=

C1,µ1/ p

2n Cψ

|a|^n/2ψ1. (2.7)

Therefore,

Lψf(·, a)_m,p,ω≤C

|µ|≤m

Rⁿωµ(x)∂^µf(x)^pdx

1/ p

, (2.8) where

C=max

|µ|≤mCµ, (2.9)

that is,

L_ψf(_·, a)_m,p,ω≤Cf_m,p,ω. (2.10) ByProposition 2.1, we extend (L_ψf(·, a)) for fixedato a continuous mapping from W_ω^m,p(Rⁿ) to itself. It is called the integral wavelet transform in weighted Sobolev space.

Theorem2.2. Ifψandϕare basic wavelets and f , gbelong toW_ω^m,p(Rⁿ), then the following estimate holds true:

L_ψf(·, a)₋L_ϕg(·, a)_m,p,ω

≤C¹|a|^n/2





ψ

2n C_ψ⁻

ϕ

2n C_ϕ

1

fm,p,ω+ ϕ1

2n

C_ϕf−gm,p,ω



,

(2.11)

whereC¹is a constant independent of f andg.

Proof. It is sufficient to prove the case f , g_∈᏿(Rⁿ).

Obviously

Rⁿωµ(·)∂^µLψf−Lϕg(·, a)^pd(·)

1/ p

=







Rⁿωµ(·) ∂^µf∗



D^−aψ¯

2n C_ψ⁻

D^−aϕ¯

2n C_ϕ



(·)

p

d(·)







1/ p

=







Rⁿωµ(x)

Rⁿ

∂^µf(x−y)



D^−aψ¯

2n C_ψ⁻

D^−aϕ¯

2n C_ϕ



(y)d y

p

dx







1/ p

≤

Rⁿ



D^−aψ¯

2n C_ψ⁻

D^−aϕ¯

2n C_ϕ



(y)

Rⁿωµ(x)∂^µf(x−y)

p

dx

1/ p

d y

≤C_2,µ

Rⁿω_µ(x)∂^µf(x)^pdx

1/ p

,

(2.12) where

C_2,µ=|a_|^n/2

ψ

2n C_ψ⁻

ϕ

2n C_ϕ

1

C_1,µ^{1/ p}. (2.13)

So

Lψf(·, a)−

Lϕf(·, a)_m,p,ω≤C¹|a|^n/2

ψ

2n C_ψ⁻

ϕ

2n C_ϕ

1

fm,p,ω, (2.14)

where

C¹₌max

|µ|≤m

C_1,µ^{1/ p}. (2.15)

Similarly we obtain Lϕf(·, a)−

Lϕg(·, a)_m,p,ω≤C¹|a|^n/2 ϕ1

2n Cϕ

f−gm,p,ω. (2.16)

By the triangle inequality we get Lψf(·, a)−

Lϕg(·, a)_m,p,ω

≤C¹_|a_|^n/2





ψ

2n Cψ

− ϕ

2n Cϕ

1

f_m,p,ω+ ϕ1

2n

Cϕ

f₋g_m,p,ω



.

(2.17)

3. Symptotical behaviour for small dilation parameter

FromTheorem 2.2the following proposition follows immediately.

Proposition3.1. Ifψis a basic wavelet and f ∈W_ω^m,p(Rⁿ), then

L_ψf(·, a)_m,p,ω=O_|a_|^n/2. (3.1) Now consider the operator

Λψf(b, a)₌ψ_a∗f(b)₌ 1 aⁿ

Rⁿ f(t)ψ b₋t

a dt, (3.2)

whereψ∈L¹(Rⁿ), f ∈L^p(Rⁿ), 1≤p <∞, and ψa(x)= 1

aⁿψ x

a , a=0. (3.3)

In the sequel, the following lemma is needed.

Lemma3.2. Let f _∈W_ω^m,p(Rⁿ),1≤p <_∞,ψ_∈L¹(Rⁿ)∩L^q(Rⁿ), with

Rⁿψ(t)dt₌1, 1 p+1

q ⁼1. (3.4)

Then

(i) (Λψf)(·, a)→f(·)inW_ω^m,p(Rⁿ)asa→0⁺;

(ii) [∂^µ(Λψf)](·, a) = [Λψ(∂^µf)](·, a) = a^−|µ|(Λ∂^µψf)(·, a) if ∂^µψ ∈ L¹(Rⁿ)

∩L^q(Rⁿ),|µ|< m,a >0.

Proof. Since (Λψf)(·, a)=(ψa∗f)(·), and for each multi-indexα,ⁿ_i=1αi ≤m such that∂^µf _∈L^p(Rⁿ),ψ_∈L^q(Rⁿ), we obtain

∂^µψa∗f(·)=

ψa∗∂^µf(·), Λψf(_·, a)₋f(_·)_m,p,ω

=

|µ|≤m

Rⁿωµ(·)∂^µΛψf(·, a)−f(·)^pd(·)

1/ p

=

|µ|≤m

Rⁿω_µ(·)Λψ

∂^µf(·, a)−

∂^µf(·)^pd(·)

1/ p

.

(3.5)

Taking into account that∂^µf ∈L^p(Rⁿ), it is easy to see that

Rⁿωµ(·)Λψ

∂^µf(·, a)−

∂^µf(·)^pd(·)

≤ω_µ∞

Rⁿ

Λ∂^µf(_·, a)₋∂^µf(_·)^pd(_·)_−→0 asa_−→0⁺.

(3.6)

Consequently, (Λψf)(·, a)→f(·) inW_ω^m,p(Rⁿ) asa→0⁺, that is, (i) is proved.

To check (ii) take fr ∈ ᏿(Rⁿ), fr → f in W_ω^m,p(Rⁿ). It is obvious that in W_ω^m−|µ|,p(Rⁿ)

∂^µΛψf_r(·, a)= Λψ

∂^µf_r(·, a)=a^−|µ|Λ∂^µψf_r(·, a). (3.7) By the continuity of the operators

Λψ:W_ω^m,pRⁿ

−→W_ω^m,pRⁿ ,

∂^µ:W_ω^m,pRⁿ

−→W_ω^m−|µ|,pRⁿ

, (3.8)

forψ_∈L^q(Rⁿ),_|µ_|< m, lettingr_{→ ∞}we get (ii).

Theorem3.3. Let f ∈W_ω^m,p(Rⁿ),1≤p <∞,ψ∈L¹(Rⁿ)∩L^q(Rⁿ)such that

Rⁿψ(t)dt₌1. (3.9)

Moreover, assume that ∂^µψ is a basic wavelet for each multi-index µ,|µ| ∈ {0,1,2, . . . , m_}, and∂^µψ_∈L^q(Rⁿ), with1/ p+ 1/q₌1. Suppose additionally that f andψare real-valued functions anda >0. Then

a→0lim⁺

1 a^|µ|+n/2

L∂^µψf(·,−a)− 1

2n

C_∂^µ_ψ∂^µf(·)_{m−|µ|,p,ω}=0. (3.10)

Proof. Obviously

L∂^µψf(·,−a)= a^n/2

2n C_∂^µ_ψ

∂^µψ_a∗f(·)

= a^n/2

2n C_∂^µ_ψ

ψ_a∗∂^µf(·)= a^|µ|+n/2

2n C_∂^µ_ψ

∂^µψ_a∗f(·).

(3.11)

Under the assumptions of the theorem, the differentiation and integration can be interchanged, and furthermore by the continuity of the operator

∂^µ:Wωm,p Rⁿ

−→W_ω^m−|µ|,pRⁿ

, (3.12)

for|µ_|< m, we get

1 a^|µ|+n/2

L_∂^µ_ψf(_·,₋a)₋ 1

2n C∂^µψ

∂^µf(_·) _{m−|µ|,p,ω}

≤ 1

2n C∂^µψ

∂^µψ_a∗f(·)−

∂^µf(·)_{m−|µ|,p,ω}

≤ C

2n C_∂^µ_ψ

ψa∗f(·)−f(·)_m,p,ω.

(3.13)

Lemma 3.2implies now that the last term in (3.13) tends to 0 asa→0⁺. Acknowledgment

This paper was supported by the National Fundamental Research Fund Grant for Natural Science, Vietnam.

References

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[2] N. M. Chuong, N. M. Tri, and L. Q. Trung,Theory of Partial Differential Equations, Science and Technology Publishing House, Hanoi, 1995.

[3] N. M. Chuong and T. N. Tri,The integral wavelet transform inL^p(Rⁿ), 1≤p≤ ∞, Fract. Calc. Appl. Anal.3(2000), no. 2, 133–140.

[4] L. H¨ormander,The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Fundamental Principles of Mathematical Sciences, vol. 256, Springer-Verlag, Berlin, 1983.

[5] Y. Meyer,Ondelettes et Op´erateurs. I[Wavelets and Operators. I], Hermann, Paris, 1990.

[6] A. Rieder,The wavelet transform on Sobolev spaces and its approximation properties, Numer. Math.58(1991), no. 8, 875–894.

Nguyen Minh Chuong: Institute of Mathematics, P.O. Box631, Bo Ho,10 000 Hanoi, Vietnam

E-mail address:[email protected]

Ta Ngoc Tri: University of Pedagogy Hanoi II, Me Linh, Vinh Phu, Vietnam